Teacher Annotated Edition - SecondaryMathsecondarymath.cmswiki.wikispaces.net/file/view/geosnbte.pdf7-2 Similar Polygons ... 13-2 Permutations and Combinations ... already know about

Study Notebook

Teacher Annotated Edition

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Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior written permission of the publisher.

Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240

ISBN: 978-0-07-890858-3MHID: 0-07-890858-2

Printed in the United States of America

1 2 3 4 5 6 7 8 9 10 009 14 13 12 11 10 09 08

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Chapter 1Before You Read ............................................. 1Key Points ........................................................ 21-1 Points, Lines, and Planes ........................... 31-2 Linear Measure and Precision ................... 51-3 Distance and Midpoints .............................. 71-4 Angle Measure ........................................... 91-5 Angle Relationships .................................. 111-6 Two-Dimensional Figures ......................... 131-7 Three-Dimensional Figures ...................... 15Tie It Together ................................................ 17Before the Test .............................................. 18

Chapter 2Before You Read ........................................... 19Key Points ...................................................... 202-1 Inductive Reasoning and Conjecture ....... 212-2 Logic ......................................................... 232-3 Conditional Statements ............................ 252-4 Deductive Reasoning ............................... 272-5 Postulates and Paragraph Proofs ............ 292-6 Algebraic Proof ......................................... 312-7 Proving Segment Relationships ............... 332-8 Proving Angle Relationships .................... 35Tie It Together ................................................ 37Before the Test .............................................. 38

Chapter 3Before You Read ........................................... 39Key Points ...................................................... 403-1 Parallel Lines and Transversals ............... 413-2 Angles and Parallel Lines ......................... 433-3 Slopes of Lines ......................................... 453-4 Equations of Lines .................................... 473-5 Proving Lines Parallel ............................... 493-6 Perpendiculars and Distance ................... 51Tie It Together ................................................ 53Before the Test .............................................. 54

Chapter 4Before You Read ........................................... 55Key Points ...................................................... 564-1 Classifying Triangles ................................ 574-2 Angles of Triangles ................................... 594-3 Congruent Triangles ................................. 614-4 Proving Congruence: SSS, SAS .............. 634-5 Proving Congruence: ASA, AAS .............. 654-6 Isosceles and Equilateral Triangles ......... 674-7 Congruence Transformations ................... 694-8 Triangles and Coordinate Proof ............... 71Tie It Together ................................................ 73Before the Test .............................................. 74

Chapter 5Before You Read ........................................... 75Key Points ...................................................... 765-1 Bisectors of Triangles ............................... 775-2 Medians and Altitudes of Triangles .......... 795-3 Inequalities in One Triangle ..................... 815-4 Indirect Proof ............................................ 835-5 The Triangle Inequality ............................. 855-6 Inequalities in Two Triangles .................... 87Tie It Together ................................................ 89Before the Test .............................................. 90

Chapter 6Before You Read ........................................... 91Key Points ...................................................... 926-1 Angles of Polygons ................................... 936-2 Parallelograms .......................................... 956-3 Tests for Parallelograms .......................... 976-4 Rectangles ................................................ 996-5 Rhombi and Squares .............................. 1016-6 Kites and Trapezoids .............................. 103Tie It Together .............................................. 105Before the Test ............................................ 106

Chapter 7Before You Read ......................................... 107Key Points .................................................... 1087-1 Ratios and Proportions ........................... 1097-2 Similar Polygons ..................................... 1117-3 Similar Triangles ..................................... 1137-4 Parallel Lines and Proportional

Parts ....................................................... 1157-5 Parts of Similar Triangles ....................... 1177-6 Similarity Transformations ...................... 1197-7 Scale Drawings and Models ................... 121Tie It Together .............................................. 123Before the Test ............................................ 124

Chapter 8Before You Read ......................................... 125Key Points .................................................... 1268-1 Geometric Mean ..................................... 1278-2 The Pythagorean Theorem and Its

Converse ................................................ 1298-3 Special Right Triangles .......................... 1318-4 Trigonometry .......................................... 1338-5 Angles of Elevation and Depression ...... 1358-6 The Law of Sines and Cosines .............. 1378-7 Vectors ................................................... 139Tie It Together .............................................. 141Before the Test ............................................ 142

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Chapter 9Before You Read ......................................... 143Key Points .................................................... 1449-1 Refl ections .............................................. 1459-2 Translations ............................................ 1479-3 Rotations ................................................ 1499-4 Compositions of Transformations ........... 1519-5 Symmetry ............................................... 1539-6 Dilations .................................................. 155Tie It Together .............................................. 157Before the Test ............................................ 158

Chapter 10Before You Read ......................................... 159Key Points .................................................... 16010-1 Circles and Circumference ................... 16110-2 Measuring Angles and Arcs ................. 16310-3 Arcs and Chords ................................... 16510-4 Inscribed Angles ................................... 16710-5 Tangents ............................................... 16910-6 Secants, Tangents, and Angle

Measures .............................................. 17110-7 Special Segments in a Circle ............... 17310-8 Equations of Circles ............................. 175Tie It Together .............................................. 177Before the Test ............................................ 178

Chapter 11Before You Read ......................................... 179Key Points .................................................... 18011-1 Areas of Parallelograms and

Triangles ............................................... 18111-2 Areas of Trapezoids, Rhombi

and Kites .............................................. 18311-3 Areas of Circles and Sectors ................ 18511-4 Areas of Regular Polygons and

Composite Figures ............................... 18711-5 Areas of Similar Figures ....................... 189Tie It Together .............................................. 191Before the Test ............................................ 192

Chapter 12Before You Read ......................................... 193Key Points .................................................... 19412-1 Representations of

Three-Dimensional Figures .................. 19512-2 Surface Areas of Prisms and

Cylinders ............................................... 19712-3 Surface Areas of Pyramids and

Cones ................................................... 19912-4 Volumes of Prisms and Cylinders ........ 20112-5 Volumes of Pyramids and Cones ......... 20312-6 Surface Areas and Volumes

of Spheres ............................................ 20512-7 Spherical Geometry .............................. 20712-8 Congruent and Similar Solids ............... 209Tie It Together .............................................. 211Before the Test ............................................ 212

Chapter 13Before You Read ......................................... 213Key Points .................................................... 21413-1 Representing Sample Spaces .............. 21513-2 Permutations and Combinations .......... 21713-3 Geometric Probability ........................... 21913-4 Simulations ........................................... 22113-5 Probabilities of Independent and

Dependent Events ................................ 22313-6 Probabilities of Mutually Exclusive

Events ................................................... 225Tie It Together .............................................. 227Before the Test ............................................ 228

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Note-Taking TipsNote-Taking TipsYour notes are a reminder of what you learned in class. Taking good notes can help you succeed in mathematics. The following tips will help you take better classroom notes.

• Before class, ask what your teacher will be discussing in class. Review mentally what you already know about the concept.

• Be an active listener. Focus on what your teacher is saying. Listen for important concepts. Pay attention to words, examples, and/or diagrams your teacher emphasizes.

• Write your notes as clear and concise as possible. The following symbols and abbreviations may be helpful in your note-taking.

Word or Phrase

Symbol or Abbreviation

Word or Phrase

Symbol or Abbreviation

for example e.g. not equal ≠

such as i.e. approximately ≈

with w/ therefore ∴

without w/o versus vs

and + angle ∠

• Use a symbol such as a star (�) or an asterisk (∗) to emphasis important concepts. Place a question mark (?) next to anything that you do not understand.

• Ask questions and participate in class discussion.

• Draw and label pictures or diagrams to help clarify a concept.

• When working out an example, write what you are doing to solve the problem next to each step. Be sure to use your own words.

• Review your notes as soon as possible after class. During this time, organize and summarize new concepts and clarify misunderstandings.

Note-Taking Don’tsNote-Taking Don’ts

• Don’t write every word. Concentrate on the main ideas and concepts.

• Don’t use someone else’s notes as they may not make sense.

• Don’t doodle. It distracts you from listening actively.

• Don’t lose focus or you will become lost in your note-taking.

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Before You Read

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Chapter 1 1 Glencoe Geometry

Before you read the chapter, respond to these statements.1. Write an A if you agree with the statement.2. Write a D if you disagree with the statement.

Study Organizer Construct the Foldable as directed at the beginning of this chapter.

Note Taking TipsNote Taking Tips • When taking notes, summarize the main ideas presented in the lesson. Summaries are useful for condensing data and realizing what is important.

• When you take notes, write descriptive paragraphs about your learning experiences.

The Tools of Geometry

Before You Read The Tools of Geometry

See students’ work.

• Collinear points are lines that run through the same point.

• Line segments that are congruent have the same measure.

• Distance on a coordinate plane is calculated using a form of the Pythagorean Theorem.

• The Midpoint Formula can be used to find the coordinates of the endpoint of a segment.

• The formula to find the area of a circle is A = πr2.

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Key Points

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Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on linear measure, one fact might be that unlike a line, a line segment can be measured because it has two endpoints. After completing the chapter, you can use this table to review for your chapter test.

Lesson Fact

1-1 Points, Lines, and Planes See students’ work.

1-2 Linear Measure

1-3 Distance and Midpoints

1-4 Angle Measure

1-5 Angle Relationship

1-6 Two-Dimensional Figures

1-7 Three-Dimensional Figures


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Points, Lines, and Planes1-1

Less

on

1-1

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What You’ll Learn Scan the text in Lesson 1-1. Write two facts you learned about points, lines, and planes as you scanned the text.

Sample answer: A line does not have thickness or1. ____________________________________________________

width. ____________________________________________________

Sample answer: Collinear points are points on the2. ____________________________________________________

same line. ____________________________________________________

New Vocabulary Write the definition next to each term.

can only be explained using examples and descriptions————————————————————————————

a location that has neither shape nor size————————————————————————————

made up of points and has no thickness or width————————————————————————————

flat surface made up of points that extends————————————————————————————

indefinitely in all directions————————————————————————————

points that lie on the same line————————————————————————————

points that lie on the same plane————————————————————————————

the set of points that two or more geometric figures————————————————————————————

have in common————————————————————————————

can be explained using undefined terms and/or other————————————————————————————

defined terms————————————————————————————

a boundless, three-dimensional set of all points————————————————————————————

undefined term

point

line

plane

collinear

coplanar

intersection

definition/defined term

space

Active Vocabulary

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Lesson 1-1 (continued)

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Main Idea Details

Model a point, line, and plane with a representative drawing. Label your drawing.

Compare undefined and defined terms by completing the table. Provide definitions and examples.

Points, Lines, and Planespp. 5–6

Intersections of Lines and Planespp. 6–7

linepoint

plane

Sample answers:

•

Term Definition Examples

defined term

explained using undefined terms or other defined terms; same as definition

space, intersection, collinear, coplanar, intersection

undefi ned term

can only be explained using examples and descriptions

points, lines, planes

Helping You Remember Recall or look in a dictionary for the meaning of the prefix co-. What does the prefix mean? How can it help you remember the meaning of collinear?

Sample answer: joint or working together; two or more points together —————————————————————————————————————————

—————————————————————————————————————————

—————————————————————————————————————————


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Linear Measure1-2

Less

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1-2

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What You’ll Learn Scan the text under the Now heading. List two things you will learn about in this lesson.

Sample answer: how to measure segments1.

Sample answer: how to calculate with measures2.

New Vocabulary Fill in each blank with the correct term or phrase.

Between any two on a line there is another

.

a portion of a with two

point A is between two other points, B and C; A, B, and C

are on the same line, and + =

method of creating without

two with the same

Vocabulary Link Congruent segments can be illustrated by real-world examples. Consider opposite sides of this book. The lines are the same size. Write some real-world examples of other congruent segments.

Sample answer: railroad tracks, edges of picture————————————————————————————

frame————————————————————————————

————————————————————————————

betweenness of points

line segment

between

construction

congruent segment

Active Vocabulary

point

line endpoints

AB AC BC

measuring tools

segments measures

geometric figures

points


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Main Idea Details

Use the model to fill in each blanks.

A B

1. Each inch is divided into _________.

2. Point B is closer to the ______ inch mark.

3. −−

AB is about ______ inches long.

Model the three types of methods to calculate measures of a line segment with an example.

Measure Line Segmentspp. 14–15

Calculate Measurespp. 15–17

eighths

1 3 −

8

1 3 −

8

Find measurements by subtracting.

Sample answer: number line with a smaller segment subtracted from a larger segment

Find measurements by writing and solving equations.

Sample answer: ST = 4x, SV = 5x - 3, TV = 15 Find the value of x and ST. 4x + 5x - 3 = 15 x = 2; ST = 8

Find measurements by adding.

Sample answer: number line with two segments that are equal to a longer segment

Helping You Remember Construction is a word used in everyday English as well as in mathematics. Look up construction in the dictionary. Explain how the everyday definition can help you remember how construction is used in mathematics.

Sample answer: The way in which something is put together; figures are put—————————————————————————————————————————

together just like buildings and structures.—————————————————————————————————————————


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Distance and Midpoints1-3

Less

on

1-3

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What You’ll Learn Skim Lesson 1-3. Predict two things that you expect to learn based on the headings and the Key Concept box.

Sample answer: how to find the distance between1. ____________________________________________________

two points ____________________________________________________

Sample answer: how to find the midpoint between2. ____________________________________________________

two points ____________________________________________________


the length of a segment with two endpoints————————————————————————————

————————————————————————————

point that is halfway between the endpoints————————————————————————————

————————————————————————————

segment, line, plane that intersects a segment at ————————————————————————————

its midpoint————————————————————————————

Vocabulary Link Midpoint can have non-mathematical meanings as well. Look up midpoint in the dictionary. Explain how the English definition can help you remember how midpoint is used in mathematics.

Sample answer: a point in time halfway between————————————————————————————

the beginning and end; a point halfway between————————————————————————————

two extremes ————————————————————————————

————————————————————————————

distance

midpoint

segment bisector

Active Vocabulary


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Main Idea Details

Fill in the organizer for the distance formula on a number line and a coordinate plane with line segment AB, with endpoints (x1, y1) and (x2, y2).

Model the solution to find the distance between the points on the coordinate plane. Use the lines to show your calculations. Round to the nearest tenth if necessary.

(x1, y1) = (-5, 3) ; (x2, y2) = ( 2, 4 )

KL = √ �� (x2 - x

1) 2 + (y2 - y

1) 2

KL = √ �� (2 - (-5) ) 2 + (4 - 3) 2

KL = √ �� 49 + 1

KL = √ �� 50 ≈ 7.1

Distance Between Two Pointspp. 25–26

Midpoint of a Segmentpp. 27–30

y

x

If a segment has endpoints with

coordinates (x1, y

1)

and (x2, y

2), you can

find the distance with the distance

formula.

Number Line

The distance between two points is the

absolute value of the difference between their coordinates.

Coordinate Plane

Distance Formula

Symbols

Words

AB = ⎪x1 - x

2⎥ or ⎪x2 - x

1⎥ AB = √ �� (x

2 - x

1)2 + (y

2 - y

1)2


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Angle Measure1-4

Less

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What You’ll Learn Skim the Examples for Lesson 1-4. Predict two things you think you will learn about angle measures.

Sample answer: how to measure angles1. ____________________________________________________

____________________________________________________

Sample answer: how to classify angles2. ____________________________________________________

____________________________________________________

New Vocabulary Write the correct term next to each definition.

angle with measure of 90º

where two rays meet to form an angle

angle with measure greater than 90º

a ray that divides an angle into two congruent angles

the region of a plane inside of an angle

angle with measure less than 90º

a ray that forms part of an angle

part of a line with one endpoint and one end that extends forever

formed by two noncollinear rays with a common endpoint

units used to measure angles

extend in two directions from a point on a line

the region of a plane outside of an angle

Active Vocabulary

right angle

vertex

obtuse angle

angle bisector

interior

acute angle

side

ray

angle

degree

opposite rays

exterior


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Details

Summarize information about angles in the graphic organizer below.

1. Draw and label an obtuse, acute, and right angle.

2. Draw and label one angle that has these 3 names:∠B, ∠ABC, and ∠3.

Model a pair of congruent angles. Use a compass and straightedge.

Drawings will vary, but both angles have the same measure. Sample answer shown.

Measure and Classify Anglespp. 36–38

Congruent Anglespp. 39–40

Helping You Remember Compare and contrast congruent segments and congruent angles.

Sample answer: Congruent segments and congruent angles have the—————————————————————————————————————————

same measure. Segments have a linear measure. Angles are measured—————————————————————————————————————————

in degrees.—————————————————————————————————————————

Main Idea

acute obtuse

right

3


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Angle Relationships1-5

Less

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1-5

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What You’ll Learn Scan Lesson 1-5. List two headings you would use to make an outline of this lesson.

Sample answer: Special Pairs of Lines1. ____________________________________________________

____________________________________________________

Sample answer: Perpendicular Lines2. ____________________________________________________

____________________________________________________

New Vocabulary Label the diagram with the correct terms. Use each term once.

Vocabulary Link Perpendicular has a nonmathematical meaning. Look up perpendicular in the dictionary. List at least five examples of things that are perpendicular in everyday life.

Sample answer: lines of corner of book, desk, room,————————————————————————————

refrigerator and floor, wall and ceiling————————————————————————————

————————————————————————————

Active Vocabulary

1

2 3

45

∠1 and ∠2 aresupplementary angles

and a linear pair.

Lines w and x areperpendicular.

∠4 and ∠5 arecomplementary anglesand adjacent angles.

∠1 and ∠3 arevertical angles.

x

w

adjacent

linear pair

vertical

complementary

supplementary

perpendicular

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Main Idea Details

Summarize angle relationships by completing the graphic organizer with the terms adjacent angles, complementary angles, vertical angles, and supplementary angles.

If m∠ABC = 2x + 12, find x so that ⎯⎯ � BC ⊥

⎯⎯ � BA .

= definition of perpendicular

= Simplify.

Pairs of Anglespp. 46–48

Perpendicular Linespp. 48–50

2x + 12 90

x

relationship due to positioning

relationship due to angle measures

adjacent angles

vertical angles

Special angle pairs

complementaryangles

supplementary angles

39

Helping You Remember Look up the nonmathematical meaning of supplementary in a dictionary. How can this definition help you to remember the meaning of supplementary angles?

Sample answer: something added to complete a deficiency; two angles added—————————————————————————————————————————

together to be a complete line—————————————————————————————————————————


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Two-Dimensional Figures1-6

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What You’ll Learn Skim the lesson. Write two things you already know about two-dimensional figures.

Sample answer: Polygons are two-dimensional1. ____________________________________________________

figures. ____________________________________________________

Sample answer: Perimeter and area of 2. ____________________________________________________

two-dimensional figures can be found. ____________________________________________________

New Vocabulary Write each term next to its definition.

where two sides of a polygon intersect

polygon with no side that passes through the figure’s interior

a convex polygon that is equilateral and equiangular

a polygon where all angles are congruent

perimeter of a circle

a polygon with n number of sides

closed figure formed by coplanar segments where two segments intersect at a common noncollinear endpoint

polygon with some side that passes through the figure’s interior

a polygon where all sides are congruent

sum of the lengths of a polygon’s sides

number of square units needed to cover a surface

Active Vocabulary

vertex of polygon

convex

regular polygon

equiangular polygon

circumference

n-gon

polygon

concave

equilateral polygon

perimeter

area


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Main Idea

Identify Polygonspp. 56–57

Perimeter, Circumference, and Areapp. 58–60

Details

Fill in the organizer about polygons.

What is a polygon?• noncollinear vertices

• closed figure

• made of coplanar segments

What is not a polygon?• open

• curved sides

• intersecting segments

Examples Nonexamples

Summarize the formulas for perimeter, circumference, and area of the figures by completing the chart.

Polygons

Area A = 1 − 2 bh A = s2 A = ℓ × w πr2

Perimeter P = a + b + c P = 4s P = 2ℓ + 2wsame as

circumfer-ence

Circumference n/a n/a n/aC =πd or C = 2πr


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Three-Dimensional Figures1-7

Less

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What You’ll Learn Scan the text in Lesson 1-7. Write two facts you learned about three-dimensional figures.

Sample answer: A polyhedron is a solid with all1. ____________________________________________________

flat surfaces that enclose a single region or space. ____________________________________________________

____________________________________________________

Sample answer: There are exactly five types of2. ____________________________________________________

regular polyhedron. ____________________________________________________

____________________________________________________


polyhedron with two bases connected by————————————————————————————

parallelogram faces————————————————————————————

polyhedron with a polygonal base and three or more————————————————————————————

triangular faces that meet at a common vertex————————————————————————————

solid with congruent parallel circular bases and————————————————————————————

curved side————————————————————————————

parallel, congruent faces that intersect faces————————————————————————————

————————————————————————————

solid with a circular base connected by a curved ————————————————————————————

surface to a single vertex ————————————————————————————

set of points equidistant from a given point————————————————————————————

————————————————————————————

Active Vocabulary

prism

pyramid

cylinder

base

cone

sphere


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Main Idea Details

Complete the concept circle by drawing in a fourth example. Then state the relationship of the 4 figures on the line below the concept circle.

The relationship is: All are prisms.————————————————————————————

Organize information about volume and surface area in the chart.

Figure

Surface Area

T = Ph +

2B

T = 1 − 2 Pℓ

+ B

T = 2πrh + 2πr 2

T = πrℓ + πr 2

T =

4πr 2

Volume V = Bh V = 1 − 3 bh V = πr2h V =

1 −

3 πr 2h

V = 4 − 3

πr3

Identify Three-Dimensional Figurespp. 67–68

Surface Area and Volumepp. 69–70

Drawingswill vary;must be aprism.

Helping You Remember A good way to remember the characteristics of geometric solids is to think of how different solids are alike. Name a way which pyramids and cones are alike.

Sample answer: Pyramids and cones both have one common vertex in which —————————————————————————————————————————

the sides or faces are attached.—————————————————————————————————————————

—————————————————————————————————————————


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Tie It Together

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Complete the graphic organizer with a term or formula from the chapter.


Vertical

ComplementaryLinear Pair

SupplementaryAdjacent Angle Pairs

Angle

Right

Obtuse

Acute

LinePoint Plane

Basic Geometry

3D Figures

Cylinder Cone

Sphere

Polyhedron

Prism

Pyramid

Volume

Surface Area

Perpendicular

Perimeter

Area Polygons

Equilateral Equiangular

Midpoint Formula

( (x1 + x

2) −

2 ,

(y1 + y

2) −

2 )

Length/Distance Formula

d = √ �� (x2 - x

1) + (y

2 - y

1)

Segment

Regular

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Before the Test

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Now that you have read and worked through the chapter, think about what you have learned and complete the table below. Compare your previous answers with these.

1. Write an A if you agree with the statement.2. Write a D if you disagree with the statement.

Math Online Visit glencoe.com to access your textbook, more examples, self-check quizzes, personal tutors, and practice tests to help you study for concepts in Chapter 1.

Are You Ready for the Chapter Test?Use this checklist to help you study.

□ I used my Foldable to complete the review of all or most lessons.

□ I completed the Chapter 1 Study Guide and Review in the textbook.

□ I took the Chapter 1 Practice Test in the textbook.

□ I used the online resources for additional review options.

□ I reviewed my homework assignments and made corrections to incorrect problems.

□ I reviewed all vocabulary from the chapter and their definitions.

Study TipsStudy Tips • Make a calendar that includes all of your daily classes. Besides writing down all

assignments and due dates, include in your daily schedule time to study, work on projects, and reviewing notes you took during class that day.


The Tools of Geometry After You Read

• Collinear points are lines that run through the same point.

D

• Line segments that are congruent have the same measure.

A

• Distance on a coordinate plane is calculated using a form of the Pythagorean Theorem.

A

• The Midpoint Formula can be used to find the coordinates of the endpoint of a segment.

A

• The formula to find the area of a circle is A = πr2.A

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Before You Read

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Note Taking TipsNote Taking Tips • When you take notes, you may wish to use a highlighting marker to emphasize

important concepts.

• When you take notes, think about the order in which the concepts are being presented.

Write why you think the concepts were presented in this sequence.

Before You Read Reasoning and Proof

See students’ work. • Listing a counterexample is a method

to prove a conjecture is true.

• A Venn diagram can illustrate a conjunction.

• A form of inductive reasoning is the Law of Detachment.

• The statement a line contains two points, is an example of a postulate.

• Supplementary angles have measures whose sum is 90º.

Reasoning and Proof

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Key Points

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Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on inductive reasoning and conjecture, one fact might be a concluding statement reached using inductive reasoning is called a conjecture. After completing the chapter, you can use this table to review for your chapter test.

Lesson Fact

2-1 Inductive Reasoning and Conjecture See students’ work.

2-2 Logic

2-3 Conditional Statements

2-4 Deductive Reasoning

2-5 Postulates and Paragraph Proofs

2-6 Algebraic Proof

2-7 Proving Segment Relationships

2-8 Proving Angle Relationships

The Reasoning and Proof


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Inductive Reasoning and Conjecture2-1

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Sample answer: Making Conjectures1. ____________________________________________________

____________________________________________________

Sample answer: Finding Counterexamples2. ____________________________________________________

____________________________________________________


____________ that uses a number of specific ___________ to

arrive at a _____________

the _____________ that is reached within inductive reasoning

an example that ____________ a ____________

Vocabulary Link Conjecture is a word that is used in everyday English. Find the definition of conjecture using a dictionary. Write how the definition of conjecture can help you remember the mathematical definition of conjecture.

Sample answer: an expression of an opinion; ————————————————————————————

A conclusion about a statement is also an expression————————————————————————————

of an opinion.————————————————————————————

inductive reasoning

conjecture

counterexample

reasoning

conclusion

examples

conclusion

disproves conjecture

Active Vocabulary

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Main Idea

Make Conjecturespp. 89–91

Find Counterexamplesp. 92

Details

Sequence the steps of making a conjecture with algebraic terms or geometric terms.

Write a statement in which you can make a true conjecture. Then write another statement in which you can make a false conjecture. Provide a counterexample of the false conjecture.

True Conjecture: Sample answer: The product of two

odd numbers is an even number.

False Conjecture: Sample answer: The product of a

negative number and a positive number is a positive

number. Sample answer: -4 � 5 = -20

Make a conjecture.

Test the conjecture.

Ensure all consecutive

terms follow the

same repeating

operation or attribute.

Look for operations or

attributes that repeat.

Helping You Remember Write a short sentence that can help you remember why it only takes one counterexample to prove that a conjecture is false.

Sample answer: A conjecture must be true in all cases, so one counterexample—————————————————————————————————————————

is enough to disprove a conjecture.—————————————————————————————————————————


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Logic2-2

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What You’ll Learn Skim the Examples for Lesson 2-2. Predict two things you think you will learn about logic.

Sample answer: how to identify the truth of a1. ____________________________________________________

statement ____________________________________________________

Sample answer: how to use Venn diagrams2. ____________________________________________________

____________________________________________________

New Vocabulary Match the term with the correct definition by drawing a line between the two.

a sentence that is either true or false

two or more statements joined by and or or

a compound statement that uses the word or

the value of a statement as either true or false

statement with the opposite meaning and opposite truth

a compound statement that uses the word and

convenient method to determine the truth value of statement

Vocabulary Link Negation is a word used in everyday English as well as in mathematics. Look up negation in the dictionary. Explain how the English definition can help you remember how negation is used in mathematics.

Sample answer: denying; Negation denies the truth of————————————————————————————

a statement.————————————————————————————

truth table

disjunction

statement

truth value

conjunction

negation

compound statement

Active Vocabulary


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Main Idea Details

Fill in the blanks to summarize negations, conjunctions, and disjunctions. Write a description of each term.

Model the situation using a Venn diagram.At Terrace Middle school, 68 students play basketball, 77 play volleyball, 19 play soccer and basketball, and 27 play all three sports. If 13 students play both volleyball and basketball, how many students play just basketball? 9

Volleyball Soccer

Determine Truth Valuespp. 97–99

Venn Diagramspp. 99–100

truth value____________

disjunction____________

negation__________

conjunction____________

9

13 27 19

Basketball

Helping You Remember Prefixes can often help you remember the meaning of words or distinguish between similar words. Use your dictionary to find the meanings of the prefixes con and dis and explain how these meanings can help you remember the difference between a conjunction and disjunction.

Sample answer: Con means together, and dis means undo or do the————————————————————————————————————————

opposite. Together means and and opposite means or.————————————————————————————————————————

statement

compound

Statement


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Conditional Statements2-3

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What You’ll Learn Skim Lesson 2-3. Predict two things you will learn based on the headings and the Key Concept box.

Sample answer: definition of a conditional1. ____________________________________________________

statement ____________________________________________________

Sample answer: definition of hypothesis and2. ____________________________________________________

conclusion ____________________________________________________


a statement that can be written in if-then form————————————————————————————

in the form if p, then q————————————————————————————

formed by exchanging the hypothesis and conclusion————————————————————————————

of the conditional————————————————————————————

the phrase immediately following the word if in a————————————————————————————

conditional statement————————————————————————————

formed by negating both the hypothesis and————————————————————————————

conclusion of the conditional————————————————————————————


conclusion of the converse of the conditional————————————————————————————

statements with the same truth value————————————————————————————

other statements based on a given conditional————————————————————————————

statement————————————————————————————

the phrase immediately following the word then in a————————————————————————————

conditional statement————————————————————————————

conditional statement

if-then statement

converse

hypothesis

inverse

contrapositive

logically equivalent

related conditionals

conclusion

Active Vocabulary


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Main Idea Details

Identify the hypothesis and conclusion of each conditional statement. Circle the hypothesis and underline the conclusion.

1. A number is divisible by 9 if the sum of its digits isdivisible by 9.

2. If the measure of an angle is less than 90 degrees, it is an acute angle.

Fill in the organizer for related conditionals.

If-Then Statementspp. 105–107

Related Conditionalspp. 107–108

Conditional Statements

Symbols: -q → -p

Words: ___________________________________if not q then not p

contrapositive;Symbols: q → p

Words: ______________________if q then p

converse;

Symbols: p → q

Words: ____________________________________

conditional

statement; if p then q

Symbols: -p → -q

Words: ______________________________

inverse;

if not p then not q

Helping You Remember When working with a conditional statement and its three related conditional, what is an easy way to remember which statements are logically equivalent?

Sample answer: The two terms that contain verse (converse and inverse) are—————————————————————————————————————————

logically equivalent. The other two terms are also logically equivalent.—————————————————————————————————————————

—————————————————————————————————————————


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Deductive Reasoning2-4

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What You’ll Learn Scan the text in Lesson 2-4. Write two facts you learned about deductive reasoning as you scanned the text.

Sample answer: Deductive reasoning uses facts,1. ____________________________________________________

rules, definitions, or properties to reach logical ____________________________________________________

conclusions. ____________________________________________________

Sample answer: One valid form of deductive2. ____________________________________________________

reasoning is the Law of Detachment. ____________________________________________________


allows you to draw conclusions from two true conditional statements when the conclusion of one statement is the hypothesis of the other

uses facts, rules, definitions, or properties to reach logical conclusions from given statements

form of deductive reasoning that states that if all the facts are true, then the conclusion reached is also true

Vocabulary Link Syllogism is a word used in everyday English as well as in mathematics. Look up syllogism in the dictionary. Explain how the English definition can help you remember how syllogism is used in mathematics.

Sample answer: logic; consists of major, minor————————————————————————————

premise, and conclusion; is analogy to drawing————————————————————————————

conclusion based on two true conditional statements————————————————————————————

————————————————————————————

_______________________

_______________________

_______________________

Law of Syllogism

deductive reasoning

Law of Detachment

Active Vocabulary


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Main Idea Details

Identity the steps to validate the two conclusions by completing the graphic organizer.

Match the portions of each statement with the correct term by drawing a line to connect the two. Then write the true conditional statement and the valid conclusion using the Law of Syllogism.

p you buy bread

q you walk to the store

r you can make toast

If you walk to the store, you can buy bread. If you buy————————————————————————————

bread, you can make toast.————————————————————————————

Law of Detachment pp. 115–117

Law of Syllogismpp. 117–118

If the light bulb is broken, the lamp will not work. Jackson’s lamp will not work.Jackson’s light bulb is broken.

Identify the hypothesis.The light bulb is broken.

Identify the conclusion.The lamp will not work.

Analyze the conclusion.Jackson’s light bulb is broken. Invalid; the lamp could be broken.

Helping You Remember A good way to remember something is to explain it to someone else. Suppose that a classmate is having trouble remembering the Law of Detachment. In your own words, explain what the Law of Detachment means?

Sample answer: a logical argument where if the first and second statements—————————————————————————————————————————

are true, you can conclude that a third statement connecting the two true—————————————————————————————————————————

statements is true—————————————————————————————————————————


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Postulates and Paragraph Proofs2-5

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Sample answer: using basic postulates about1. ____________________________________________________

points, lines, and planes ____________________________________________________

Sample answer: writing paragraph proofs2. ____________________________________________________

____________________________________________________


a that is accepted as without

a logical in which each statement you make is

by a statement that is as

a logical of that link the

to what you are trying to prove

a or conjecture that has been

writing a to explain why a for a

given situation is

same as __________________

same as

postulate

proof

deductive argument

theorem

paragraph proof

informal proof

axiom

Active Vocabulary

statement true

proof

argument

supported accepted

true

chain statements

given

provenstatement

paragraph conjecture

true

paragraph proof

postulate


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Main Idea Details

Write a postulate of points, lines, and planes. Then model the postulate.

Through any three noncollinear points, there is————————————————————————————

exactly one plane. ————————————————————————————

Sequence the steps in the proof process by completing the organizer.

Steps in the Proof Process

Points, Lines, and Planespp. 125–126

Paragraph Proofs pp. 126–127

Sample answer:

Step 1: List the ___________________; draw a ________

if possible.

given information picture

Step 2: State the _________ or conjecture to

be ________.

theorem

proven

Step 3: Create a _____________________ by forming

a _____________ of statements linking the given to what you are trying to prove.

deductive argument

logical chain

Step 4: Justify each ____________ with a reason.

Reasons include definitions, algebraic ____________,

postulates, and ___________.

statement

properties

theorems

Step 5: State what has been ________.proven


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Algebraic Proof2-6

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What You’ll Learn Skim the lesson. Write two things you already know about algebraic proof.

Sample answer: properties of real numbers1. ____________________________________________________

____________________________________________________

Sample answer: how to solve an equation2. ____________________________________________________

____________________________________________________

Review Vocabulary Write the definition next to the term. (Lesson 2-4)

uses facts, rules, definitions, and properties to reach————————————————————————————

logical conclusions from given statements————————————————————————————

New Vocabulary Match the term with its definition by drawing a line to connect the term.

same as formal proof

contains statements and reasons in two columns

uses a group of algebraic steps to solve problems and justify each step.

Vocabulary Link Proof is a word that is used in everyday English. Find the definition of proof using a dictionary. Explain how the English definition can help you remember how proof is used in mathematics.

Sample answer: evidence sufficient enough to————————————————————————————

establish something as true; Evidence is the same as————————————————————————————

justification with postulates.————————————————————————————

deductive reasoning

algebraic proof

two-column proof

formal proof

Active Vocabulary


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Main Idea Details

Write a 2-column proof by completing each step.

Given: 8 − 3 + x = 6 - 1 − 3 x Prove: x = 2 1 − 2

Match the property with its example by drawing a line to connect the example.

Symmetric Property If ST = WX, and WX = YZ, then ST = YZ.

Reflective Property If ∠C = ∠D, then ∠D = ∠C.

Transitive Property ∠A = ∠A

Algebraic Proofpp. 134–135

Geometric Proofp. 136

Statements Reasons

8 − 3 + x = 6 - 1 −

3 x Given

3 ( 8 − 3 + x) = 3 (6 - 1 −

3 x) Multiplication Prop of

Equality

8 + 3x = 18 - x Distributive Property

8 + 3x + x = 18 - x + x Addition Prop of Equality

8 + 4x = 18 Simplify.

8 + 4x - 8 = 18 - 8 Subtraction Prop of Equality

4x = 10 Substitution Prop of Equality

4 − 4 x = 10 −

4 Division Prop of Equality

x = 10 − 4 or 2 1 −

2 Substitution Prop of

Equality


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Proving Segment Relationships2-7

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What You’ll Learn Scan the text in Lesson 2-7. Write two facts you learned about proving segment relationships.

Sample answer: You can use a ruler or segment1. ____________________________________________________

addition to show congruent segments. ____________________________________________________

Sample answer: There are postulates about2. ____________________________________________________

congruence segments. ____________________________________________________

Review Vocabulary Write the definition next to each term. (Lessons 2-1 and 2-3)

reasoning that uses a number of specific examples to————————————————————————————

arrive at a conclusion————————————————————————————

the conclusion that is reached within inductive————————————————————————————

reasoning————————————————————————————

an example that disproves a conjecture————————————————————————————

a statement that can be written in if-then form————————————————————————————

formed by exchanging the hypothesis and conclusion————————————————————————————

of the conditional————————————————————————————


conclusion of the converse of the conditional————————————————————————————


conclusion of the conditional————————————————————————————

inductive reasoning

conjecture

counterexample

conditional statement

converse

contrapositive

inverse

Active Vocabulary


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Main Idea Details

Use the Model and Fill in each blank to summarize the Segment Addition Postulate.

If P, Q, and R are __________, then point __ is between __

and __ if and only if ____ + ____ = PR.

Complete the graphic organizer to prove the statement below.

Given: −−−

QR � −−

ST ; −−

ST � −−−

UV Prove: −−−

QR � −−−

UV

Ruler Postulatepp. 142–143

Segment Congruencepp. 143–144

collinear Q P

R PQ QR

−−−

QR � −−

ST and −−

ST � −−−

UV

QR = ST and ST = UVby the

____________________.congruence

definition of

QR = UV by the____________________________________.PropertyTransitive

−− QR �

−−− UV by the

.of congruence

definition

Helping You Remember A good way to keep the names straight in your mind is to associate something in the name of the postulate with the content of the postulate. How can you use this idea to distinguish between the Rule Postulate and the Segment Addition Postulate?

Sample answer: There are two words in Ruler Postulate, and three words in—————————————————————————————————————————

Segment Addition Postulate. Ruler Postulate has two points in the statement,—————————————————————————————————————————

and Segment Addition Postulate mentions three points.—————————————————————————————————————————


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Proving Angle Relationships2-8

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Sample answer: Supplementary and1. ____________________________________________________

Complementary Angles ____________________________________________________

Sample answer: Congruent Right Angles 2. ____________________________________________________

____________________________________________________

Review Vocabulary Fill in each blank with the correct term or phrase. (Lessons 1-4 and 1-5)

two __________ angles with measures that have a sum of

_____________

angles with a common ______

two __________ angles with measures that have a sum of

______________

the region of a plane _______ an angle

the region of a plane _________ an angle

an angle with a measure of _______________________

an angle with a measure of _________________________

complementary angles

adjacent angles

supplemetary angles

interior

exterior

acute angle

obtuse angle

adjacent

90 degrees

side

180

inside

outside

less than 90 degrees

Active Vocabulary

adjacent

degrees

greater than 90 degrees


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Main Idea Details

Compare and contrast the Angle Addition Postulate and the Segment Addition Postulate by completing the Venn diagram.

Write a two-column proof to prove the statement by completing the chart.Given: m∠2 = 90 ∠1 � ∠3Prove: m∠1 = 45

Supplementary and Complementary Anglespp. 149–150

Congruent Anglespp. 151–153

Segment Addition PostulateAngle Addition Postulate

add adjacent angles

add collinear segments

sum of parts is measure of larger figure

Statements Reasons

1. m∠2 = 90∠1 � ∠3 1. Given

2. m∠3 = m∠1 2. Definition of congruent

3. m∠1 + m∠3 + m∠2 = 180

3. Definition of complementary angles

4. m∠1 + m∠1 + 90 = 180 4. Substitution

5. m∠1 + m∠3 + 90 – 90 = 180 – 90 5. Subtraction Property

6. m∠1 + m∠1 = 90 6. Substitution

7. m∠1 + m∠1 − 2 = 90 ÷ 2 7. Division Property

8. m∠1 = 45 8. Substitution

1 2 3


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Tie It Together

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Complete the graphic organizer with a term from the chapter.

Reasoning and Proof

the process of justifying algebraic steps with

properties of real numbersalgebraic proof

the process of justifying geometric statements with

postulates or theoresgeometric proof

If p -> q is a true statement, and p is true, then q is true.Law of Detachment

If p -> q and q -> r, then p -> r.Law of Syllogism

the process of justifying statements with

previously accepted statements

deductive reasoningthe process of

justifying statements based on past observations

inductive reasoning

Logic

Statements

p ∧ qconjunction

q ∨ pdisjunction

…ThenconclusionIf-Then Statements p -> q

conditional statements

q -> pconverse

~q -> ~pcontrapositive

~p -> pinverse

If…hypothesis

~pnegation

001-026_GEOSNEM_890858.indd Sec1:37001-026_GEOSNEM_890858.indd Sec1:37 6/14/08 12:43:26 AM6/14/08 12:43:26 AM

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Before the Test

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Study TipsStudy Tips • Make up acronyms to remember lists or sequences. PEMDAS is one acronym for

remembering the order of operations (parentheses, exponents, multiply and divide, add and subtract). (Please Excuse My Dear Aunt Sally).

Reasoning and Proof After You Read

• Listing a counterexample is a method to prove a conjecture is true.

D

• A Venn diagram can illustrate a conjunction. A

• A form of inductive reasoning is the Law of Detachment. D

• The statement a line contains two points, is an example of a postulate.

A

• Supplementary angles have measures whose sum is 90°.D

Reasoning and Proof


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Before You Read

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Before you read the chapter, think about what you know about the topic. List three things you already know about parallel and perpendicular lines in the first column. Then list three things you would like to learn about them in the second column.


Note Taking TipsNote Taking Tips • When you take notes, preview the lesson and make generalizations about what

you think you will learn. Then compare that with what you actually learned after each lesson.

• Before each lesson, skim through the lesson and write any questions that come to mind in your notes.

As you work through the lesson, record the answer to your question.

K

What I know…

W

What I want to find out…


Parallel and Perpendicular Lines


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Key Points

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Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on parallel lines and transversals, one fact might be parallel planes are planes that do not intersect. After completing the chapter, you can use this table to review for your chapter test.


Lesson Fact

3-1 Parallel Lines and Transversals See students’ work.

3-2 Angles and Parallel Lines

3-3 Adding and Subtracting Polynomials

3-4 Equations of Lines

3-5 Proving Lines Parallel

3-6 Perpendiculars and Distance


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Parallel Lines and Transversals3-1

Less

on

3-1

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What You’ll Learn Skim the Examples for Lesson 3-1. Predict two things you think you will learn about parallel lines and transversals.

Sample answer: how to identify and classify 1. ____________________________________________________

parallels and transversals ____________________________________________________

Sample answer: how to classify angle pairs2. ____________________________________________________

____________________________________________________

New Vocabulary Use the figure to complete each statement.

∠ 2 and ∠ 7 are .

∠ 5 and ∠ 6 are .

Vocabulary Link Skew is a word used in everyday English as well as in mathematics. Look up skew in the dictionary. Explain how the English definition can help you remember how skew is used in mathematics.

Sample answer: turn aside or swerve; Skew lines look ————————————————————————————

like they turn aside from one another.————————————————————————————

parallel lines

transversal

interior angles

exterior angles

consecutive interior angles

alternative interior angles

alternate exterior angles

corresponding angles

Active Vocabularyt

a

b

1

3

5

7

2

4

6

8

∠ 1 and ∠ 7 are .exterior angles

t is a .transversal

∠ 2 and ∠ 6 are .corresponding angles

a and b are .parallel lines

∠ 4 and ∠ 6 are .consecutive interior angles


interior angles

∠ 4 and ∠ 5 are .alternate interior angles


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Main Idea Details

Summarize information about parallel lines and skew lines in the graphic organizer below.

Model a transversal t which intersects two or more parallel lines. Identify consecutive interior angles, alternative interior and exterior angles, and corresponding angles.consecutive interior angles: ∠ 3 and ∠ 5, ∠ 4 and ∠ 6alternate interior angles:∠ 3 and ∠ 6, ∠ 4 and ∠ 5alternate exterior angles:∠ 1 and ∠ 8, ∠ 2 and ∠ 7corresponding angles:∠ 1 and ∠ 5, ∠ 2 and ∠ 6,∠ 3 and ∠ 7, ∠ 4 and ∠ 8

Relationships Between Lines and Planespp. 171–172

Transversal Angle Pair Relationshipspp. 172–173

What are parallel lines?coplanar lines that do not intersect

How are parallel lines and skew lines similar?Both types of lines never intersect.Parallel

LinesHow are parallel lines and skew lines different?Parallel lines are on the same plane and there is a symbol, ||. Skew lines are on different planes and there is no symbol.

What are skew lines?noncoplanar lines that do not intersect

Skew Lines

t

a

b

1

3

5

7

2

4

6

8

Helping You Remember Look up meaning of the prefix trans- in the dictionary. Write down 4 words that have trans- as a prefix. How can the meaning of the prefix help you remember the meaning of transversal?

Sample answer: Trans means across; transatlantic, Transamerica, transport, —————————————————————————————————————————

transparent; A transversal cuts across two or more parallel lines.—————————————————————————————————————————


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Angles and Parallel Lines3-2

Less

on

3-2

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What You’ll Learn Scan Lesson 3-2. List two headings you would use to make an outline of this heading.

Sample answer: Parallel Lines and Angle Pairs 1. ____________________________________________________

____________________________________________________

Sample answer: Algebra and Angle Measures2. ____________________________________________________

____________________________________________________

Review Vocabulary Match the term with the correct definition by drawing a line between the two. (Lesson 3-1)

lie in the regions not between parallel lines that are cut by a transversal

a line that intersects 2 or more coplanar lines at 2 different points

lie on the same side of both the parallel lines and the transversal

coplanar lines that never intersect

lie between parallel lines that are cut by a transversal

interior lines that lie on the same side of a transversal

lines that are not on the same plane and never intersect

parallel lines

skew

interior angles

exterior angles



tranversal

Active Vocabulary


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Main Idea

NonexampleSample answer:

∠ 3 and ∠ 7 are not congruent because p and q are not parallel.

Details

Fill in the organizer about the Corresponding Angles Postulate.

Fill in each blank to summarize the Perpendicular Transversal Theorem.

The Perpendicular Transversal Theorem states that in a

, if a line is to one of two

, then it is to the other.

Parallel Lines and Angle Pairspp. 178–179

Algebra and Angle Measurespp. 180

Why is it useful?If you know an angle measure, you will know that the corresponding angle has the same measure. Consecutive interior angles will be supplementary.

DefinitionIf two parallel lines are cut by a transversal, each pair of corresponding angles is congruent.

ExampleSample answer:

∠ 3 � ∠ 7 because they are corresponding angles.

Corresponding Angles Postulate

t

p

q

1

3

5

7

2

4

6

8

t

p

q

1

3

5

7

2

4

6

8

plane perpendicular

parallel lines perpendicular

Helping You Remember How can you use an everyday meaning of the adjective alternate to help you remember the types of angle pairs for two lines and a transversal?

Sample answer: One definition of alternate says first on one side and then the —————————————————————————————————————————

other. Alternating angles occur on opposite sides of a transversal.—————————————————————————————————————————


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Slopes of Lines3-3

Less

on

3-3

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What You’ll Learn Scan the text in Lesson 3-3. Write two facts you learned about slopes of lines as you scanned the text.

Sample answer: Slopes can be positive, negative, 1. ____________________________________________________

zero, or undefined. ____________________________________________________

Sample answer: Parallel lines have equal slopes.2. ____________________________________________________

____________________________________________________

Review Vocabulary Fill in each blank with the correct term or phrase. (Lesson 1-1)

made up of with no thickness or width

a without shape or size

can be explained using terms

can only be explained using and


ratio of the change along the y-axis to the change ————————————————————————————

along the x-axis between any two points on the line————————————————————————————

————————————————————————————

————————————————————————————

same as slope; describes how a quantity y changes in ————————————————————————————

relationship to a quantity x————————————————————————————

————————————————————————————

————————————————————————————

line

point

defined term

undefined term

slope

rate of change

Active Vocabulary

points

location

undefined

examples descriptions


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Main Idea Details

Complete the organizer to summarize the steps of finding the slope of a line. Then fill in each blank of the example.

Fill in each blank for the Parallel and Perpendicular Lines Postulates.

Slopes of Parallel Lines: Two non-vertical will

have the same if they are . Vertical lines

are parallel.

Slopes of Perpendicular Lines: Two

lines are if the of their

is . Vertical and horizontal lines are

perpendicular.

Slope of a Linepp. 186–188

Parallel and Perpendicular Linespp. 189–190

Step 1:Label the cordinates as

(x1, y

1) and (x

2, y

2).

Step 3: Simplify.

Find the Slope of a Line

Step 2:Substitute the

coordinates into the

m = y

2 - y

1 −

x2

- x1

slope formula.

54321

1 2 3 4 5-1-1-2-3-4-5

-2-3-4-5

x

y

Step 1: Substitute (-2, -1) for (x1, y1) and (2, 5) for (x2, y2).Step 2: Use the formula. Step 3: Simplify.

5 - (-1)

− 2 - (-2)

6 − 4 =

3 −

2

Find the slope of a line.

slope

lines

parallel

always

nonvertical

perpendicular product

slopes always

Helping You Remember Suppose 2 non-vertical lines have slopes which are reciprocals of each other. Are they perpendicular? Explain.

Sample answer: Reciprocals have a product of 1, not -1 so the lines are not —————————————————————————————————————————

perpendicular.—————————————————————————————————————————

-1


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Less

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Equations of Lines 3-4

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What You’ll Learn Skim Lesson 3-4. Predict two things that you expect to learn based on the headings and the Key Concept Box.

Sample answer: how to write equations of lines1. ____________________________________________________

____________________________________________________

Sample answer: how to solve real-world problems 2. ____________________________________________________

by writing equations ____________________________________________________

Review Vocabulary Match the term with its definition by drawing a line to connect the two. (Lesson 2-5)

a logical chain of statements that link the given to what you are trying to prove

a statement or conjecture that has been proven

a logical argument in which each statement you make is supported by a statement that is accepted as true

a statement that is accepted as true without proof

New Vocabulary Write the correct equation next to each term.

Vocabulary Link Slope is a word used in everyday English as well as in mathematics. Look up slope in the dictionary. Explain how the English definition can help you remember how slope is used in mathematics.

Sample answer: steepness; The slope of a line is the ————————————————————————————

steepness of the line.————————————————————————————

————————————————————————————

————————————————————————————

postulate

proof

deductive argument

theorem

point-slope form

slope-intercept form

Active Vocabulary

y = mx + b

y - y1 = m(x - x

1)


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Main Idea Details

Complete the chart on how to write an equation of a line.

Writing an Equation of a LineForms Procedures

Given: slope and y-intercept

Sample answer: Use slope-intercept form. Substitute m for slope and b for the y-intercept.

Given: slope and point on the line

Sample answer: Use point-slope form. Substitute m for slope and a point for (x

1, y

1)

and then solve for y.

Given: two points Sample answer: Use the slope formula to find m, then use point-slope form.

Given: for a horizontal line, when m = 0

Sample answer: Use the point-slope form where y = b.

Given: equation of one line and one set of points for ⊥ and ‖ lines

Sample answer: Solve the equation for y, then find m. Use the opposite reciprocal of m to write the equation.

Fill in each blank to write the equations.Alicia wants to paint for about 5 hours. One art studio charges an $8 fee plus $2 per hour. The second studio does not charge a fee and charges $4 an hour. Write two equations to represent the cost of each studio. Which studio should she choose?Fee + Hourly fee × x for Number of Hours = Total Cost

+ x = y

+ x = yThe studio charges a better deal for up to 4 hours.

Write Equations of Linespp. 196–198

Write Equations to Solve Problemsp. 199

8

0 4

2

first


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Proving Lines Parallel3-5

Less

on

3-5

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What You’ll Learn Skim the lesson. Write two things you already know about proving lines parallel.

Sample answer: Parallel lines have the same 1. ____________________________________________________

slope. ____________________________________________________

____________________________________________________

Sample answer: Corresponding angles are 2. ____________________________________________________

congruent, so knowing their measures could ____________________________________________________

prove lines parallel. ____________________________________________________

Review Vocabulary Write the correct term next to each definition. (Lesson 3-1)

nonadjacent interior angles that lie on opposite sides of a transversal

nonadjacent exterior angles that lie on opposite sides of a transversal

angles that lie on the same side of the transversal and on the same side of parallel lines

coplanar lines that do not intersect

line that intersects two or more parallel lines at two different points

Where a transversal cuts through two parallel lines, these angles lie in the same region between parallel lines.

Where a transversal cuts through two parallel lines, these angles lie in the same two regions that are not between the two parallel lines.

Active Vocabulary

alternative interiorangles

alternate exteriorangles


parallel lines

transversal

interior angles

exterior angles


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Main Idea Details

Use the model to cross out the congruence statement in the concept circle that does not belong.

Fill in each blank to summarize the lesson about proving lines are parallel.

When two lines are cut by a ,

the angle pairs formed are either or

. When a pair of lines form angles that

do not meet this condition, the lines be parallel.

Identify Parallel Linespp. 205–207

Prove Lines Parallelp. 208

1 3

5 7

2 4

6 8

∠1 � ∠8 ∠2 � ∠4

∠2 � ∠7∠6 � ∠7

parallel transversal

congruent

supplementary

cannot

Helping You Remember A good way to remember something new is to draw a picture. How can a sketch help you to remember the Parallel Postulate?

Sample answer: Draw a line with a ruler and choose a point not on the line. Try —————————————————————————————————————————

to draw lines through the point that are parallel, and you will see there is only —————————————————————————————————————————

one way to do this.—————————————————————————————————————————

—————————————————————————————————————————


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Perpendiculars and Distance3-6

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Sample answer: how to find the distance between 1. ____________________________________________________

a line and point ____________________________________________________

Sample answer: how to find the distance between 2. ____________________________________________________

parallel lines ____________________________________________________

Review Vocabulary Write the definition next to each term. (Lesson 2-4)

collinear lines that never intersect————————————————————————————

noncoplanar lines that never intersect————————————————————————————


The between two lines when

measured along a line to the lines is

always .

Vocabulary Link Equidistance is a word that is used in everyday English. Find the definition of equidistance using a dictionary. Explain how the English definition can help you remember how equidistance is used in mathematics.

Sample answer: always the same distance; ————————————————————————————

Equidistance means the same thing in mathematics ————————————————————————————

and everyday life.————————————————————————————

————————————————————————————

parallel lines

skew

equidistance

Active Vocabulary

distance parallel

perpendicular

the same


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Main Idea Details

Summarize the steps to find the distance between a point p and a line ℓ on a coordinate plane.

Fill in each blank to find the distance between the parallel lines.

1. y = 3x – 7 The lines are units apart.

y = 3x + 4

2. y = –2x – 8 The lines are units apart.

y = –2x + 2

Distance from a Point to a Linepp. 213–216

Distance Between Parallel Linespp. 216–217

Find the Distance between a Point and a Line.

Step 1: Write an for the line ℓ.

Step 2: Write an for the perpendicular line

that goes through point p

Step 3: Use to find the

coordinates where line ℓ the perpendicular line.

Step 4: Use the to find the distance between point P and the point of intersection of the two lines.

equation

Distance Formula

equation

system of equation

11

10

Helping You Remember Use your dictionary to find the meaning of the Latin root aequus. List three words that are derived from this root and give meaning to each.

Sample answer: Aequus means even, fair, or equal. Equinox means one of the ————————————————————————————————————————

two times a year when day and night are of equal length. Equity means being ————————————————————————————————————————

just or fair. Equivalent means being equal in value or meaning.————————————————————————————————————————

intersects


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Tie It Together

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lines slope

opposite

reciprocal

slopes

perpendicular

equal

slopes

parallel

vertical line

x = a

horizontal line

y = a

positive

negative

zero

undefined

perpendicular

coplanar lines that intersect to form a right angle


parallel

coplanar lines that never intersect


alternate interior angles


skew lines

lines that never intersect and are not coplanar

equations

point-slope form

y - y1 = m(x - x

1)

slope-intercept form

y = mx + b y

x

y

x

y

x

y

x


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Before the Test

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Review the ideas you listed in the table at the beginning of the chapter. Cross out any incorrect information in the first column. Then complete the table by filling in the third column.









Study TipsStudy Tips • Designate a place to study at home that is free of clutter and distraction. Try to study at

about the same time each afternoon or evening so that it is part of your routine.

K

What I know…

W


L

What I learned…




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Before You Read

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Before you read the chapter, think about what you know about congruent triangles. List three things you already know about them in the first column. Then list three things you would like to learn about them in the second column.


Note Taking TipsNote Taking Tips • Remember to always take notes on your own. Don’t use someone else’s notes as they may not make sense.

• When you take notes, listen or read for main ideas. Then record those ideas in a simplified form for future reference.

K

What I know…

W



Congruent Triangles


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Key Points

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Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on congruent triangle, one fact might be that in two congruent triangles all of the side and angles of one triangle are congruent to the corresponding parts of the other triangle. After completing the chapter, you can use this table to review for your chapter test.

Congruent Triangles

Lesson Fact

4-1 Classifying Triangles See students’ work.

4-2 Angles of Triangles

4-3 Congruent Triangles

4-4 Proving Triangles Congruent–SSS, SAS

4-5 Proving Triangles Congruent –ASA, AAS

4-6 Isosceles and Equilateral Triangles

4-7 Congruence Transformations

4-8 Triangles and Coordinate Proof


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Classifying Triangles4-1

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What You’ll Learn Scan Lesson 4-1. Predict two things that you expect to learn based on the headings and the Key Concept box.

Sample answer: how to classify triangles by their1. ____________________________________________________

sides ____________________________________________________

Sample answer: how to classify triangles by their2. ____________________________________________________

angles ____________________________________________________

New Vocabulary Label the diagram with the correct terms.

acute triangle

equiangular triangle

equilateral triangle

isosceles triangle

obtuse triangle

right triangle

scalene triangle

Active Vocabulary

58° 50°

72°

60°

60°60°

120°

13

11 7

isosceles triangle acute triangle right triangle

equiangular triangle obtuse triangle

scalene triangle equilateral triangle


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Main Idea Details

Classify the triangle below as acute, equiangular, obtuse, or right.

Classify each triangle below as equilateral, isosceles, or scalene.

Classify Triangles by Anglespp. 235–236

Classify Triangles by Sidespp. 236–237

90°

right triangle

2 in. 2 in.

2 in.

12 cm

12 cm

7 cm

isosceles equilateral

Helping You Remember A good way to remember a new mathematical term is to relate it to a nonmathematical definition of the same word. How is the use of the word acute, when used to describe acute pain, related to the use of the word acute when used to describe an acute angle or an acute triangle?

Sample answer: An acute pain is a sharp, focused pain. An acute angle is less —————————————————————————————————————————

than 90˚, so you can think of it as a sharp, focused angle that is not spread out.—————————————————————————————————————————

—————————————————————————————————————————

noAcute: Does the triangle have 3

acute angles?

no

Equiangular: Does the triangle

have 3 congruent acute angles?

no

Obtuse: Does the triangle have an angle greater than 90˚?

yesRight: Does the triangle have 1

right angle?


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Angles of Triangles4-2

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What You’ll Learn Scan the text under the Now heading. List two things you will learn about in the lesson.

Sample answer: how to apply the Triangle 1. ____________________________________________________

Angle-Sum Theorem ____________________________________________________

Sample answer: how to apply the Exterior Angle2. ____________________________________________________

Theorem ____________________________________________________


an extra line or segment drawn in a figure to help analyze geometric relationships

one of two angles inside a triangle that is not adjacent to an angle outside the triangle

an angle formed by one side of a triangle and the extension of an adjacent side

a method of showing statements to be true using boxes and arrows to show the logical progression of an argument

Vocabulary Link Auxiliary is a word that is used in everyday English. Find the definition of auxiliary using a dictionary. Explain how the English definition can help you remember how auxiliary line is used in mathematics.

Sample answer: Auxiliary means offering or providing ————————————————————————————

help; giving assistance or support. So, it makes sense————————————————————————————

that an auxiliary line is a line drawn in a figure to————————————————————————————

help you analyze geometric relationships.————————————————————————————

————————————————————————————

remote interior angle

auxiliary line

exterior angle

flow proof

Active Vocabulary


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Main Idea Details

Fill in each box to complete the flow proof.Given: ∠A � ∠D ∠B � ∠EProve: ∠C � ∠F

Proof:

Solve for angles 1 and 2 in the figure below.

Triangle Angle-Sum Theorempp. 244–245

Exterior Angle Theorempp. 246–247

Angle-Sum Theorem

∠A � ∠D ∠B � ∠E

m∠A + m∠B + m∠C = 180 m∠D + m∠E + m∠F = 180

m∠A + m∠B + m∠C = m∠D + m∠E + m∠F

m∠A + m∠B + m∠C = m∠A + m∠B + m∠F

m∠C = m∠F

∠C = ∠F

GivenGiven

Angle-Sum Theorem

Substitution

Substitution

Subtraction

Def. of congruent angles

m∠1 = 115

m∠2 = 65

45°

70°

2 1


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Congruent Triangles4-3

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What You’ll Learn Scan the lesson. Write two things you already know about congruent triangles.

Sample answer: The corresponding parts of1. ____________________________________________________

congruent triangles are congruent. ____________________________________________________

____________________________________________________

Sample answer: If the corresponding parts of two2. ____________________________________________________

triangles are congruent, then the triangles are ____________________________________________________

congruent. ____________________________________________________


a triangle with 3 congruent acute angles————————————————————————————

a triangle with at least 2 congruent sides ————————————————————————————

a triangle with no congruent sides————————————————————————————


If two geometric figures have exactly the same shape and

size, they are ______________.

In two , all of the parts of one

polygon are congruent to the or

matching parts of the other polygon.

equiangular triangle

isosceles triangle

scalene triangle

congruent

congruent polygons

corresponding parts

Active Vocabulary


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Main Idea Details

If �ABC � �DEF, complete the diagram to solve for x.

Name each property of triangle congruence.

If �ABC � �DEF, then �DEF � �ABC.

�RST � �RST

If �ABC � �DEF and �DEF � �LMN, then �ABC � �LMN.

Congruent and Corresponding Partspp. 253–254

Prove Triangles Congruentpp. 255–256

40

3221 9x - 4

Write a congruence relationship that will

help solve the problem. Solve the equation for x.

x = 4

Use the congruencerelationship to

to write an equation.

9x - 4 = 32DF � AC��

Check

Transitive Property

Symmetric Property

Reflexive Property

Helping You Remember Suppose a classmate is having trouble writing congruence statements for triangles because he thinks he has to match up three pairs of sides and three pairs of angles. How can you help him understand how to write correct congruence statements more easily?

Sample answer: Name one of the triangles using its three vertices. Name—————————————————————————————————————————

the other by listing corresponding vertices in the same order as the first.—————————————————————————————————————————


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Proving Triangles Congruent – SSS, SAS4-4

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Sample answer: SSS1. ____________________________________________________

____________________________________________________

Sample answer: SAS2. ____________________________________________________

____________________________________________________

Review Vocabulary Use the diagram to write an example for each of the following. (Lesson 4-2)

Sample answer: ⎯⎯

� YZ

————————————————————————————

Sample answer: ∠W and ∠X————————————————————————————

Sample answer: ∠XYZ————————————————————————————

New Vocabulary Write the definition next to the term.

the angle formed by two adjacent sides of a polygon————————————————————————————

————————————————————————————

————————————————————————————

————————————————————————————

auxiliary line

two remote interior angles to angle XYZ

exterior angle

included angle

Active Vocabulary


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Main Idea Details

Use the coordinate grid below to graph two triangles that are congruent. Explain how you could show they are congruent using the SSS postulate. Then show they are equivalent.

Sample answer: Use the vertices of the triangles and————————————————————————————

the distance formula to find the length of each side.————————————————————————————

If the corresponding sides have the same lengths, the————————————————————————————

triangles are congruent by the SSS postulate.————————————————————————————

Use the SAS Postulate to write a triangle congruence statement for the figure below.

_____________________

SSS Postulatepp. 262–264

SAS Postulatepp. 264–266

2

4

6

8

2-2

-2

-4

-6

-8

-4-6-8 4 6 8

�DEF

D( , )

E( , )

F( , )

�QRS � �TUS

�ABC

A( , )

B( , )

C( , )


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Proving Triangles Congruent – ASA, AAS4-5

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What You’ll Learn Skim the Examples for Lesson 4-5. Predict two things you think you will learn about proving triangles congruent.

Sample answer: how to prove triangles congruent 1. ____________________________________________________

using the ASA Postulate ____________________________________________________

____________________________________________________

Sample answer: how to prove triangles congruent2. ____________________________________________________

using the AAS Theorem ____________________________________________________

____________________________________________________

Review Vocabulary Write the name of the postulate that could be used to prove �EFG � �EHG. (Lesson 4-4)

New Vocabulary Write the correct term next to the definition.

the side located between two consecutive angles of a polygon

________________________

________________________

________________________

6 cm6 cm

10 cm 10 cm

SAS Postulate

SSS Postulate

included side

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Main Idea Details

Fill in each blank with the appropriate terms in the following statements.

In order to apply the ASA Postulate to prove two triangles are congruent, two pairs of and the of one triangle must be

to the corresponding parts of a second triangle.

In the figure above, because ∠ � ∠ , ∠ � ∠ , and � , you know that � � � .

Use the AAS Theorem to write a triangle congruence statement for the figure below.

_________________

ASA Postulatepp. 273–274

AAS Theorempp. 274–276

70° 50°

70°50°


included side

congruent

A B

N C AN BC

UNA KCB

75°

75°71°

71°

�PRS � �SYP

Helping You Remember Summarize what is needed to prove triangle congruence using the four methods learned so far.

SSSThree pairs of corresponding sides are congruent.

SASTwo pairs of corresponding sides and their included angles are congruent.

ASATwo pairs of corresponding angles and their included sides are congruent.

AASTwo pairs of corresponding angles and their corresponding non-included sides are congruent.


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Isosceles and Equilateral Triangles4-6

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What You’ll Learn Scan the text in Lesson 4-6. Write two facts you learned about isosceles and equilateral triangles as you scanned the text.

Sample answer: If two sides of a triangle are 1. ____________________________________________________

congruent, then the angles opposite those sides ____________________________________________________

are congruent. ____________________________________________________

Sample answer: An equilateral triangle is also an 2. ____________________________________________________

equiangular triangle. ____________________________________________________

____________________________________________________


a triangle with at least two congruent sides

a triangle with three congruent sides

a triangle with one angle greater than 90˚


base angle

vertex angle

Active Vocabulary

vertex angle

base angle

isosceles triangle

equilateral triangle

obtuse triangle


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Main Idea Details

Describe the Isosceles Triangle Theorem and its converse in the boxes below. Then write a conditional statement using the figures to illustrate each theorem.

Solve for x in equilateral triangle QRS.

__________________

Properties of Isosceles Trianglespp. 283–284

Properties of Equilateral Trianglespp. 284–286

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Example: If −−

LM � −−

LN , then ∠1 � ∠2.

Converse of Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Example: If ∠1 � ∠2, then −−

RS � −−

RT .

2

1

(6x - 2)°

x = 10 1 − 3

Helping You Remember If a theorem and its converse are both true, you can often remember them most easily by combining them into an “if-and-only-if” statement. Write such a statement for the Isosceles Triangle Theorem and its converse.

Sample answer: A triangle is isosceles if and only if two angles of the triangle —————————————————————————————————————————

are congruent.—————————————————————————————————————————

21


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Classifying Triangles4-7

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Sample answer: how to identify real-world1. ____________________________________________________

transformations ____________________________________________________

____________________________________________________

Sample answer: how to verify congruence of two2. ____________________________________________________

fi gures after a transformation ____________________________________________________

____________________________________________________

New Vocabulary Match the term with its definition by drawing a line to connect the two.

a transformation in which the position of the image may differ from that of the preimage, but the two figures remain congruent

a type of transformation that is a flip over a line

also known as a rigid transformation

the end result of a geometric transformation

an operation that maps an original geometric figure onto a new figure

a type of transformation that is a turn around a fixed point

a type of transformation that is a slide of a figure

the original figure in a geometric transformation

congruence transformation

image

isometry

preimage

reflection

rotation

transformation

translation

Active Vocabulary


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Main Idea Details

Write the type of transformation of triangle ABC (the preimage) in each box.

Complete each statement to verify the congruence.

Identify Congruence Transformationpp. 294–295

Verify Congruencep. 296

translation

reflectionrotation

y

x

′

′ ′

′ ′

′ ′

′ ′

y

x

2

4

6

-6

-4

-2

-2-4-6

RS =

ST =

RT =

R'S' =

S'T' =

R'T' =

Helping You Remember Geometric reflections, translations, and rotations are also known as flips, slides, and turns. Describe how these terms appropriately illustrate the corresponding transformations.

Sample answer: In a reflection, points are flipped across a line. In a translation, —————————————————————————————————————————

points are slid the same distance and direction. In a rotation, points are turned —————————————————————————————————————————

about a common point.—————————————————————————————————————————

3

5

4

3

5

4


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Triangles and Coordinate Proof4-8

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Sample answer: how to position and label1. ____________________________________________________

triangles for use in coordinate proofs ____________________________________________________

____________________________________________________

Sample answer: how to write coordinate proofs2. ____________________________________________________

____________________________________________________

____________________________________________________


an operation that maps an original geometric figure onto a new figure

a type of transformation that is a slide of a figure

also known as a rigid transformation

a type of transformation that is a flip over a line

the end result of a geometric transformation

a type of transformation that is a turn around a fixed point

the original figure in a geometric transformation


a method of using figures in the coordinate plane ————————————————————————————

and algebra to prove geometric concepts————————————————————————————

————————————————————————————

————————————————————————————

________________________

________________________

________________________

________________________

________________________

________________________

________________________

coordinate proof

transformation

translation

isometry

reflection

image

rotation

preimage

Active Vocabulary


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Main Idea Details

Complete the boxes below to illustrate the four key steps in placing a triangle on a coordinate grid to aid in coordinate proof.

Complete each statement showing the side lengths of triangle RST. Round to the nearest hundredth if necessary. Classify the triangle.

Position and Label Trianglespp. 301–302

Write Coordinate Proofspp. 302–303

y

xStep 1 Usethe origin as avertex.

Step 3Keep the triangle in the first quadrant

2

2 4 6

4

6

Step 4 Usecoordinatesthat makecomputationssimple.

Step 2 Place at least one side on an axis.

y

x

23

5

1

1 2 3 4 5 6 7

4

67 RS �

RT �

TS �

Triangle RST is .

6.71

6.32

5

scalene

Helping You Remember Many students find it easier to remember mathematical formulas if they can put them into words in a compact way. Describe how you can use coordinate proof to help you remember the midpoint formula.

Sample answer: When you plot two points on the coordinate grid, the midpoint —————————————————————————————————————————

is halfway between the x- and y-coordinates of the endpoints. Add the —————————————————————————————————————————

x-coordinates and add the y-coordinates. Divide each by 2.—————————————————————————————————————————


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Tie It Together

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Congruent Triangles

Transformations Types Angles

Triangles

Congruence

Obtuse

RightAcute

Equiangular

IsoscelesEquilateral

Scalene

Translations

Classified by Angles

Classified by Sides

Interior

Reflexive

Symmetric

TransitiveAAS

ASA

SAS

SSSProperties Theorems

Exterior = Sum

of remote

interiors

Sum of 360°

Sum of 180°

Exterior

Rotations

Reflections


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Before the Test

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Study TipsStudy Tips • Be an active listener in class. Take notes, circle or highlight information that your

teacher stresses, and ask questions when ideas are unclear to you.

K

What I know…

W


L

What I learned…


Congruent Triangles


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Before You Read

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Note Taking TipsNote Taking Tips • When you take notes, include personal experiences that relate to the lesson

and ways in which what you have learned will be used in your daily life.

• When you take notes, write questions you have about the lessons in the margin of your notes.

Then include the answers to these questions as you work through the lesson.

Relationships in Triangles

Before You Read Relationships in Triangles


• Concurrent lines do not intersect and stay the same distance apart.

• To find the incenter of a triangle, draw a circle within the triangle.

• Every triangle has 3 medians that are concurrent.

• The largest angle in a triangle is opposite the longest side.

• The Hinge Theorem is a way of proving triangle relationships.


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Key Points

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Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on medians and altitudes of triangles, one fact might be lines containing the altitudes of a triangle are concurrent, intersecting at a point called the orthocenter. After completing the chapter, you can use this table to review for your chapter test.


Lesson Fact

5-1 Bisectors of Triangles See students’ work.

5-2 Medians and Altitudes of Triangles

5-3 Inequalities of One Triangle

5-4 Indirect Proof

5-5 The Triangle Inequality

5-6 Inequalities in Two Triangles

5-7 Congruence Transformations

5-8 Triangles and Coordinate Proof


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Bisectors of Triangles5-1

Less

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5-1

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What You’ll Learn Skim Lesson 5-1. Predict two things that you expect to learn based on the headings and figures in the lesson.

Sample answer: how to use the Perpendicular 1. ____________________________________________________

Bisector Theorems ____________________________________________________

____________________________________________________

Sample answer: how to use the Circumcenter 2. ____________________________________________________

Theorem ____________________________________________________

____________________________________________________


the place where three or more intersecting lines meet

a segment, line, or plane that intersects a segment at its midpoint and is perpendicular to the segment

the point of intersection of the perpendicular bisectors of the three sides of a triangle

three or more lines that intersect at a common point

the point of intersection of the angle bisectors of the three angles of a triangle

Vocabulary Link Look up the definition of bisect. Use it to write the meaning of an angle bisector.

Sample answer: Bisect means to cut or divide into ___________________________________________________

two equal parts. So, an angle bisector divides an ___________________________________________________

angle into two equal angles.___________________________________________________

point of concurrency

perpendicular bisector

concurrent lines

circumcenter

incenter

Active Vocabulary


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Main Idea Details

Describe the Perpendicular Bisector Theorem in the box below. Then write a conditional statement using the figures to illustrate the theorem.

Solve for x in the figure below.

Perpendicular Bisector Theorem

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Perpendicular Bisectorspp. 322–324

Angle Bisectorspp. 324–326

1 2

Example: If −−

LO is a perpendicular bisector of −−

MN ,

then −−−

MO = −−

ON .

x =

Helping You Remember A good way to remember theorems and postulates in geometry is to explain them to other classmates in your own words. How would you describe the Angle Bisector Theorem to a classmate who is having difficulty understanding the theorem

Sample answer: Any point on the bisector of an angle is the same distance —————————————————————————————————————————

from both sides of the angle.—————————————————————————————————————————

—————————————————————————————————————————

5x + 2

8x - 7

90°

90°

3


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Medians and Altitudes of Triangles5-2

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What You’ll Learn Skim the Examples for Lesson 5-2. Predict two things you think you will learn about medians and altitudes of triangles.

Sample answer: how to use the Centroid Theorem1. ____________________________________________________

____________________________________________________

Sample answer: how to find the orthocenter on 2. ____________________________________________________

a coordinate plane ____________________________________________________

Review Vocabulary Label the diagram with the correct term. (Lesson 5-1)


a segment with endpoints being a vertex of a triangle ___________________________________________________

and the midpoint of the opposite side___________________________________________________

the point of concurrency of the medians of a triangle___________________________________________________

___________________________________________________

a segment of a triangle from a vertex that is ___________________________________________________

perpendicular to the line containing the opposite ___________________________________________________

of the triangle___________________________________________________

the point of concurrency of the lines containing the ___________________________________________________

altitudes of a triangle___________________________________________________

median

centroid

altitude

orthocenter

Active Vocabulary


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Main Idea Details

Point O is the centroid of triangle VTR. Use the Centroid Theorem to complete each statement.

TO = TW WR =

VS = VO 2 RS =

Name the orthocenter of �RST.

Medianspp. 333–335

Altitudespp. 335–337

2 − 3

3 −

2

VW

RT

y

x ( , )-1 4

Helping You Remember A good way to remember something is to explain it to someone else. Suppose that a classmate is having trouble remembering whether the center of gravity of a triangle is the orthocenter, the centroid, the incenter, or the circumcenter of the triangle. Suggest a way to remember which point it is.

Sample answer: Associate the term centroid with center of gravity. —————————————————————————————————————————

The centroid is the point of balance for any triangle.—————————————————————————————————————————

—————————————————————————————————————————

—————————————————————————————————————————


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Inequalities in One Triangle5-3

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Sample answer: how to recognize and apply 1. ____________________________________________________

properties of inequalities to the measures of the ____________________________________________________

angles of a triangle ____________________________________________________

Sample answer: how to recognize and apply 2. ____________________________________________________

properties of inequalities to the relationships ____________________________________________________

between the angles and sides of a triangle ____________________________________________________

Review Vocabulary Match the term with its definition by drawing a line to connect the two. (Lessons 5-1 and 5-2)

the place where three or more intersecting lines meet

the point of concurrency of the medians of a triangle

a segment of a triangle from a vertex that is perpendicular to the line containing the opposite of the triangle

the point of concurrency of the lines containing the altitudes of a triangle

a segment with endpoints being a vertex of a triangle and the midpoint of the opposite side

a segment, line, or plane that intersect a segment at its midpoint and is perpendicular to the segment

three or more lines that intersect at a common point


the point of intersection of the angle bisectors of the three angles of a triangle


orthocenter

altitude

incenter

point of concurrency

median

circumcenter

concurrent lines

centroid

Active Vocabulary


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Main Idea Details

Describe the Exterior Angle Inequality Theorem in your own words. Then use the figure below to write two inequalities.

Sample answer: The measure of an exterior angle of ___________________________________________________

a triangle is greater than the measure of either remote ___________________________________________________

interior angle; m∠1 < m∠A, m∠1 < m∠B___________________________________________________

List the angles of �LMN in order from least to greatest.

Angle Inequalitiespp. 342–343

Angle-Side Inequalitiespp. 343–345

1

Helping You Remember Explain how the Exterior Angle Inequality Theorem is related to the Exterior Angle Theorem, and why the Exterior Angle Inequality Theorem must be true if the Exterior Angle Theorem is true.

Sample answer: The Ext. Angle Theorem says that the measure of an exterior —————————————————————————————————————————

angle of a triangle is equal to the sum of the measures of the two remote —————————————————————————————————————————

interior angles, while the Ext. Angle Inequality Theorem says that the measure —————————————————————————————————————————

of an exterior angle is greater than the measure of either remote interior angle.—————————————————————————————————————————

If a number is equal to the sum of two positive numbers, it must be greater —————————————————————————————————————————

than each of those two numbers.—————————————————————————————————————————

10.2

7.511.6

∠L, ∠N, ∠M


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Indirect Proof5-4

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Sample answer: Indirect Algebraic Proof1. ____________________________________________________

____________________________________________________

Sample answer: Indirect Proof with Geometry2. ____________________________________________________

____________________________________________________


Indirect reasoning is a method of thinking that assumes that a conclusion is and then showing that this assumption leads to a contradiction.

In an indirect proof, you temporarily assume that what you are trying to prove is false. By showing this assumption to be logically impossible, you prove your assumption false and the true.

To construct a proof by contradiction, the first step is to that the conclusion you want to

prove is false.

Vocabulary Link Think of other times you have encountered the word indirect in mathematics. Describe any similarities to an indirect proof.

Sample answer: indirect measurement; You don’t ___________________________________________________

directly measure a quantity just as you don’t directly ___________________________________________________

prove a conclusion.___________________________________________________

___________________________________________________

indirect reasoning

indirect proof

proof by contradiction

Active Vocabulary

original conclusion

false

assume


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Main Idea Details

Complete the table below showing the steps involved in constructing an indirect proof.

How to Write an Indirect ProofStep 1 Sample answer: Identify the conclusion you

are asked to prove. Make the assumption that this conclusion is false by assuming that the opposite is true.

Step 2 Sample answer: Use logical reasoning to show that this assumption leads to a contradiction of the hypothesis, or some other fact, such as a definition, postulate, theorem, or corollary.

Step 3 Sample answer: Point out that because the assumption leads to a contradiction, the original conclusion, what you were asked to prove, must be true.

Suppose you want to prove that the sum of interior angles of a triangle is equal to 180º. What assumption would you make to form an indirect proof of this statement?

Indirect Algebraic Proofpp. 351–353

Indirect Proof with Geometrypp. 353–354

Helping You Remember A good way to remember a new concept in mathematics is to relate it to something you have already learned. How is the process of indirect proof related to the relationship between a conditional statement and its contrapositive?

Sample answer: The contrapositive of a statement p → q is the statement —————————————————————————————————————

~q → ~p. In an indirect proof of a conditional statement p → q, you assume —————————————————————————————————————

that q is false and show that this implies that p is false. That is, you show that —————————————————————————————————————

~q → ~p is true, which is logically equivalent to p → q.—————————————————————————————————————

1 2

3

Assume: m∠1 + m∠2 + m∠3 ≠ 180


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The Triangle Inequality5-5

Less

on

5-5

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What You’ll Learn Scan the text in Lesson 5-5. Write two facts you learned about the triangle inequality.

Sample answer: You can use the triangle 1. ____________________________________________________

inequality to identify possible triangles given their ____________________________________________________

side lengths. ____________________________________________________

Sample answer: The sum of two sides of a triangle 2. ____________________________________________________

must be greater than the length of the third side. ____________________________________________________

____________________________________________________

Review Vocabulary Label each figure with the correct term. (Lessons 5-1 and 5-2)

Active Vocabulary

3

3

55

4

4


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Main Idea Details

If the measures of two sides of a triangle are 4 centimeters and 9 centimeters, what is the least possible whole number measure for the third side?

Complete the two-column proof.Given:

−− FI �

−− FJ

Prove: FI + FH > HJ

The Triangle Inequalitypp. 360–361

Proofs Using the Triangle Inequalityp. 362

Step 1: Use the triangle inequality to write three inequalities for a triangle with sides 4, 9, and x centimeters.

4 + 9 > x 4 + x > 9 9 + x > 4

Step 2: Solve each the inequalities.

4 + 9 > x

x < 13

4 + x > 9

x > 5

9 + x > 4x > - 5

Step 3: Use the inequalities to solve the problem.

x = 6 centimeters

Statements Reasons1.

−−

FI � −−

FJ 1. Given

2. FI = FJ 2. Def of cong. seg.

3. FJ + FH > HJ 3. Triangle Ineq.

4. FI + FH > HJ 4. Substitution

Helping You Remember A good way to remember a new theorem is to state it informally in different words. How could you restate the Triangle Inequality Theorem?

Sample answer: The sides that connect one vertex of a triangle to another is —————————————————————————————————————————

a shorter path between the two vertices than the path that goes through the —————————————————————————————————————————

third vertex.—————————————————————————————————————————


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Inequalities in Two Triangles5-6

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What You’ll Learn Skim the lesson. Write two things you already know about inequalities in two triangles.

Sample answer: The Hinge Theorem is also called 1. ____________________________________________________

the SAS Inequality Theorem. ____________________________________________________

____________________________________________________

Sample answer: You can prove triangle 2. ____________________________________________________

relationships using the Hinge Theorem. ____________________________________________________

____________________________________________________

Review Vocabulary Write the correct term next to each definition. (Lessons 5-1 and 5-2)

a segment of a triangle from a vertex that is perpendicular to the line containing the opposite of the triangle

a segment, line, or plane that intersect a segment at its midpoint and is perpendicular to the segment


the point of concurrency of the medians of a triangle

Vocabulary Link Describe how the hinge of a door can be used to illustrate the Hinge Theorem.

Sample answer: The width of the door frame and the ___________________________________________________

door itself form the two sides. If you open two of the ___________________________________________________

same size doors at different angles, the door that is ___________________________________________________

open to a greater angle is farther from the door frame.___________________________________________________

altitude


circumcenter

centroid

Active Vocabulary


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Main Idea Details

Find the range of possible values for x in the figure below.

Describe how you could use the Hinge Theorem to complete the proof below.Given: AC = CDProve: BD > AB

Hinge Theorempp. 367–370

Prove Relationships in Two Trianglespp. 370–371

(3x - 5)°61°

10 10

1412

Write an inequality using the Converse of the Hinge Theorem.

Write an inequality given the angle is less

than 180º.61 < 3x - 5 3x - 5 < 180

Solve for x. Solve for x.x > 22 x > 61 2 −

3

What are the possible values of x ?

22 < x < 61 2 − 3

Sample answer: You know that 2 sides of �BDC are congruent to 2 sides of �ABC. Since the included angle of �BDC is greater, the Hinge Theorem says that the third side is greater than the third side of �ABC. So, BD > AB. The distance between the points increases or decreases accordingly.

Helping You Remember A good way to remember something is to think of it in concrete terms. How can you illustrate the Hinge Theorem with everyday objects?

Sample answer: Put two pencils on a desktop so that the erasers touch. —————————————————————————————————————————

As you increase or decrease the measure of the angle formed by the pencils, —————————————————————————————————————————

the distance between the points increases of decreases accordingly.—————————————————————————————————————————


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Tie It Together

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Complete each graphic organizer with a term, description, diagram, or theorem from the chapter.


Triangle Inequalities

Exterior Angles The measure of an exterior angle is greater than the measure of either remote interior angle.

Angle-Side The largest angle is across from the longest side. The smallest angle is across from the shortest side.

Triangle Inequality

The longest side is across from the largest angle. The shortest side is across from the smallest angle.

Hinge If two pairs of sides in two triangles are congruent, and the included angle of one is larger than the included angle of the other, then the third side of the first triangle is longer than the third side of the second triangle.

Segment DefinitionPoint

of Concurrency

Property of

PointDiagram

perpendicularbisector

a segment that bisects the side of a triangle at a right angle

circumcenter center of the circumscribed circle

angle bisector a segment that bisects the angles of a triangle

incenter center of an inscribed circle

median a segment that connects a vertex of a triangle with the midpoint of the opposite side

centroid center of gravity of the triangle

altitude a segment from the vertex of a triangle that intersects the opposite side at a right angle

orthocenter the point of concurrency of the altitudes of a triangle


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Before the Test

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Study TipsStudy Tips • Get a good nights rest before a test. Students that take the time to sleep usually do

better than students who stay up late cramming.

Relationships in Triangles After You Read

• Concurrent lines do not intersect and stay the same distance apart.

D

• To find the incenter of a triangle, draw a circle within the triangle.

A

• Every triangle has 3 medians that are concurrent. A

• The largest angle in a triangle is opposite the shortest side.

D

• The Hinge Theorem is a way of proving triangle relationships.

A



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Before You Read

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Note Taking TipsNote Taking Tips • When you take notes, look for written real-world examples in your everyday

life. Comment on how writers use statistics to prove or disprove points of view and discuss

the ethical responsibilities writers have when using statistics.

• When you take notes, include visuals. Clearly label the visuals and write captions when needed.

Quadrilaterals

Before You Read Quadrilaterals


• The sum of the interior angles of a convex polygon is (n – 1) × 180.

• A diagonal cuts a parallelogram into 2 congruent triangles.

• A rhombus is a quadrilateral.

• If a parallelogram has 1 right angle, then it has 4 right angles.

• A trapezoid can have 0 or 1 set of parallel sides.


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Key Points

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Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on rectangles, one fact might be a parallelogram with four right angles is a rectangle. After completing the chapter, you can use this table to review for your chapter test.

Quadrilaterals

Lesson Fact

6-1 Angles of Polygons See students’ work.

6-2 Parallelograms

6-3 Tests for Parallelograms

6-4 Rectangles

6-5 Rhombi and Squares

6-6 Trapezoids and Kites


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Angles of Polygons6-1

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Sample answer: how to find the sum of the1. ____________________________________________________

interior angles of a polygon ____________________________________________________

____________________________________________________

Sample answer: how to find the number of sides2. ____________________________________________________

of a polygon given its interior angle measures ____________________________________________________

____________________________________________________


a segment in a polygon that connects any two————————————————————————————

nonconsecutive vertices————————————————————————————

Name all of the diagonals in polygon ABCDE.

−− AC ,

−− AD ,

−− BD ,

−− BE ,

−− CE

Vocabulary Link Look up how television screens are measured. What does it mean for a television to have a 32-inch screen?

Sample answer: The screens are measured along the————————————————————————————

diagonal; 32-inch diagonal.————————————————————————————

————————————————————————————

diagonal

Active Vocabulary


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Main Idea Details

Complete the following table for convex polygons. For middle column, find the number of triangles you can divide the polygon into by drawing all the possible diagonals from one vertex.

polygonnumberof sides

number of triangles

sum of interior angle measures

triangle 3 1 180°

hexagon 6 4 720°

octagon 8 6 1080°

nonagon 9 7 1260°

n-gon n n - 2 (n - 2)180°

Find the value of x in the figure below.

x = 15x = 15

Polygon Interior Angles Sumpp. 389–392

Polygon Exterior Angles Sumpp. 392–393

(8x + 5)°(6x)°

80°

65°

Helping You Remember A good way to remember a new mathematical idea or formula is to relate it to something you already know. How can you use your knowledge of the Angle Sum Theorem (for a triangle) to help you remember the Interior Angle Sum Theorem?

Sample answer: The sum of the measures of the interior angles of a triangle—————————————————————————————————————————

is 180°. Each time another side is added, the polygon can be subdivided into —————————————————————————————————————————

one more triangle, so 180° is added to the interior angle sum.—————————————————————————————————————————


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Parallelograms6-2

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Sample answer: about the properties of1. ____________________________________________________

parallelograms ____________________________________________________

____________________________________________________

Sample answer: about the properties of the2. ____________________________________________________

diagonals of parallelograms ____________________________________________________

____________________________________________________

New Vocabulary Write the definition of the term.

a quadrilateral with both pairs of opposite sides————————————————————————————

parallel————————————————————————————

————————————————————————————

Fill in each blank using parallelogram ABCD.

−−

AB ||

−−− BE �

−−− AD �

−−− BC ||

−− AE �

parallelogram

−−

CD

−−

DE

−−

BC

−−

AD

−−

EC

Active Vocabulary


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Main Idea Details

Complete the table using Theorems 6.3, 6.4, 6.5, and 6.6 in the Student Edition.

Properties of ParallelogramsTheorem Property

6.3 If a quadrilateral is a parallelogram, then its

opposite sides are .


opposite angles are .


opposite sides are .

6.6 If a parallelogram has one right angle, then it

has four .

Find the value of z in the parallelogram below.

Sides and Angles of Parallelogramspp. 399–400

Diagonals of Parallelogramspp. 401–402

congruent

congruent

supplementary

right angles

9z - 4

4z + 21

z = 5

Helping You Remember A good way to remember new theorems in geometry is to relate them to theorems you learned earlier. Name a theorem about parallel lines that can be used to remember the theorem that says, “If a parallelogram has one right angle, it has four right angles.”

Perpendicular Transversal Theorem—————————————————————————————————————————


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Tests for Parallelograms6-3

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What You’ll Learn Skim the Examples for Lesson 6-3. Predict two things you think you will learn about tests for parallelograms.

Sample answer: how to recognize the conditions 1. ____________________________________________________

that ensure a quadrilateral is a parallelogram ____________________________________________________

____________________________________________________

Sample answer: how to prove that a set of points2. ____________________________________________________

forms a parallelogram in the coordinate plane ____________________________________________________

____________________________________________________

Review Vocabulary Fill in each blank with the correct term or phrase. (Lessons 6-1 and 6-2)

A diagonal is a segment in a polygon that connects any two

vertices.

A parallelogram is a quadrilateral with both pairs of

parallel.

Fill in each blank to review the properties of parallelograms.

The sides of a parallelogram are congruent.

The opposite angles of a parallelogram are .

Consecutive angles in a parallelogram are .

If a parallelogram has one angle, then it

has four angles.

diagonal

parallelogram

Active Vocabulary

nonconsecutive

opposite sides

opposite

congruent

supplementary

right

right


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Main Idea Details

Complete the table using Theorems 6.9, 6.10, 6.11, and 6.12 in the student book for any quadrilateral ABCD.

Properties of Parallelograms

Theorem Property

6.9If both pairs of opposite sides of ABCD are

, then ABCD is a parallelogram.

6.10If both pairs of opposite angles of ABCD are


6.11If the diagonals of ABCD

each other, then ABCD is a parallelogram.

6.12If one pair of opposite sides of ABCD is both


Find the slope of each line segment to verify that WXYZ is a parallelogram.

Conditions for Parallelogramspp. 409–411

Parallelograms on the Coordinate Planepp. 412–413

congruent

congruent

parallel and congruent

bisect

y

x

slope of −−−

WX = _________ 1 − 8

slope of −−

XY = _________5

slope of −−

YZ = _________ 1 − 8

slope of −−−

WZ = _________5

Helping You Remember A good way to remember a large number of mathematical ideas is to think of them in groups. How can you state the conditions as one group about the sides of a quadrilateral that guarantee it is a parallelogram?

Sample answer: both pairs of opposite sides parallel, both pairs of opposite —————————————————————————————————————————

sides congruent, or one pair of opposite sides both parallel and congruent—————————————————————————————————————————


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Rectangles6-4

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Sample answer: Properties of Rectangles1. ____________________________________________________

____________________________________________________

Sample answer: Prove that Parallelograms are2. ____________________________________________________

Rectangles ____________________________________________________

New Vocabulary Write the definition of the term.

a parallelogram with four right angles————————————————————————————

————————————————————————————

Fill in each blank using rectangle ABCD.

−− AB �� __________

−− AC � __________

m∠ADC = __________

m∠BCD = __________

−−−

AD � __________

rectangle

−−

CD

−−

BD

90

90

−−

BC

Active Vocabulary


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Main Idea Details

Cross out the incorrect quadrilateral to complete the Venn diagram and illustrate the relationship between rectangles and parallelograms. Then write the definition of the correct quadrilateral in the space provided.

Use the Distance Formula to determine whether or not parallelogram RSTU is a rectangle.

1010

Properties of Rectanglespp. 419–420

Prove That Parallelograms Are Rectanglespp. 420–421

rectangleparallelogram

rectangleparallelogram

y

xSU ≈ _________

RT ≈ _________10.2

RSTU is not a rectangle.

Helping You Remember It is easier to remember a large number of geometric relationships and theorems if you are able to combine some of them. How can you combine the two theorems about diagonals that you studied in this lesson?

Sample answer: A parallelogram is a rectangle if and only if its diagonals are—————————————————————————————————————————

congruent.—————————————————————————————————————————


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Rhombi and Squares6-5

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What You’ll Learn Scan the text in Lesson 6-5. Write two facts you learned about rhombi and squares.

Sample answer: The diagonals of a rhombus are1. ____________________________________________________

perpendicular. ____________________________________________________

____________________________________________________

Sample answer: If a quadrilateral is both a2. ____________________________________________________

rectangle and a rhombus, then it is a square. ____________________________________________________

____________________________________________________

New Vocabulary Label the diagrams with the correct terms.

Vocabulary Link A square is a shape that you learn at a very early age. Name some everyday items that are shaped like a square. Can you name any items that are shaped like a rhombus?

Sample answer for square: tile on a floor————————————————————————————

————————————————————————————

Sample answer for rhombus: the diamonds in a————————————————————————————

deck of cards————————————————————————————

rhombus

square

rhombus and

square

rhombus

Active Vocabulary


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Main Idea Details

Use the properties of rhombi to solve for x in rhombus ABCD.

Describe how you could prove that RS2 = RV2 + SV2 in rhombus RSTU.

Properties of Rhombi and Squarespp. 426–428

Prove That Quadrilaterals Are Rhombi or Squarespp. 428–430

AD = 8x - 11

DC = 5x + 13

Check

What property of rhombi can you use tohelp you solve for x?

Use the property to write an equation.

– = +

Solve for x.

=

Sample answer: ∠RVS is a

right angle because the

Use the Pythagoren Theorem.

diagonals are perpendicular.


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Trapezoids and Kites6-6

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What You’ll Learn Skim the lesson. Write two things you already know about trapezoids and kites.

Sample answer: If a quadrilateral has only one set 1. ____________________________________________________

of parallel sides, then it is a trapezoid. ____________________________________________________

____________________________________________________

____________________________________________________

Sample answer: The diagonals of a kite are2. ____________________________________________________

perpendicular to each other. ____________________________________________________

____________________________________________________

____________________________________________________


the segment that connects the midpoints of the legs of the trapezoid

a quadrilateral with exactly two pairs of consecutive congruent sides

the nonparallel sides of a trapezoid

the parallel sides of a trapezoid

the angles formed by the base and one of the legs of a trapezoid

a quadrilateral with exactly one pair of parallel sides

a trapezoid with congruent legs

base angles

bases

isosceles trapezoid

kite

legs of a trapezoid

midsegment of a trapezoid

trapezoid

Active Vocabulary


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Main Idea Details

Complete the flow proof below.Given: ABCD is an isosceles trapezoid

with bases −−

AB and −−−

CD .Prove: ∠BDC � ACD

Solve for x if −−−

MN is a midsegment of trapezoid RSTU.

Properties of Trapezoidspp. 435–438

Properties of Kitespp. 438–439

ABCD is an isosceles trapezoid with bases −−

AB and −−

CD .

Given

−−

AB � −−

AB −−

AC � −−

BD −−

AD � −−

BC

Reflexive Prop. Diagonals are cong. Def. isoc. trap.

�BDC � �ACD ∠BDC � ∠ACD

SSS CPCTC

x

16.5

14.4

18.6x =

Helping You Remember A good way to remember a new geometric theorem is to relate it to one you already know. Name and state in words a theorem about triangles that is similar to the theorem in this lesson about the median of a trapezoid.

Sample answer: Triangle Midsegment Theorem; A midsegment of a triangle is—————————————————————————————————————————

parallel to one side of the triangle, and its length is one half the length of that—————————————————————————————————————————

side.—————————————————————————————————————————


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Tie It Together

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Complete the graphic organizer with properties of the figure from the chapter.

Quadrilaterals

Quadrilaterals

Kites

Two pairs of consecutive sides are congruent.Diagonals are perpendicular.One pair of opposite angles are congruent.

IsoscelesTrapezoids

One pair of sides is congruent.Both pairs of base angles are congruent.Diagonals are congruent.

Trapezoids

One pair of sides is parallel.

Parallelograms

Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. One pair of opposite sides are both congruent and parallel.

Squares

Rectangles

All angles are right angles.Diagonals are congruent.

Rhombi

All sides are congruent.Diagonals are perpendicular.


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Before the Test

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Study TipsStudy Tips • When you are preparing to read new material, scan the text first, briefly looking over

headings, highlighted text, pictures, and call out boxes. Think of questions that you can search the text for as you read.

Quadrilaterals

Quadrilaterals After You Read

• The sum of the interior angles of a convex polygon is (n – 1) × 180.

D

• A diagonal cuts a parallelogram into 2 congruent triangles.

A

• A rhombus is a quadrilateral. A

• If a parallelogram has 1 right angle, then it has 4 right angles.

A

• A trapezoid can have 0 or 1 set of parallel sides. D


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Before You Read

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Before you read the chapter, think about what you know about proportions and similarity. List three things you already know about proportions and similarity in the first column. Then list three things you would like to learn about them in the second column.


Note Taking TipsNote Taking Tips • When you take notes, write instructions on how to do something presented in

each lesson. Then follow your own instructions to check them for accuracy.

• When you take notes, be sure to describe steps in detail. Include examples of questions you might ask yourself during problem solving.

Proportions and Similarity

K

What I know…

W




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Key Points

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Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on similar polygons, one fact might be similar polygons have the same shape but not the same size. After completing the chapter, you can use this table to review for your chapter test.


Lesson Fact

7-1 Ratios and Proportions See students’ work.

7-2 Similar Polygons

7-3 Similar Triangles

7-4 Parallel Lines and Proportional Parts

7-5 Parts of Similar Triangles

7-6 Similarity Transformations

7-7 Scale Drawings and Models


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Ratios and Proportions7-1

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7-1

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What You’ll Learn Skim the Examples for Lesson 7-1. Predict two things you think you will learn about ratios and proportions.

Sample answer: how to write and use ratios1. ____________________________________________________

____________________________________________________

____________________________________________________

Sample answer: how to use cross products to2. ____________________________________________________

make predictions ____________________________________________________

____________________________________________________


the numbers a and d in the proportion a − b = c −

d

an equation stating that two ratios are equal

a ratio that compares three or more numbers

the product of the extremes, ad, and the product of the means, bc in the proportion a −

b = c −

d

a comparison of two quantities using division

the numbers b and c in the proportion a − b = c −

d

Vocabulary Link Look up the meaning of the word proportional. How does this relate to a proportion in mathematics?

Sample answer: properly related in size or scale————————————————————————————

————————————————————————————

————————————————————————————

cross product

extended ratio

extremes

means

proportion

ratio

Active Vocabulary


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Main Idea Details

The ratio of the measures of the angles in �DEF is 4:5:9. Find the measures of the angles.

_______ : _______ : _______

_________________________

_________________________

_________________________

A quality control engineer randomly samples 120 car stereos coming off an assembly line. He finds that 3 of them have cosmetic blemishes. How many car stereos would you expect to have cosmetic blemishes in a production run of 8000 car stereos?

_________________________

_________________________

Write and Use Ratiospp. 457–458

Use Properties of Proportionspp. 458–459

9x5x4xStep 1: Multiply the ratio by x to represent the unknown angle measures.

Step 2: Write an equation using the extended ratio and the Triangle Sum Theorem.

Step 3: Solve for x.

Step 4: Find the angle measures.

4x + 5x + 9x = 180

3 −

120 = x

− 8000

40°, 50°, 90°

Write a proportion.

Solve the proportion. 200 stereos

Helping You Remember Sometimes it is easier to remember a mathematical idea if you put it into words without using any mathematical symbols. How can you use this approach to remember the concept of equality of cross products?

Sample answer: The product of the means equals the product of the extremes.—————————————————————————————————————————

—————————————————————————————————————————

x = 10


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Similar Polygons7-2

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Sample answer: how to identify similar figures1. ____________________________________________________

____________________________________________________

____________________________________________________

Sample answer: how to use similar figures to find2. ____________________________________________________

missing measures ____________________________________________________

____________________________________________________


an equation stating that two ratios are equal————————————————————————————

a ratio that compares three or more numbers————————————————————————————

the numbers b and c in the proportion a −

c =

c −

d

————————————————————————————

a comparison of two quantities using division————————————————————————————


the scale factor between two similar polygons

polygons that have the same shape but not necessarily the same size

the ratio of the lengths of the corresponding sides of two similar polygons

Active Vocabulary

proportion

extended ratio

extremes

ratio

similarity ratio

similar polygons

scale factor


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PDF 2nd


Main Idea Details

Model two similar triangles by writing the side lengths of �XYZ. Then tell the scale factor from �HIJ to �XYZ.

Sample answer is given.

Solve for y in the similar trapezoids below.

Identify Similar Polygonspp. 465–466

Use Similar Figurespp. 467–468

20

12

16 16

9.6

12.8

scale factor: _____________1.25

9

27

18

2y + 1

y = _____________2.5

Helping You Remember A good way to remember a new mathematical vocabulary term is to relate it to words used in everyday life. The word scale has many meanings in English. Give three phrases that include the word scale in a way that is related to proportions.

Sample answer: scale drawing, scale model, scale of miles (on a map)—————————————————————————————————————————

—————————————————————————————————————————

—————————————————————————————————————————


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Similar Triangles7-3

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Sample answer: how to identify similar triangles1. ____________________________________________________

using the AA Postulate, SSS Theorem, and SAS ____________________________________________________

Theorem ____________________________________________________

____________________________________________________

Sample answer: how to use similar triangles to2. ____________________________________________________

solve problems ____________________________________________________

____________________________________________________

____________________________________________________

Review Vocabulary Fill in each blank with the correct term or phrase. (Lessons 7-1, 7-2)

The extremes are the numbers ___________ and ___________

in the proportion a − b = c −

d .

The means are the numbers ___________ and ___________ in

the proportion a − b = c −

d .

A proportion is an equation stating that __________________

are equal.

The scale factor is the ratio of the lengths of the

____________________ sides of two similar polygons.

A ratio is a comparison of two quantities using ____________.

Similar polygons have the same ____________ but not

necessarily the same ____________.

a d

b c

corresponding

two ratios

division

shape

size

extremes

means

proportion

scale factor

ratio

similar polygons

Active Vocabulary


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Main Idea Details

Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

�ORS ∼ �YVT by AA similarity. ____________________________________________________

Complete the table to show the properties of similarity in Theorem 7.5.

Identify Similar Trianglespp. 474–477

Use Similar Trianglespp. 477–478

77°

65°

77°

65°

Theorem 7.5Properties of Similarity

Reflexive Property �ABC ∼ �ABC

Symmetric Property If �ABC ∼ �DEF, then �DEF ∼ �ABC.

Transitive Property If �ABC ∼ �DEF, and �DEF ∼ �LMN, then �ABC ∼ �LMN.

Helping You Remember A good way to remember something is to explain it to somone else. Suppose one of your classmates is having trouble understanding the difference between SAS for congruent triangles and SAS for similar triangles. How can you explain the difference to him or her?

Sample answer: In SAS for congruence, corresponding sides are congruent, —————————————————————————————————————————

while in SAS for similarity, corresponding sides are proportional. In both, the —————————————————————————————————————————

included angles are congruent.—————————————————————————————————————————


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Parallel Lines and Proportional Parts7-4

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Sample answer: Proportional Parts Within1. ____________________________________________________

Triangles ____________________________________________________

Sample answer: Proportional Parts with Parallel2. ____________________________________________________

Lines ____________________________________________________

Review Vocabulary Label the diagrams with the correct terms. (Lesson 7-1)


extreme

mean

extended ratio

Active Vocabulary

mean

extreme

extended ratio

3 − 6 =

29 − 58

4 : 8 : 11

midsegment


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Main Idea Details

Use the Triangle Proportionality Theorem to find RT.

Model Corollary 7.1 by drawing two transversals on the parallel lines below and writing a proportion. Sample answers are given.

Proportional Parts Within Trianglespp. 484–486

Proportional Parts with Parallel Linespp. 486–488

3.6

6.3 8.4

Step 1: Use the Triangle Proportionality Theorem to write a proportion.

Step 3: Solve the proportion for RT.

Step 2: Substitute the known values into the proportion.

______________________ MQ

− QS

= RT −

RS

______________________RT = 4.8______________________ 3.6

− 6.3

= RT −

8.4

______________________ AB

− BC

= XY −

YZ

m

k

l

Helping You Remember A good way to remember a new mathematical term is to relate it to other mathematical vocabulary that you already know. What is an easy way to remember the definition of midsegment using other geometric terms?

Sample answer: A midsegment is a segment that connects the midpoints of —————————————————————————————————————————

two sides of a triangle.—————————————————————————————————————————

—————————————————————————————————————————

—————————————————————————————————————————


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Parts of Similar Triangles7-5

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What You’ll Learn Scan the text in Lesson 7-5. Write two facts you learned about parts of similar triangles.

Sample answer: If two triangles are similar, the1. ____________________________________________________

lengths of corresponding altitudes are ____________________________________________________

proportional to the lengths of corresponding sides. ____________________________________________________

____________________________________________________

Sample answer: You can use special segments in2. ____________________________________________________

similar triangles to solve for missing measures. ____________________________________________________

____________________________________________________

____________________________________________________

Review Vocabulary Label the diagrams with the correct terms. (Lessons 5-1 and 5-2)

50° 50°

altitude

angle bisector

median

angle bisector

altitude

median

Active Vocabulary


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Main Idea Details

Use Theorem 7.8 in the student book to solve for BD in the similar triangles below.

Model the Triangle Angle Bisector Theorem by drawing an angle bisector and writing a proportion. Sample answer:

Special Segments of Similar Trianglespp. 495–497

Triangle Angle Bisector Theoremp. 498

15 16 20

Step 1: Use Theorem 7.8 to write a proportion. ______________________

BD −

US = AD

− ST

______________________ BD

− 16

= 15

− 20

______________________BD = 12

Step 2: Substitute the known values into the proportion.

Step 3: Solve the proportion for BD.

______________________ RU

− SU

= RT −

ST

Helping You Remember A good way to remember a large amount of information is to remember key words. What key words will help you remember the features of similar triangles that are proportional to the lengths of the corresponding sides?

Sample answer: perimeters, altitudes, angle bisectors, medians—————————————————————————————————————————

—————————————————————————————————————————


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Similarity Transformations7-6

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What You’ll Learn Skim the lesson. Write two things you already know about similarity transformations.

Sample answer: A similarity transformation is a1. ____________________________________________________

transformation that maps a figure onto a similar ____________________________________________________

figure. ____________________________________________________

____________________________________________________

Sample answer: You can use SAS Similarity to2. ____________________________________________________

verify that two figures are similar after a dilation. ____________________________________________________

____________________________________________________

____________________________________________________


Dilations are performed with respect to a _______________ called the center of dilation.

A dilation is a transformation that _______________ or

_______________ the original figure proportionally.

A dilation with a scale factor ____________________ produces an enlargement.

The scale factor of a dilation describes the _______________ of the dilation.

A similarity transformation is a transformation that maps a

figure onto a _______________ figure.

A dilation with a scale factor _____________________ produces a reduction.

Active Vocabulary

fixed point

enlarges

reduces

greater than 1

extent

similar

between 0 and 1

center of dilation

dilation

enlargement

scale factor of a dilation

similarity transformation

reduction


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Main Idea Details

Model a reduction and an enlargement by drawing a similar triangle on each coordinate grid. Write the scale factor of each dilation. Sample answer:

Reduction Enlargement

Graph the original figure and its dilated image on a sheet of graph paper. Verify that the dilation is a similarity transformation. See students’ work. The transformation is an enlargement with a scale factor of 2.

Preimage: A(0, 0), B(2, 3), C(4, 1)

Image: A′(0, 0), B′(4, 6), C′(8, 2)

Identify Similarity Transformationspp. 505–506

Verify Similarityp. 507

x

y

' '

'

x

y

' '

'

scale factor: __________0.5 scale factor: __________1.5

Helping You Remember A good way to develop a higher understanding of a concept is to compare and contrast it with other similar topics. Compare and contrast similarity transformations with the congruency transformations that you learned about in chapter 6.

Sample answer: Both types of transformations deal with taking a geometric —————————————————————————————————————————

figure (preimage) and mapping it onto an image. With congruency —————————————————————————————————————————

transformations, the image is congruent to the preimage. With similarity —————————————————————————————————————————

transformations, the image is similar to the preimage.—————————————————————————————————————————


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Scale Drawings and Models7-7

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What You’ll Learn Scan the text in Lesson 7-7. Write two facts you learned about scale drawings and models.

Sample answer: A scale model is an object with1. ____________________________________________________

lengths proportional to the object it represents. ____________________________________________________

____________________________________________________

Sample answer: Scale factors are always written2. ____________________________________________________

so that the model length in the ratio comes first. ____________________________________________________

____________________________________________________


the ratio of a length on a scale model or drawing to————————————————————————————

the actual length of the object being modeled or drawn————————————————————————————

a drawing with lengths proportional to the object————————————————————————————

that it represents————————————————————————————

an object with lengths proportional to the object that————————————————————————————

it represents————————————————————————————

New Vocabulary Scale drawings and models have a wide range of applications. For instance, a roadmap is a real world example of a scale drawing. Can you think of other real world examples of scale drawings and scale models?

Sample answers: blueprints, toy models, photographs ————————————————————————————

————————————————————————————

scale

scale drawing

scale model

Active Vocabulary


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Main Idea Details

The scale on a map is 2 centimeters : 15 kilometers. If the distance between two cities on the map is 9 centimeters, find the actual distance between the cities.

Suppose a model airplane has a wingspan of 8 inches. The actual plane has a wingspan of 48 feet.1. What is the scale of the model? ______ in. : ______ ft2. How many times as long as the actual plane is the

model? ______________

Scale Modelspp. 512–513

1 6

Helping You Remember A good way to remember something is to explain it to someone else. Suppose one of your classmates is having trouble understanding how to use the scale on a map to calculate actual distances. How can you explain the procedure to her?

Sample answer: Use a ruler to measure the distance on the map. Set up a —————————————————————————————————————————

proportion to solve for the actual distance. Use the scale of the map (map —————————————————————————————————————————

distance : actual distance) to write one ratio. Use the ratio measured distance : —————————————————————————————————————————

unknown actual distance as the other. Solve.—————————————————————————————————————————

Method 1 Method 2

Write a proportion using the scale.

Write an equation using the number of kilometers per centimeters.

______________________ 2 cm

− 15 km

= 9 cm

− x km

______________________

a = 15

− 2 m

Solve the proportion. Solve the equation. Substitute 9 for m.

______________________x = 67.5 km ______________________a = 67.5 km

1 − 72

Check Check


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Tie It Together

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Complete the graphic organizer with a phrase or formula from the chapter.


Definition

corresponding angles congruent corresponding sides proportional

Definition

transformation in which a figure is reduced or enlarged

Definition

an object with lengths proportional to the object it represents

Equivalent Proportions

a −

b =

c −

d ,

b −

a =

d −

c ,

a −

c =

b −

d ,

c −

a =

d −

b

Ratio

Algebra

Similar Triangles

Cross Products

If a −

b =

c −

d ,

then ad = bc.

• Triangle Proportionality

• Triangle Midsegment

• Proportional Parts of Parallel Lines

• Congruent Parts of Parallel Lines

• Special Segments Proportionality

• Triangle Angle Bisector

DilationsScale

Drawings

TheoremsSimilar Polygons

Theorems

AASSSSAS

Properties

ReflexiveSymmetricTransitive


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Before the Test

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Study TipsStudy Tips • Complete reading assignments before class. Write down or circle any questions you may

have about what was in the text.

K

What I know…

W


L

What I learned…




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Before You Read

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Before you read the chapter, think about what you know about right triangles and trigonometry. List three things you already know about them in the first column. Then list three things you would like to learn about them in the second column.


Note Taking TipsNote Taking Tips • When searching for the main idea of a lesson, ask yourself, “What is this

paragraph or lesson telling me?”

• When you take notes, include definitions of new terms, explanations of new concepts, and examples of problems.

K

What I know…

W



Right Triangles and Trigonometry


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Key Points

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Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on geometric mean, one fact might be the geometric mean between two numbers is the positive square root of their product. After completing the chapter, you can use this table to review for your chapter test.


Lesson Fact

8-1 Geometric Mean See students’ work.

8-2 The Pythagorean Theorem and Its Converse

8-3 Special Right Triangles

8-4 Trigonometry

8-5 Angles of Elevation

8-6 The Law of Sines and Law of Cosines

8-7 Vectors


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Geometric Mean8-1

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Sample answer: how to find the geometric mean1. ____________________________________________________

of two numbers ____________________________________________________

____________________________________________________

Sample answer: how to solve problems involving 2. ____________________________________________________

right triangles using the geometric mean ____________________________________________________

____________________________________________________


an equation stating that two ratios are equal

the numbers a and d in the proportion a − b = c −

d

the numbers b and c in the proportion a − b = c −

d

New Vocabulary Complete each statement by filling in the blank with the correct term or phrase or writing the correct formula.

The geometric mean of two positive numbers a and b is the

number x such that .

The measure of the altitude drawn from the vertex of a right

triangle to its hypotenuse is the geometric mean between

the measures of the two segments of the ________________.

The geometric mean of two positive numbers a and b can be

calculated using the expression .

proportion

means

extremes

geometric mean

Active Vocabulary

hypotenuse

x = √ �� ab

a − x = x −

b


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Main Idea Details

Find the geometric mean between 8 and 18.

__________________ definition of geometric mean

__________________ Substitute for a and b.

__________________ Multiply.

__________________ Simplify.

Model Theorem 8.2, the Geometric Mean (Altitude) Theorem, by drawing a segment on right triangle DEF and writing a proportion. Sample answers are given.

Geometric Meanp. 531

Geometric Means in Right Trianglespp. 531–534

x = √ �� ab

x = √ �� 8·18

x = √ �� 144

x = 12

90°

Helping You Remember A good way to remember a new mathematical concept is to relate it to something you already know. How can the meaning of mean in a proportion help you to remember how to find the geometric mean between two numbers?

Sample answer: Write a proportion in which the two means are equal. This —————————————————————————————————————————

common mean is the geometric mean between the two extremes.—————————————————————————————————————————

—————————————————————————————————————————

_______________________FK = √ �� DK · KE


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The Pythagorean Theorem and Its Converse8-2

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What You’ll Learn Scan the text in Lesson 8-2. Write two facts you learned about the Pythagorean Theorem and its converse as you scanned the text.

Sample answer: In a right triangle, the sum of the 1. ____________________________________________________

squares of the lengths of the legs is equal to the ____________________________________________________

square of the length of the hypotenuse. ____________________________________________________

Sample answer: The converse of the Pythagorean 2. ____________________________________________________

Theorem can be used to help you determine ____________________________________________________

whether a triangle is a right triangle given the ____________________________________________________

measures of all three sides. ____________________________________________________


The geometric mean of two ________________ numbers a

and b can be calculated using the expression x = √ �� ab .

The measure of the ________________ drawn from the

vertex of the right triangle to its hypotenuse is the

geometric mean between the measures of the two

segments of the hypotenuse.


a set of three nonzero whole numbers a, b, and c,————————————————————————————

such that a2 + b2 = c2

————————————————————————————

————————————————————————————

geometric mean

Pythagorean triple

altitude

positive

Active Vocabulary


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Main Idea Details

Circle each set of numbers that is a Pythagorean triple.

1. 6, 8, 10 2. 7, 21, 25 3. 5, 12, 13

4. 24, 45, 51 5. 14, 48, 50 6. 10, 15, 17

7. 21, 72, 75 8. 16, 30, 36 9. 15, 36, 39

Fill in each blank to complete the converse of the Pythagorean Theorem.

The Pythagorean Theorempp. 541–544

Converse of the Pythagorean Theorempp. 544–545

Helping You Remember Many students who studied geometry long ago remember the Pythagorean Theorem as the equation a2 + b2 = c2, but cannot tell you what this equation means. A formula is useless if you don’t know what it means and how to use it. How could you help someone who has forgotten the Pythagorean Theorem remember the meaning of the equation a2 + b2 = c2?

Sample answer: Draw a right triangle. Label the lengths of the two legs as a —————————————————————————————————————————

and b and the length of the hypotenuse as c.—————————————————————————————————————————

a c

b

Converse of the Pythagorean TheoremWords If the sum of the ____________ of the lengths of

the shortest sides of a triangle is equal to the

square of the length of the ____________ side,

then the triangle is a right triangle.

Symbols If a2 + b2 = c2, then _________________________.

squares

�ABC is a right triangle

longest


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Special Right Triangles8-3

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Sample answer: how to use the properties of 1. ____________________________________________________

45º-45º-90º triangles ____________________________________________________


30º-60º-90º triangles ____________________________________________________

New Vocabulary Quadrilateral ABCD is a square with side lengths s. Fill in each blank to show what you know about the figure.

AB = ________

m∠ADC = ________

m∠ACD = ________

AC = ________

Triangle LMN is equilateral with side lengths 2s. Fill in each blank to show what you know about the figure.

ON = ________

m∠OLN = ________

m∠LNO = ________

LO = ________

s

2s

s

90°

45°

s √ � 2

s

30°

60°

√ � 3

Active Vocabulary

127-140_GEOSNC08_890858.indd 131127-140_GEOSNC08_890858.indd 131 6/7/08 6:40:26 AM6/7/08 6:40:26 AM

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Main Idea Details

Fill in each blank and label the figure to illustrate the 45º-45º-90º Triangle Theorem.

Solve for q and s in the figure below.

Properties of 45º-45º-90º Trianglespp. 552–553

Properties of 30º-60º-90º Trianglespp. 553–555

Helping You Remember Some students find it easier to remember mathematical concepts in terms of specific numbers rather than variables. How can you use specific numbers to help you remember the relationship between the lengths of the three sides in a 30º-60º-90º triangle?

Sample answer: Draw a 30º-60º-90º triangle. Label the length of the shorter leg —————————————————————————————————————————

as 1. Then the length of the hypotenuse is 2, and the length of the longer leg —————————————————————————————————————————

is √ � 3 . Just remember 1, 2, √ � 3 .—————————————————————————————————————————

—————————————————————————————————————————

—————————————————————————————————————————

45°

45°

�

�

�2

60°

15s

q

45º-45º-90º Triangle Theorem

In a 45º-45º-90º triangle, the legs ℓ

are ______________ and the length

of the hypotenuse h is ___________

times the length of a leg.

congruent

√ � 2

q = ____________7.5

7.5 √ � 3 s = ____________


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Trigonometry8-4

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What You’ll Learn Skim the Examples for Lesson 8-4. Predict two things you think you will learn about trigonometry.

Sample answer: how to find the sine, cosine, and 1. ____________________________________________________

tangent ratios for angles in a right triangle ____________________________________________________

____________________________________________________

Sample answer: how to use trigonometric ratios to 2. ____________________________________________________

find lengths ____________________________________________________

____________________________________________________


the ratio of the opposite leg to the hypotenuse of a right triangle

the measure of ∠A if sin A is known

the study of triangle measurement

the measure of ∠A if tan A is known

the ratio of the adjacent leg to the hypotenuse of a right triangle

the measure of ∠A if cos A is known

a ratio of the lengths of two sides of a right triangle

the ratio of the opposite leg to the adjacent leg of a right triangle

______________________

______________________

______________________

______________________

______________________

______________________

______________________

______________________

sine

inverse sine

trigonometry

inverse tangent

cosine

inverse cosine

trigonometric ratio

tangent

Active Vocabulary


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Main Idea Details

Complete the statements to show the trigonometric ratios for angles H and K.

sin H =

sin K =

cos H =

cos K =

tan H =

tan K =

Use a calculator to find m∠F to the nearest tenth.

Trigonometric Ratiospp. 562–564

Use Inverse Trigonometric Ratiospp. 564–566

Helping You Remember How can the co in cosine help you remember the relationship between the sines and the cosines of the two acute angles of a right triangle?

Sample answer: The co in cosine comes from complement, as in —————————————————————————————————————————

complementary angles. The cosine of an acute angle is equal to the sine of its —————————————————————————————————————————

complement.—————————————————————————————————————————

16

9 m∠F ≈_________34.2

k �

h

h − �

k − �

k − �

h − �

h − k

k − h


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Angles of Elevation and Depression8-5

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What You’ll Learn Skim the lesson. Write two things you already know about angles of elevation and depression.

Sample answer: An angle of elevation is the angle 1. ____________________________________________________

formed by a horizontal line and an observer’s line ____________________________________________________

of sight to an object above the horizontal line. ____________________________________________________

Sample answer: You can use inverse trigonometric2. ____________________________________________________

ratios to solve for angles of elevation and ____________________________________________________

depression if you know two of the side lengths. ____________________________________________________


the angle formed by a horizontal line and an ————————————————————————————

observer’s line of sight to an object above the ————————————————————————————

horizontal line————————————————————————————

the angle formed by a horizontal line and an ————————————————————————————

observer’s line of sight to an object below the ————————————————————————————

horizontal line————————————————————————————

Vocabulary Link What are some real-world examples of angles of elevation and depression?

Sample answer: When a plane takes off, its climb ————————————————————————————

forms an angle of elevation. When a plane lands, it ————————————————————————————

comes in at an angle of depression.————————————————————————————

angle of elevation

angle of depression

Active Vocabulary


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Main Idea Details

A plane takes off and climbs at an angle of elevation of 35º. How high above the ground is the plane after traveling 250 yards? Model the situation with a diagram and solve. Sample answer is given.

Solve for x in the figure below.

Angles of Elevation and Depressionpp. 574–575

Two Angles of Elevation or Depressionpp. 575–576

Helping You Remember A good way to remember something is to explain it to someone else. Suppose a classmate finds it difficult to distinguish between angles of elevation and angles of depression. What are some hints you can give her to get it right every time?

Sample answers: (1) The angle of depression and the angle of elevation are —————————————————————————————————————————

both measured between the horizontal and the line of sight. (2) The angle of —————————————————————————————————————————

depression is always congruent to the angle of elevation in the same diagram. —————————————————————————————————————————

(3) Associate the word elevation with the word up and the word depression —————————————————————————————————————————

with the word down.—————————————————————————————————————————

20°

25°

20 m

x

35°

250 ydh

sin 35° = h −

250

250(sin 35°) = h

143.4 ≈ h

about 143.4 yards

x ≈ __________22.9 m


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The Law of Sines and Law of Cosines8-6

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Sample answer: Law of Sines1. ____________________________________________________

Sample answer: Law of Cosines2. ____________________________________________________


the angle formed by a horizontal line and an observer’s line of sight to an object above the horizontal line

the ratio of the opposite leg to the adjacent leg of a right triangle

the angle formed by a horizontal line and an observer’s line of sight to an object above the horizontal line

the ratio of the adjacent leg to the hypotenuse of a right triangle


You can use the Law of Sines to solve a ________________ if

you know the measures of ________________ and

________________ (AAS). If given ASA, use the Triangle

Angle Sum Theorem to find the measure of the

________________.

You can use the Law of Cosines to solve a triangle if you

know the measures of two ________________ and the

__________________ (SAS).

tangent

angle of depression

cosine

angle of elevation

Law of Sines

Law of Cosines

Active Vocabulary

triangle

any side

two angles

third angle

sides

included angle


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Main Idea Details

Complete the statement below to illustrate the Law of Sines for �ABC.

Use the Law of Cosines to solve for x to the nearest tenth.

Law of Sinespp. 582–583

Law of Cosinespp. 583–585

Helping You Remember Many students remember mathematical equations and formulas better if they can state them in words. State the Law of Sines in your own words without using variables or mathematical symbols.

Sample answer: In any triangle, the ratio of the sine of an angle to the length of —————————————————————————————————————————

the opposite side is the same for all three angles.—————————————————————————————————————————

—————————————————————————————————————————

—————————————————————————————————————————

Law of Sines

If �ABC has lengths a, b, and c, representing the length of the sides opposite the angles with measures A, B, and C, then

sin A

− a =

sin B −

b =

sin C −

c

c b

a

18

15x

12

_____________x ≈ 41.4°

.


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Vectors8-7

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What You’ll Learn

______________________

______________________

______________________

______________________

______________________

______________________

______________________

______________________

Skim the Examples for Lesson 8-7. Predict two things you think you will learn about vectors.

Sample answer: how to write vectors in 1. ____________________________________________________

component form ____________________________________________________

Sample answer: how to find the direction and 2. ____________________________________________________

magnitude of a vector ____________________________________________________


a description of a vector in terms of its horizontal change x and vertical change y from its initial point to its terminal point

the angle that is formed with the x-axis or any other horizontal line

the length of the vector from its initial point to its terminal point

a way to find the sum of two vectors using a parallelogram

the sum of two vectors

a vector that is placed in the coordinate plane with its initial point at the origin

a way to find the sum of two vectors using a triangle

a quantity that has both magnitude and direction

component form

direction

magnitude

parallelogram method

resultant

standard position

triangle method

vector

Active Vocabulary


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Main Idea Details

Write the component form of ⎯⎯⎯ � ST .

Model the parallelogram method and the triangle method below by drawing two vectors and showing the resultant. Sample answers are shown.

Helping You Remember A good way to remember a new mathematical term is to relate it to a term you already know. You learned about scale factors when you studied similarity and dilations. How is the idea of a scalar related to scale factors?

Sample answer: A scalar is the term used for a constant (a specific real —————————————————————————————————————————

number) when working with vectors. A vector has both magnitude and —————————————————————————————————————————

direction, while a scalar is just a magnitude. Multiplying a vector by a positive —————————————————————————————————————————

scalar changes the magnitude of the vector, but not the direction, so it —————————————————————————————————————————

represents a change in scale.—————————————————————————————————————————

Describe Vectorspp. 593–594

Vector Additionpp. 594–596

Parallelogram Method Triangle Method

y

x

⟨7, 6⟩


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Tie It Together

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Right Triangles

Notation Diagram/Examples

Similar Right Triangles

Geometric Mean a −

x = x −

b

3 −

x = x −

15 ; x2 = 45; x = 3 √ � 5

Right Triangle Geometric Mean Theorem

x − h =

h −

y ;

c −

b =

b −

x

PythagorasPythagorean Theorem a2 + b2 = c2

Pythagorean InequalityTheorems

c2 < a2 + b2: acute

c2 > a2 + b2: obtuse

Special Right Triangles45°-45°-90° ℓ = ℓ, h = ℓ √ � 2

30°-60°-90° h = 2s and ℓ = s √ � 3

TrigonometryRatios sin A =

opp −

hyp

cos A = adj

− hyp

tan A = opp

− adj

Law of Sines

Law of Cosines

sin A

− a =

sin B −

b =

sin C −

c

a2 = b2 + c2 - 2bc cos A

a b

cy x

h

a

b

c

a

c

b

45°

45°

�

�

� 2

60°

30°

s2s

s 3

adj hyp

opp

b c

a


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Before the Test

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Study TipsStudy Tips • While note-taking use abbreviations to use less time and room. Write neatly and place a

question mark by any information that you do not understand.

K

What I know…

W


L

What I learned…




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Before You Read

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Note Taking TipsNote Taking Tips • At the end of each lesson, write a summary of the lesson, or write in your own

words what the lesson was about.

• As you read each lesson, list examples of ways the new knowledge has been or will be in your daily life.

Transformations and Symmetry

Before You Read Transformations and Symmetry


• The orientation of an image that undergoes a reflection stays the same.

• The image after a 180º rotation is equal to the original image.

• An image after 2 reflections in parallel lines is equal to a translation.

• Vectors can be used to define translations.

• Dilations can have positive or negative scale factors.


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Key Points

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Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on symmetry, one fact might be the number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. After completing the chapter, you can use this table to review for your chapter test.


Lesson Fact

9-1 Reflections See students’ work.

9-2 Translations

9-3 Rotations

9-4 Compositions of Transformations

9-5 Symmetry

9-6 Dilations


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Reflections9-1

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What You’ll Learn Scan the text in Lesson 9-1. Write two facts you learned about reflections as you scanned the text.

Sample answer: To reflect a polygon in a line, 1. ____________________________________________________

reflect each of the polygon’s vertices. Then ____________________________________________________

connect the reflected vertices with line segments. ____________________________________________________

Sample answer: To reflect a point in the x-axis, 2. ____________________________________________________

multiply its y-coordinate by –1. ____________________________________________________

Review Vocabulary Write the definition next to the term. (Lesson 7-2)

polygons that have the same shape but not ————————————————————————————

necessarily the same size————————————————————————————

New Vocabulary Label the diagram with the correct term.

Vocabulary Link When you look into a mirror, you see your reflection. Explain how this relates to reflections in geometry.

Sample answer: Each point of your image in the ————————————————————————————

mirror appears to be the same distance from each ————————————————————————————

point on the preimage.————————————————————————————

Active Vocabulary

line of reflection

similar polygons


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Main Idea Details

Reflect �ABC in the line shown.

Describe the similarities and differences between reflecting a point in the x-axis and reflecting a point in the y-axis.

Similarities Sample answer: Both methods result in a——————————————————————

point being equidistant from one of the axes on a ————————————————————————————

coordinate grid.————————————————————————————

————————————————————————————

Differences Sample answer: To reflect a point in the——————————————————————

x-axis, you multiply its y-coordinate by -1. To ————————————————————————————

reflect a point in the y-axis, you multiply its ————————————————————————————

x-coordinate by -1.————————————————————————————

Draw Reflectionspp. 615–616

Draw Reflections in the Coordinate Planepp. 616–618

'

'

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Helping You Remember Sometimes it is helpful to put a geometric concept into your own words to help you remember it. Explain in your own words how you can reflect a point (x, y) in the line y = x.

Sample answer: To reflect a point (x, y) in the line y = x, interchange the —————————————————————————————————————

coordinates. The reflected image is the point (y, x). —————————————————————————————————————


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Translations9-2

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What You’ll Learn Skim the Examples for Lesson 9-2. Predict two things you think you will learn about translations.

Sample answer: how to draw a translation given 1. ____________________________________________________

a figure and a translation vector ____________________________________________________

____________________________________________________

Sample answer: how to perform translations in the 2. ____________________________________________________

coordinate plane ____________________________________________________

____________________________________________________


a quantity that has both magnitude and direction

the length of the vector from its initial point to its terminal point

a description of a vector in terms of its horizontal change x and vertical change y from its initial point to its terminal point

a vector that is placed in the coordinate plane with its initial point at the origin


A translation maps each point to its image along a , called the translation vector, such that

• each joining a point and its image has the same as the vector, and

• this segment is also to the vector.

vector

magnitude

component form

standard position

translation vector

parallel

vector

segment

length

Active Vocabulary


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Main Idea Details

Draw the translation of �RST along the translation vector.

Complete the following table to demonstrate how to translate a point in the coordinate plane.

WordsTo translate a point along vector (a, b), add a to the x-coordinate and b to the

y-coordinate.

Symbols (x, y) → (x + a, y + b)

Example The image of point R(2, 9) translated along

vector (1, -4) is (3, 5).

Draw Reflectionspp. 615–616

Draw Reflections in the Coordinate Planepp. 616–618

'

'

'

v⇀

Helping You Remember A good way to remember a new mathematical term is to relate it to an everyday meaning of the same word. How is the meaning of translation in geometry related to the idea of translation from one language to another?

Sample answer: When you translate from one language to another, you carry —————————————————————————————————————————

over the meaning from one language to another. When you translate —————————————————————————————————————————

a geometric figure, you carry over the figure from one position to another —————————————————————————————————————————

without changing its basic orientation.—————————————————————————————————————————

.


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Rotations9-3

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Sample answer: what it means to rotate a point 1. ____________________________________________________

about a fixed point ____________________________________________________

____________________________________________________

Sample answer: how to draw rotations clockwise 2. ____________________________________________________

and counterclockwise in the coordinate plane ____________________________________________________

____________________________________________________


A rotation about a , called the center of rotation, through an angle of x˚ maps a point to its image such that

• if the point is the center of rotation, then the image and preimage are the , or

• if the point is not the center of rotation than the image and preimage are the same from the center of rotation. The measure of the angle of rotation formed by the preimage, center of rotation, and image points is x.

Vocabulary Link Describe some real-world examples of rotations that you have encountered.

Sample answers: tires on a car or a bike, Ferris ————————————————————————————

wheel, CD or DVD player————————————————————————————

————————————————————————————

————————————————————————————

center of rotation

angle of rotation

fixed point

same point

distance

Active Vocabulary


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Main Idea Details

Draw a 90˚ counterclockwise rotation of �ABC about point D.

Fill in the boxes to describe the rotation of �DEF about the origin. Include the angle of rotation and the direction.

Draw Rotationspp. 632–633

Draw Rotations in the Coordinate Planepp. 633–634

preimage

180˚ clockwise or 180˚ counter-clockwise



'

'

'

y

x

'

'

'

'

''

'

'

'

8642

−4−6−8

−2−4−6−8 2 4 6 8

Helping You Remember A good way to help you remember a concept in geometry is to explain it to someone else. Suppose a classmate is having difficulty remembering how to find the coordinates of a point after a 180˚ rotation about the origin. How would you explain it to him?

Sample answer: Multiply the x- and y-coordinates by –1.—————————————————————————————————————————

—————————————————————————————————————————


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Compositions of Transformations9-4

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Sample answer: Glide Reflections1. ____________________________________________________

____________________________________________________

Sample answer: Compositions of Two Reflections2. ____________________________________________________

____________________________________________________


a line such that the points of a preimage and its image are the same distance from the line

the object that determines the distance and direction that a preimage is slid to create its image

the point about which a preimage is turned to create its image

the amount a preimage is turned to create its image


the result when a transformation is applied to a figure and then another transformation is applied to its image

the composition of a translation followed by a reflection in a line parallel to the translation vector

center of rotation

line of reflection

angle of rotation

translation vector

Active Vocabulary

composition of transformations

glide reflection


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Main Idea Details

Describe the composition of transformations that lead to �A�B�C�.

State the type of transformation that is achieved by each of the following compositions.

Compositions of Transformations

Glide Refl ection Translation Rotations

the composition of a reflection

and a translation

the composition of two reflections in

parallel lines

the composition of two reflections

in intersecting lines

Glide Reflectionspp. 641–642

Compositions of Two Reflectionspp. 642–644

translated down 10 units, then

refl ected in the y-axis

y

x

''''

'

''

''

8642

−4−6−8

−2−4−6−8 2 4 6 8

Helping You Remember Theorem 9.1 in the student text states that the composition of two or more isometries is also an isometry. Describe what this means in your own words, and state a property of congruence that leads to this theorem.

Sample answer: An isometry is a transformation that preserves congruence. —————————————————————————————————————————

So, Theorem 9.1 states that the composition of two or more refl ections, —————————————————————————————————————————

translations, or rotations, results in an image that is congruent to its preimage. —————————————————————————————————————————

This is suggested by the Transitive Property of Congruence.—————————————————————————————————————————


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Symmetry9-5

Less

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9-5

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Sample answer: how to identify line and rotational 1. ____________________________________________________

symmetries in two-dimensional figures ____________________________________________________

Sample answer: how to identify line and rotational 2. ____________________________________________________

symmetries in three-dimensional figures ____________________________________________________


also called the axis of symmetry

also called the point of symmetry

the number of times a figure maps onto itself as it rotates from 0˚ to 360˚

the characteristic of a figure if there exists a rigid motion that maps the figure onto itself

the smallest angle through which a figure can be rotated so that it maps onto itself

the characteristic of a figure if the figure can be mapped onto itself by a reflection in a line

the characteristic of a figure if the figure can be mapped onto itself by a rotation between 0˚ and 360˚

Vocabulary Link Symmetry is a word that is used in everyday English. Find the definition of symmetry using a dictionary. Describe how the definition of symmetry can help you remember the mathematical definition of symmetry.

Sample answers: corresponding in shape, size, and ————————————————————————————relative position on opposite sides of a dividing line; ————————————————————————————A figure has symmetry when it can be mapped onto ————————————————————————————itself, which results in a figure with the same shape ————————————————————————————and size.————————————————————————————

Active Vocabulary

line of symmetry

center of symmetry

order of symmetry

symmetry

magnitude of symmetry

line symmetry

rotational symmetry


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Main Idea Details

Circle the figures that have rotational symmetry.

1. 2.

3. 4.

State whether the figure below has plane symmetry, axis symmetry, both, or neither.

Symmetry in Two-Dimensional Figurespp. 653–654

Symmetry in Three-Dimensional Figuresp. 655 Step 1: Can the figure be mapped onto

itself by a reflection in a plane? yes

Step 3: What kind of symmetry does the figure have? both

Step 2: Can the figure be mapped onto itself by a rotation between 0˚ and 360˚ in a line? yes

Helping You Remember What is an easy way to remember the order and magnitude of the rotational symmetry of a regular polygon?

Sample answer: The order is the same as the number of sides. To find the —————————————————————————————————————————

magnitude, divide 360˚ by the number of sides.—————————————————————————————————————————


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Dilations9-6

Less

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What You’ll Learn Skim the lesson. Write two things you already know about dilations.

Sample answer: A dilation is a similarity 1. ____________________________________________________

transformation that enlarges or reduces a figure ____________________________________________________

proportionally. ____________________________________________________

Sample answer: To find the coordinates of an 2. ____________________________________________________

image after a dilation centered at the origin, ____________________________________________________

multiply the coordinates by the scale factor of the ____________________________________________________

dilation. ____________________________________________________

New Vocabulary State whether the dilation from figure B to B′ is an enlargement or a reduction. Then find the scale factor of the dilation.

type of dilation: ________________

scale factor: ________________

Vocabulary Link Describe how the pupils of your eyes serve as a real-world example of dilation in different lighting conditions.

Sample answer: The pupil enlarges about the center ————————————————————————————

in low light to allow more light to enter. It reduces in ————————————————————————————

bright light to allow less light in.————————————————————————————

Active Vocabulary

enlargement

2.5

6

4

'


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Main Idea Details

Draw the image of �ABC under a dilation with center D and scale factor 2.5.

Model a reduction by giving the coordinates of segment R′S′ and plotting it on the coordinate grid. Tell what scale factor you used in your reduction. Sample answer: scale: 0.5

Draw Dilationspp. 660–662

Dilations in the Coordinate Planep. 662

' '

'

y

x'

'

89

765

1234

−1−2−3−4−5−6−7−8−9

Helping You Remember A good way to remember a new concept in geometry is to explain it to someone else in your own words. How would you describe the process of dilating a point (x, y) about the origin by a scale factor k?

Sample answer: Multiply the x-coordinate and y-coordinate of the point by k to —————————————————————————————————————————

produce the image (kx, ky).—————————————————————————————————————————

—————————————————————————————————————————


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Tie It Together

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Transformations

Reflection

Rotation

DilationTranslation

Composition

y-axis(x, y) → (-x, y)

x-axis(x, y) → (x, -y)

y = x(x, y) → (-y, x)

(x, y) → (x + a, y + b) (x, y) → (kx, ky)

90°(x, y) → (-y, x)

180°(x, y) → (-x, -y)

Glide Reflection 270°(x, y) → (y, -x)


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Before the Test

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Study TipsStudy Tips • On test day, look over the entire test to get an idea of its length and scope so that you

can pace yourself. Answer what you know first, skipping over material you do not know. When finished, go back and check for errors. Don’t change an answer unless you are certain you are correct.


Transformation and Symmetry After You Read

• The orientation of an image that undergoes a reflection stays the same.

D

• The image after a 180° rotation is equal to the original image.

D

• An image after 2 reflections in parallel lines is equal to a translation.

A

• Vectors can be used to define translations. A

• Dilations can have positive or negative scale factors. A


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Before You Read

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Before you read the chapter, think about what you know about circles. List three things you already know about them in the first column. Then list three things you would like to learn about them in the second column.


Note Taking TipsNote Taking Tips • When you take notes, record real-life examples of how you can use fractions,

decimals, and percents, such as telling time and making change.

• When you take notes, listen or read for main ideas. Then record those ideas for future reference.

K

What I know…

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Circles


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Key Points

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Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on measuring angles and arcs, one fact might be a central angle is an angle with a vertex in the center of the circle. After completing the chapter, you can use this table to review for your chapter test.

Circles

Lesson Fact

10-1 Circles and Circumference See students’ work.

10-2 Measuring Angles and Arcs

10-3 Arcs and Chords

10-4 Inscribed Angles

10-5 Tangents

10-6 Secants, Tangents, and Angle Measures

10-7 Special Segments in a Circle

10-8 Equations of a Circle


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Circles and Circumference10-1

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Sample answer: how to identify parts of a circle1. ____________________________________________________

——————————————————————————

Sample answer: how to find the circumference of2. ____________________________________________________

a circle ____________________________________________________


the distance around the circle

a segment with endpoints at the center and on the circle

the locus or set of all points in a plane equidistant from a given point

two coplanar circles that have the same center

a chord which passes though the center of a circle and is made up of collinear radii

an irrational number which by definition is the ratio of the circumference of a circle to the diameter of the circle

descriptor give to a polygon which is drawn inside a circle such that all of its vertices lie on the circle

the name used to describe the given point in the definition of a circle

a segment with endpoints on the circle

descriptor given to a circle which is drawn about a polygon such that the circle contains all of the vertices of the polygon

two circles with congruent radii

circumference

radius

circle

concentric circles

diameter

pi (π)

inscribed

center

chord

circumscribed

congruent circles

Active Vocabulary


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Main Idea Details

Compare and contrast the pairs of special segments of a circle in the diagram below.

The circumference of a circle is 234 inches. Determine the radius and the diameter of the circle to the nearest hundredth inch.

A diameter is a chord that must go through the center of the circle.

The diameter of a circle is made up of two collinear radii, thus its measure is twice that of the radius. In addition, you can say that the radius is half of the diameter.

Start withC = πd, and fill

in what you know.

234 = πd

Determine the radius.

r = 1 − 2 (74.48)

r = 37.24 in.

Segments in a circlepp. 683–685

Circumferencepp. 685–686

A radius is not a chord because a radius does not have both endpoints on the circle.

Solve for the unknown.

234 = πdd ≈ 74.48 in.

Helping You Remember Look up the origins of the two parts of the word diameter in your dictionary. Explain the meaning of each part and give a term you already know that shares the origin of that part.

Sample answer: The first part comes from dia, which means across or through, —————————————————————————————————————————

as in diagonal. The second part comes from metron, which means measure.—————————————————————————————————————————

—————————————————————————————————————————

Chord

DiameterRadius


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Measuring Angles and Arcs10-2

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What You’ll Learn Scan the text in Lesson 10-2. Write two facts you learned about measuring angles and arcs as you scanned the text.

Sample answer: A circle has 360 degrees.1. ____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

Sample answer: An arc is a portion of a circle2. ____________________________________________________

defined by two endpoints. ____________________________________________________

____________________________________________________

____________________________________________________

New Vocabulary Label the diagram with the terms listed at the left.

central angle

arc

minor arc

major arc

semicircle

congruent arcs

adjacent arcs

Active Vocabulary

x

y

x

y

congruent arcs

major arc

central angle

minor arc

semicircle

adjacent arcs


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Main Idea Details

Given the m∠EJD = 15, find each measure in � J in the order specified. Justify your answer.

Find the length of � BC and � BDC . The radius of � A is 5 centimeters. What is the relationship between the two arc lengths?

Relationship: The sum of the arcs is equal to the

circumference of the circle.

Angles and Arcspp. 692–694

Arc Lengthp. 695

100°

m � BC = 8.73 cm

m � BDC = 22.69 cm

m � ED = 15

The measure of a

minor arc is equal

to the measure of

its central angle.

m � EDH = 105

The measure is

equal to the sum of

two adjacent arcs:

90 + 15.

m � GH = 75

Because EHG is a

semicircle, you

can do 180 - 105.

m � EFG = 180

It is a semicircle

because EG is a

diameter.


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Arcs and Chords10-3

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Sample answer: Arcs and Chords1. ____________________________________________________

——————————————————————————

Sample answer: Bisecting Arcs and Chords2. ____________________________________________________

——————————————————————————

Review Vocabulary In the circle provided, draw a chord which is not a diameter using a regular line, a diameter using a heavy line, and a radius using a dashed line. Identify 3 arcs that were created on the circle. (Lessons 10-1 and 10-2)Sample answers are given.

Details

Find the value of x.

What is PO? What is MN?

What realtionship exists between � PO and � MN ?

The arcs are congruent.————————————————————————————

Arcs

� FE

� DCB

� DC

Active Vocabulary

3x + 7 = 5x - 9

x = 8

31

Use the definition of congruence to write an equation. Solve the equation for x.

31

3x + 7

5x - 9

Arcs and Chordspp. 701–702

Main Idea


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Details

Use the diagram below to state Theorems 10.3 and 10.4, from the student book, in your own words.

Theorem 10.3 � AB is a diameter of the circle and � NP is a

chord, therefore, � AB is a perpendicular bisector of � NP .

Theorem 10.4 Because NO is congruent to OP, AB is

its perpendicular bisector, which means � AB is a

diameter of the circle.

Bisecting Arcs and Chordspp. 702–704

Helping You Remember Writing a mathematical concept in your own words can help you better remember the concept. Describe Theorem 10.5, from the student book, in your own words.

Sample answer: If two chords in a circle are congruent, and you draw a—————————————————————————————————————————

segment perpendicular to the 1st chord with the center of the circle as the—————————————————————————————————————————

other endpoint, it will be the same length as the segment perpendicular from—————————————————————————————————————————

the 2nd chord to the center.—————————————————————————————————————————

—————————————————————————————————————————

Main Idea


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Inscribed Angles10-4

Less

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What You’ll Learn Skim Lesson 10-4. Predict two things that you expect to learn based on the headings and figures.

Sample answer: three ways that an angle can1. ____________________________________________________

be inscribed in a circle ____________________________________________________

Sample answer: the definition of an inscribed2. ____________________________________________________

polygon ____________________________________________________

Review Vocabulary Complete the chart below. (Lesson 6-1)


an angle which has a vertex on a circle and sides that————————————————————————————

contain chords on the circle————————————————————————————

an arc which has endpoints on the sides of an inscribed————————————————————————————

angle and lies in the interior of an inscribed angle————————————————————————————

inscribed angle

intercepted arc

Active Vocabulary

Regular Polygons

5-sidespentagon

6-sideshexagon

sum of interior angles720°


measure of each angle

120°

4-sidesquadrilateral


108°



90°


60°


3-sidestriangle


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Main Idea Details

Find the indicated measure for each figure.1. 2.

Provide details for Theorem 10.9 in the student book by defining each boldface word.

Inscribed Anglespp. 709–711

Angles of Inscribed Polygonspp. 711–712

124°

22°

Helping You Remember Describe how you could make a sketch that would help you remember the relationship between the measure of an inscribed angle and the measure of its intercepted arc.

Sample answer: Draw a diameter of a circle to divide it into two semicircles. —————————————————————————————————————————

Inscribe an angle in one of the semicircles; this angle will intercept the other—————————————————————————————————————————

semicircle. From your sketch, you can see that the inscribed angle is a right—————————————————————————————————————————

angle. The measure of the semicircle arc is 180°, so the measure of the—————————————————————————————————————————

inscribed angle is half the measure of its intercepted arc.—————————————————————————————————————————

m � RS = 44m∠BAC = 62

a polygon with 4 sides

a polygon which is drawn inside a circle such that all of its vertices lie on the circle

two angles whose sum is 180°

any two angles in a quadrilateral that are not consecutive

If a quadrilateral is inscribed in a circle, then its opposite angles

are supplementary.


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Tangents10-5

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What You’ll Learn Skim the Examples for Lesson 10-5. Predict two things you think you will learn about tangents.

Sample answer: how to identify common tangents1. ____________________________________________________

____________________________________________________

Sample answer: how to use tangents to find2. ____________________________________________________

measures ____________________________________________________

Review Vocabulary Use the Pythagorean Theorem to determine if the triangles with the specified sides are right triangles. (Lesson 8-2)


a in the same plane as a

that the circle in

exactly point

the point where a line intersects a

Vocabulary Link Use a dictionary to look up the origin of the word tangent. How can the origin of this word help you remember the definition of a tangent line?

Sample answer: The word tangent comes from the————————————————————————————

Latin word tangere, which means “to touch.” A line————————————————————————————

that is tangent to a circle just “touches” the circle in————————————————————————————

one point.————————————————————————————

tangent

point of tangency

a = 9, b = 40, c = 41

yes

a = 4, b = 5, c = 41

no

a = 6, b = 8, c = 10

yes

Active Vocabulary

line

circle intersects

one

tangent

circle


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Main Idea Details

Find the value of x. Assume that segments that appear to be tangent are tangent. Show your work in each box as indicated.

Fill in the missing measurements in the diagram below in the order specified. Justify each measurement.

Identify the length of each side of the right triangle.

Leg 1 = Leg 2 =

Hyp =

Use the Pythagorean Theorem to write an equation, then solve for x.

x2 + 122 = (x + 8)2

x2 + 144 = x2 + 16x + 64

80 = 16x

x = 5

x

x + 8

12

Tangentspp. 718–720

Circumscribed Polygonsp. 721

128

xx

1. The measure of this segment is 12 because of Theorem 10.11.



3. The measure of this segment is 7 because of segment addition.

11

12

4


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Secants, Tangents, and Angle Measures10-6

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What You’ll Learn Skim the lesson. Write two things you already know about secants, tangents, and angle measures.

Sample answer: The measure of an inscribed1. ____________________________________________________

angle is half the measure of its intercepted arc. ____________________________________________________

Sample answer: A tangent line intersects a circle2. ____________________________________________________

at one point. ____________________________________________________

Review Vocabulary Determine the measure of each angle in the diagram. Justify your answers. (Lesson 1-5)


a line which intersects a circle in exactly two points————————————————————————————

————————————————————————————

Details

Find the value of x. Show your work in each box as indicated.

secant

Intersections On or Inside a Circlepp. 727–729

Angle 1 is 55° because it is supplementary to the 125° angle.

Angle 2 is 125° because it is a vertical angle with the given angle.

Angle 3 is 55° because it is a vertical angle with angle 1.

Active Vocabulary

12

3125°

Main Idea

Determine the measure of ∠CAF.136° 62°

x°

Determine the measure of ∠EAF.

m∠EAF = 1 − 2 ( 62 + 136 ) = 99

81


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Label each diagram and explain how you would determine the measure of ∠1 in each diagram.

Intersections Outside a Circlepp. 729–731

intercepted arc tangent

secant1

The measure of ∠1 is one half the measure of the intercepted arc.

intercepted arc secant

intercepted arc secant

The measure of ∠1 is one half the measure of the difference of the intercepted arc.

1

Main Idea

Helping You Remember Some students have trouble remembering the difference between a secant and a tangent. What is an easy way to remember which is which?

Sample answer: A secant cuts a circle, while a tangent just touches it at one—————————————————————————————————————————

point. You can associate tangent with touches because they both start with t. —————————————————————————————————————————

Then associate secant with cuts.—————————————————————————————————————————


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Special Segments in a Circle10-7

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Sample answer: how to fi nd the measure of1. ____________________________________________________

segments that intersect inside a circle ____________________________________________________

____________________________________________________

Sample answer: how to fi nd the measure of2. ____________________________________________________

segments that intersect outside a circle ____________________________________________________

____________________________________________________

____________________________________________________

Review Vocabulary Write the definition next to each term. (Lessons 10-1, 10-5, and 10-6)

a segment with endpoints on the circle————————————————————————————

a line which intersects a circle in exactly two points————————————————————————————

the point where a line intersects a circle————————————————————————————

New Vocabulary Match each term with its definition by drawing a line to connect the two.

a segment formed when two chords intersect inside a circle

a segment of a secant line that has exactly one endpoint on the circle

a segment of a tangent line that has exactly one endpoint on the circle

a segment of a secant line that has an endpoint which lies in the exterior of the circle

chord

secant

tangent

tangent segment

secant segment

external secant segment

chord segment

Active Vocabulary


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Main Idea Details

Use Theorem 10.15 from the student book to find x.

Compare and contrast Theorem 10.16 with Theorem 10.15 from the student book.

How are they the same?Both define a relationship between the segments created when secants/chords intersect. Both relationships equate the products of segments.

How are they different?10.16 equates the products of the two pieces of the divided chords. 10.16 equates the products of the whole segment with the part of the segment outside the circle.

Segments Intersecting Inside a Circlepp. 736–737

Segments Intersecting Outside a Circlepp. 738–739

RP = 12, PS = 3x + 5, QP = 15, PT = 24

RP � PS = QP � PT

� �=

=

=

12 3x + 5 15 24

36036x + 60

x8 1 −

3


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Equations of Circles10-8

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What You’ll Learn Skim the lesson. Write two things you already know about equations of circles.

Sample answer: The Pythagorean Theorem is 1. ____________________________________________________

a2 + b2 = c2. ____________________________________________________

Sample answer: The Distance Formula is2. ____________________________________________________

√ �� (x1 - x

2)2 + (y

1 - y

2)2 . ____________________________________________________

Review Vocabulary Use the Distance Formula to find the distance between the given pairs of points on the coordinate plane. (Lesson 1-5)

New Vocabulary Fill in the blank with the correct term or phrase.

a locus that satisfies more than one set of conditions————————————————————————————

————————————————————————————

Vocabulary Link Previously, you wrote the equation of a line when given the graph of the line. Compare and contrast this process with writing the equation of a circle given the graph of the circle.

Sample answer: Both a line and a circle have a————————————————————————————

general form. When writing equations of each, you————————————————————————————

must identify two important characteristics of the————————————————————————————

graph. In a line, you need the y-intercept and the ————————————————————————————

slope. In a circle, you need the center and the radius.————————————————————————————

compound locus

(1, 2) and (9, 11)

√ �� 145

(9, 11) and (-7, -7)

2 √ �� 145

(1, 2) and (-7, -7)

√ �� 145

Active Vocabulary


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Main Idea Details

Write the equation of a circle that has a center at (1, 3) and passes though (6, -3).

Graph the circle given by the equation (x + 4)2 + (y - 2)2 = 16.

Equation of a Circlepp. 744–745

Graph Circlespp. 745–746

y

x

Rewrite the equation in standard form.

(x - (-4))2 + (y - 2)2 = 16

Find the length of the radius using the center and the point of the circle.

√ �� (6 - 1) 2 + (-3 - 3) 2 = √ �� 5 2 + (-6) 2 = √ �� 61

Determine h, k, and r2.

h = 1 k = 3 r2 = 61

Write the equation of the circle.

(x - 1 )2 + (y - 3 )2 = 61

Use the center and radius to identify four points on the circle.

(-8, 2), (0, 2),

(-4, 6), (-4, -2)

Identify the center. (-4, 2)

Identify the radius. 4


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Tie It Together

PDF 2nd


Name each part of the circle shown below.

Circles

�

�

�

001-026_GEOSNEM_890858.indd Sec9:177001-026_GEOSNEM_890858.indd Sec9:177 6/17/08 8:53:49 PM6/17/08 8:53:49 PM

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Before the Test

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Study TipsStudy Tips • To prepare to take lecture notes, make a column to the left about 2 inches wide. Use

this column to write additional information from your text, place question marks, and to summarize information.

Circles

K

What I know…

W


L

What I learned…



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Before You Read

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Before you read the chapter, think about what you know about areas of parallelograms and triangles. List three things you already know about them in the first column. Then list three things you would like to learn about them in the second column.


Note Taking TipsNote Taking Tips • Write down questions that you have about what you are reading in the lesson. Then record the answer to each question as you study the lesson.

• A visual (graph, diagram, picture, chart) can present information in a concise, easy-to-study format.

Clearly label your visuals and write captions when needed.

K

What I know…

W



Areas of Parallelograms and Triangles


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Key Points

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Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on areas of circles and sectors, one fact might be a sector of a circle is a region of a circle bounded by a central angle and its intercepted arc. After completing the chapter, you can use this table to review for your chapter test.

Areas of Parallelograms and Triangles

Lesson Fact

11-1 Areas of Parallelograms and Triangles


11-2 Areas of Trapezoids, Rhombi, and Kites

11-3 Areas of Circles and Sectors

11-4 Areas of Regular Polygons and Composite Figures

11-5 Areas of Similar Figures


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Less

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Areas of Parallelograms and Triangles11-1

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What You’ll Learn Skim Lesson 11-1. Predict two things you expect to learn based on the headings and the Key Concept box.

Sample answer: how to use the formula for finding 1. ____________________________________________________

the area of a parallelogram ____________________________________________________


the area of a triangle ____________________________________________________

Review Vocabulary Define parallelogram in your own words. (Lesson 6-2)

Sample answer: a quadrilateral with both pairs of ————————————————————————————

opposite sides parallel————————————————————————————

New Vocabulary Match each term with its definition.

the perpendicular distance between any two parallel bases

any side of a triangle

any side of a parallelogram

the length of an altitude drawn to a given base

Vocabulary Link Formula is a word that is used in everyday English. This lesson introduces two new formulas. Find the definition of formula using a dictionary. Explain how its English definition can help you understand the meaning of formula in mathematics.

Sample answer: A formula is a fixed procedure for————————————————————————————

doing something. In mathematics, it is a rule for ————————————————————————————

finding a value (usually expressed with algebraic ————————————————————————————

symbols).————————————————————————————

base of a parallelogram

height of a parallelogram

base of a triangle

height of a triangle

Active Vocabulary

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Main Idea Details

Find the perimeter and area of the parallelogram below. Round to the nearest tenth if necessary.

The perimeter measures cm.

The height of the parallelogram measures cm.

The area of the parallelogram measures cm2.

Use the organizer below to determine how the area of a triangle is related to the area of a rectangle.

Areas of Parallelogramspp. 763–764

Areas of Trianglespp. 765–766

64

8

176

Now you have two ‘s.

How do these triangles relate to the rectangle?

Area of a rectangle?

Draw a diagonal.

Helping You Remember A good way to remember a new formula in geometry is to relate it to a formula you already know. How can you use the formula for the area of a rectangle to help you remember the formula for the area of a parallelogram?

Sample answer: Due to the Area Addition Postulate, a parallelogram can be —————————————————————————————————————————

made into a rectangle, making their area formulas exactly the same.—————————————————————————————————————————

22 cm

10 cm

6 cm

A = bh

Conclusion: The rectangle cut in half gives two triangles. Therefore, the area of a triangle is equal to one half the area of a rectangle.Area of Triangle = 1 −

2 bh


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Areas of Trapezoids, Rhombi, and Kites 11-2

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What You’ll Learn Scan the text in Lesson 11-2. Write two facts you learned about trapezoids, rhombi, and kites as you scanned the text.

Sample answer: the definition of a trapezoid 1. ____________________________________________________

____________________________________________________


the area of a trapezoid ____________________________________________________

Review Vocabulary Fill in each blank with the correct term or phrase. (Lessons 6-2, 6-6, and 11-1)

The is any side of a triangle.

A quadrilateral with both pairs of opposites sides is known as a(n) .

A(n) is a quadrilateral with exactly one pair of parallel sides.

New Vocabulary Define height of a trapezoid in your own words.

Sample answer: The height of a trapezoid is the ————————————————————————————

perpendicular distance between its two bases.————————————————————————————

Vocabulary Link Area is a word that is used in everyday English. Find the definition of area using a dictionary. Explain how its English definition can help you understand its meaning in mathematics.

Sample answer: Area is any particular extent of space ————————————————————————————

or surface. In mathematics, it is a quantitative ————————————————————————————

measure of a plane or curved surface.————————————————————————————

parallelogram

trapezoid

base of a triangle

Active Vocabulary

base of a triangle

parallelogram

trapezoid


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Main Idea Details

Derive the formula for the area of a trapezoid.

In the formula for finding the area of a rhombus or kite, A = 1 −

2 d1d2, what do d1 and d2 represent?

the lengths of the diagaonals————————————————————————————

Find the area of the kite.

The area of the kite is .

Areas of Trapezoidspp. 773–774

Areas of Rhombi and Kitespp. 775–776

90 in2

Helping You Remember A good way to remember a new geometric formula is to state it in words. Write a short sentence that tells how to find the area of a trapezoid in a way that is easy to remember.

Sample answer: To find the area of a trapezoid, add the bases, multiply their —————————————————————————————————————————

sum by its height, then divide that product in half.—————————————————————————————————————————

18 in.

10 in.

These two congruent trapezoids together make one parallelogram.

Area of parallelogram = (b1 + b2)h

Take a basic trapezoid and make a glide reflection of it.

h h

b1 b2

b2 b1

Therefore, the formula for the area of a trapezoid = 1 −

2 (b

1 + b

2)h.


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Areas of Circles and Sectors11-3

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What You’ll Learn Scan the text under the Now heading. List two things you will learn about areas of circles and sectors.

Sample answer: how to find the area of circles1. ____________________________________________________

____________________________________________________

Sample answer: how to find the area of sectors of 2. ____________________________________________________

circles ____________________________________________________

Review Vocabulary Write the correct term next to each definition. (Lessons 10-2, 11-1, and 11-2)

any side of a triangle

an angle with a vertex in the center of a circle and with sides that contain two radii of the circle

the length of an altitude drawn to a given base of a triangle

the distance between the two bases of a trapezoid

a portion of a circle defined by two endpoints

any side of a parallelogram

New Vocabulary Define the sector of a circle in your own words.

Sample answer: A sector of a circle is a region of ————————————————————————————

a circle bounded by a central angle and its ————————————————————————————

intercepted arc.————————————————————————————

base of a triangle

central angle


height of a trapezoid

arc


Active Vocabulary


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Main Idea

Areas of Circles pp. 782–783

Areas of Sectorspp. 783–784

Details

Find the area of each circle. Round to the nearest tenth.

1. 2.

Use the organizer below and fill in the parts of the proportion that can be used to find the area of a sector.

=

Use the completed proportion to find the area of the shaded sector. Round to the nearest tenth.

A ≈

254.3 cm2 153.9 in2

A

The ratio of the area A of a sector to the area of the whole circle, πr2, is equal to the ratio of the degree measure of the intercepted arc x to 360.

πr2 360

x

Helping You Remember A good way to remember something is to explain it to someone else. Suppose your classmate Joelle is having trouble remembering which formula is for circumference and which is for area. How can you help her?

Sample answer: Circumference is measured in linear units, while area is —————————————————————————————————————————

measured in square units, so the formula containing r2 must be one for area. —————————————————————————————————————————

7.7 square units

18 cm 7 in.

4 units

55°


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Areas of Regular Polygons and Composite Figures

11-4

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What You’ll Learn Skim the Examples for Lesson 11-4. Predict two things you think you will learn about areas of regular polygons and composite figures.

Sample answer: how to identify segments and 1. ____________________________________________________

angles in regular polygons ____________________________________________________

Sample answer: how to use the formula for the 2. ____________________________________________________

area of a regular polygon ____________________________________________________

Review Vocabulary Answer the questions about the elements of the diagram below. (Lesson 10-2)

1. Angle A is what type of angle?

2. The portion of the circle that is defined by endpoints B and C is known as a(n) .


the radius of the circle that is circumscribed about the regular polygon

the center of the circle in which the regular polygon is inscribed

has its vertex at the center of the polygon and its sides pass through consecutive vertices of the polygon

a segment drawn perpendicular to a side of a regular polygon

apothem

radius of a regular polygon

center of a regular polygon

central angle of a polygon

Active Vocabulary

central angle

arc


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Main Idea Details

Identify the parts of the regular polygon.

Using the formula A = 1 − 2 aP, where A is the area of a

regular polygon, a is the apothem, and P is the perimeter, find the area of the regular octagon below.

The area of the regular octagon is .

Areas of Regular Polygonspp. 791–793

Areas of Composite Figurespp. 793–794

Helping You Remember Rolando is having trouble remembering when to subtract an area when finding the area of a composite figure. How can you help him remember?

Sample answer: Subtraction is usually needed when part of the figure is —————————————————————————————————————————

shaded and part is not shaded.—————————————————————————————————————————

396 in2

11 in.9 in.

What is the center?V

Name a central angle. Sample answer: ∠QVR

Name a radius.Sample answer: −−

VR

Name an apothem.Sample answer:−−

VW


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Areas of Similar Figures11-5

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Sample answer: Areas of Similar Figures1. ____________________________________________________

____________________________________________________

Sample answer: Scale Factors in Similar Figures2. ____________________________________________________

____________________________________________________

Review Vocabulary Define similar polygons in your own words. (Lesson 7-2)

Sample answer: Two polygons are similar if all their ————————————————————————————

corresponding angles are congruent, and all of their ————————————————————————————

corresponding sides are proportional.————————————————————————————Fill in each blank with the correct term or phrase. (Lessons 10-1 and 11-3)

A slice of pie would be an example of a(n) .

A figure or shape enclosed by another geometric shape or figure is .

A(n) is the part of the interior of a circle bound by a chord and an arc.

Vocabulary Link Similar is a word that is used in everyday English. Find the definition of similar using a dictionary. Explain how its English definition can help you understand its meaning in mathematics.

Sample answer: Similar means to have a likeness or ————————————————————————————

a resemblance. In mathematics, it has to do with the ————————————————————————————

specifi c likeness of fi gures or polygons.————————————————————————————

segment of a circle

sector of a circle

inscribed

sector of a circle

inscribed

segment of a circle

Active Vocabulary


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Main Idea Details

For the pair of similar triangles below, find the area of �DEF.

Complete the missing parts of the proportion below.

Areas of Similar Figures pp. 802–803

Scale Factors and Missing Measures in Similar Figurespp. 803–804

If you know the areas of two similar geometric shapes, you can use them to find the scale factor between them.

Area of WXYZ = 45 cm2

Area of ABCD = 20 cm2

Use k as the scale factor.

area of ABCD −− area of WXYZ

= k2

= k2 4 − 9 = k

2

Helping You Remember Anthony thinks that similar figures are equal figures. How can you explain to him that similar and equal are not the same?

Sample answer: Tell him that similar figures will have the same shape, but they —————————————————————————————————————————

will not be the same size. Equal figures, otherwise known as congruent figures, —————————————————————————————————————————

have both the same shape and the same size.—————————————————————————————————————————

9 yd

6 yd

A = 3 yd2

A = 6.75 yd2

D

FA

C B

E

The scale factor,

k = . 2 − 3 20

− 45

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Tie It Together

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Complete the graphic organizer with a formula from the chapter and identify the variables in the diagram.

Areas of Polygons and Circles

Figure Formula Diagram

Parallelogram A = bh

Triangle A = 1 − 2 bh

Trapezoid A = 1 − 2 h(b

1 + b

2)

RhombusKite

A = 1 − 2 d

1d

2

Circle A = πr2

Sector A = x −

360 πr2

Regular Polygon A = 1 − 2 aP

b

h

b

h

b2

b1

h

d2d1

r

x°

r

a


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Before the Test

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Study TipsStudy Tips • If possible, rewrite your notes. Not only can you make them clearer and neater,

rewriting them will help you remember the information.

Areas of Polygons and Circles

K

What I know…

W


L

What I learned…



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Before You Read

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Before you read the chapter, think about what you know about surface area and volume. List three things you already know about extending surface area and volume in the first column. Then list three things you would like to learn about in the second column.


Note Taking TipsNote Taking Tips • When you take notes, listen or read for main ideas. Then record those ideas for future reference.

• To help you organize data, create study cards when taking notes, recording and defining vocabulary words, and explaining concepts.

K

What I know…

W



Extending Surface Area and Volume


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Key Points

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Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on volumes of pyramids and cones, one fact might be pyramids use 1 −

3 of the area of base in the formula for volume. After completing the chapter, you can

use this table to review for your chapter test.


Lesson Fact

12-1 Representations of Three-Dimensional Figures


12-2 Surface Areas of Prisms and Cylinders

12-3 Surface Areas of Pyramids and Cones

12-4 Volumes of Prisms and Cylinders

12-5 Volumes of Pyramids and Cones

12-6 Surface Areas and Volumes of Spheres

12-7 Spherical Geometry

12-8 Congruent and Similar Solids


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Representations of Three-Dimensional Figures12-1

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What You’ll Learn Skim the Examples for Lesson 12-1. Predict two things you think you will learn about representations of three-dimensional figures.

Sample answer: how to sketch a solid when given 1. ____________________________________________________

its dimensions ____________________________________________________

Sample answer: how to identify cross sections of 2. ____________________________________________________

various solids ____________________________________________________

Review Vocabulary Define arc of a circle in your own words. (Lesson 11-3)

Sample answer: a portion of a circle defi ned by two ————————————————————————————

endpoints————————————————————————————

New Vocabulary Write the correct term next to each definition.the intersection of a solid and a plane

a corner view of a three-dimensional geometric solid on two-dimensional paper

Vocabulary Link Cross section is an expression that is used in everyday English. Either find the definition of cross section using a dictionary or think of how you have heard it used outside of geometry. Explain any similarities between its everyday use and its use in mathematics.

Sample answer: Cross section is often used to refer ————————————————————————————

to a sample section or selection of a certain ————————————————————————————

population during a study of characteristics or ————————————————————————————

opinions of people. In mathematics, it refers to a ————————————————————————————

specifi c view or section when studying the ————————————————————————————

characteristics of solid fi gures.————————————————————————————

Active Vocabulary

cross section

isometric view


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Main Idea Details

Complete the graphic organizer below.

Name the different cross sections of a cylinder. State how the plane would have to intersect the cylinder to get that particular cross section.

circle; the plane intersects the cylinder1. ____________________________________________

parallel to the circular base ____________________________________________

ellipse; the plane intersects the cylinder2. ____________________________________________

at an angle ____________________________________________

rectangle; the plane intersects the cylinder3. ____________________________________________

perpendicular to the circular base ____________________________________________

Draw Isometric Viewspp. 823–824

Investigate Cross Sectionsp. 824

two different ways to draw an isometric sketch of a solid.

Draw an isometric view from the dimensions.

Draw an isometric view from an orthographic drawing.

Helping You Remember Look up the word isometry in the dictionary. Compare its definition with the definition of corner view and perspective view. Why are corner views considered isometric views, but three-dimensional views not considered isometric views?

Sample answer: An isometry is a distance preserving transformation. Corner —————————————————————————————————————————

views represent the actual distances of fi gures, while three-point perspective —————————————————————————————————————————

views represent the apparent distances of fi gures.—————————————————————————————————————————


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Surface Areas of Prisms and Cylinders12-2

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Sample answer: Lateral Areas and Surface Areas 1. ____________________________________________________

of Prisms ____________________________________________________

Sample answer: Lateral Areas and Surface Areas 2. ____________________________________________________

of Cylinders ____________________________________________________

New Vocabulary Label the elements of this oblique pentagonal prism with the correct terms.

1. Arrow “A” is pointing to a(n) .

2. Arrow “B” is pointing to .

3. Arrow “C” is pointing to a(n) .

4. Arrow “D” is pointing to a(n) .

5. Arrow “E” is pointing to a(n) .

Fill in each blank with the correct term or phrase

A(n) of a cylinder is the segment with endpoints that are centers of the circular bases.

The of a prism is the sum of the areas of the lateral faces.

altitude

lateral edge

lateral face

base edge

bases

lateral area

axis

lateral edge

base edge

bases

altitude

lateral face

Active Vocabulary

axis

lateral area


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Main Idea Details

Complete the diagram to show the parts of the area formulas for prisms.

Given the base edges of the prism above are: 8, 6, 5, and 5 inches, and that the height of the prism is 10 inches, find its lateral area. 240 in2

The cylindrical canister below has a surface area of 33 in2. Use the formula S = 2πrh + 2πr2.

What is the approximate height of the canister? 2 in.

Lateral Areas and Surface Areas of Prismspp. 830–831

Lateral Areas and Surface Areas of Cylinderspp. 832–833

Helping You Remember A good way to remember a new mathematical term is to relate it to an everyday use of the same word. How can the way the word lateral is used in sports help you remember the meaning of the lateral area of a solid?

Sample answer: In football, a pass thrown to the side is called a lateral pass. In —————————————————————————————————————————

geometry, the lateral area is the “side area,” or the sum of the areas of all the —————————————————————————————————————————

side surfaces of the solid (the surfaces other than the bases).—————————————————————————————————————————

1.5 in.

h

The lateral areaL is L = Ph.

The surface areaS is S = L + 2B.

P = perimeter of the base

B = area of the base

h = height of the prism


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Surface Areas of Pyramids and Cones12-3

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What You’ll Learn Scan the text under the Now heading. List two things you will learn about surface areas of pyramids and cones.

Sample answer: how to fi nd lateral areas and 1. ____________________________________________________

surface areas of pyramids ____________________________________________________

____________________________________________________

Sample answer: how to fi nd lateral areas and 2. ____________________________________________________

surface areas of cones ____________________________________________________

____________________________________________________

Review Vocabulary Label the following solids as right or oblique. (Lesson 12-1)

1. 2. 3.

1.

2.

3.


the height of each lateral face of a regular pyramid

a figure where the axis of a cone is also its altitude

a figure with a base that is a regular polygon and the altitude has an endpoint at the center of the base

a figure where the axis of a cone is not its altitude

regular pyramid

slant height

right cone

oblique cone

right

right

oblique

Active Vocabulary


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Main Idea Details

Use the formulas below to find the lateral and the surface area of the regular pyramid pictured.

Lateral area, L = 1 − 2 P�, where P is the

perimeter of the base and � is the slant height.

L =

Surface area, S = 1 − 2 P� + B, where

P is the perimeter of the base, � is the slant height, and B is the area of the base.

S =

Follow the steps to find the surface area of a cone.

1. Use the Pythagorean Theorem to find the slant height of the cone. Round your answer to the nearest tenth. 21.2 cm

2. Use the formula above to find the cone’s surface area. Round to the nearest centimeter. 228 cm2

Lateral Area and Surface Area of Pyramidspp. 838–840

Lateral Area and Surface Area of Conespp. 840–842

99 cm2

121.4 cm2

cone heighth = 21 cm

radiusr = 3 cm

slant height� = ?

surface area

S = πr� + πr2

Helping You Remember One way to remember a new formula is to relate it to a formula you already know. Explain how the formulas for the lateral areas of a pyramid and cone are similar.

Sample answer: Both formulas say that the lateral area includes the distance —————————————————————————————————————————

around the base multiplied by the slant height.—————————————————————————————————————————

8 cm

3 cm

2.8 cm

� = 9 cm


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Volumes of Prisms and Cylinders12-4

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Sample answer: how to use the formula for fi nding 1. ____________________________________________________

the volume of a prism ____________________________________________________

____________________________________________________

Sample answer: how to apply Cavalieri’s Principle 2. ____________________________________________________

____________________________________________________

____________________________________________________

Review Vocabulary Fill in each blank with the correct term or phrase. (Lessons 12-1 through 12-3)

What shape describes the cross section of this sphere?

A(n) of a cylinder is the segment with endpoints that are centers of the circular bases.

When the axis of a cone is not its altitude this represents a(n) cone.

The of a prism is the sum of the areas of the lateral faces.

This type of paper is used to draw a(n) view of a solid.

isometric

circle

axis

lateral area

oblique

Active Vocabulary

circle

axis

oblique

lateral area

isometric


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Main Idea Details

Complete the steps to finding the volume of the prism below.

Use the formula V = πr2h to find the volume of the cylinder shown.

V = 502.4 cm3

Volume of Prismsp. 847

Volume of Cylinderspp. 848–849

Use the Theorem to find the area of the triangular base, B. Round to the nearest cubic inch. B = 50 in2

To find the volume, V of the prism, multiply the base, B by the of the prism.

Pythagorean

height, h

V =

Helping You Remember A good way to remember a mathematical concept is to explain it to someone else. Suppose that your younger sister, who is in eighth grade, is having trouble understanding why square units are used to measure area, but cubic units are needed to measure volume. How can you explain this to her in a way that will make it easy for her to understand and remember the correct units to use?

Sample answer: Area measures the amount of space inside a two-dimensional —————————————————————————————————————————

fi gure, while volume measures the amount of space inside a three-dimensional —————————————————————————————————————————

fi gure. A two-dimensional fi gure can be covered with small squares, which —————————————————————————————————————————

represent square units. A three-dimensional fi gure can be fi lled with small —————————————————————————————————————————

cubes, which represent cubic units.—————————————————————————————————————————

10 cm

4 cm

14.2 in.

10 in. 10 in.

16 in.

800 in3


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Volumes of Pyramids and Cones12-5

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What You’ll Learn Skim the lesson. Write two things you already know about volumes of pyramids and cones.

Sample answer: There are different types of 1. ____________________________________________________

pyramids based upon the shape of the base. ____________________________________________________

____________________________________________________

Sample answer: Some cones are straight up and2. ____________________________________________________

down (right cones), and some are slanted in one ____________________________________________________

direction or another (oblique cones). ____________________________________________________

Review Vocabulary Fill in each blank with the correct term or phrase. (Lessons 11-1 through 11-5)

A. B.

C. D.

Polygon A represents a(n) .

Polygon B represents a(n) .

The arrow in C is pointing to the .

Figure D represents an example of a(n) .

πr2 is the formula for the .

1 − 2 (b1 + b2)h is the formula for the .

area of a trapezoid


cubic unit

area of a circle

trapezoid

parallelogram

trapezoid

parallelogram


cubic unit

area of a circle

area of a trapezoid

Active Vocabulary


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Main Idea Details

Use the pyramid shown to answer the following questions.

1. What is the height of the pyramid? 9 ft

2. In the formula for finding the volume

of a pyramid, V = 1 − 3 Bh, what does

the B represent? the pyramid’s base

3. Given that B = 17.5 ft2, find the volume of this pyramid to the nearest tenth of a cubic foot. 52.5 ft3

Use the diagram of a cone to fill in the missing measurements in the chart. Round your answer to the nearest tenth, as needed.

Volume of Pyramidsp. 857

Volume of Conesp. 858

Helping You Remember Many students find it easier to remember mathematical formulas if they can put them in words. Use words to describe in one sentence how to find the volume of any pyramid or cone.

Sample answer: Multiply the area of the base by the height and divide that —————————————————————————————————————————

product by 3.—————————————————————————————————————————

—————————————————————————————————————————

—————————————————————————————————————————

Volume

V = 1 − 3 π r 2 h

h = 19 cm

r = 6.6 cm

V = 866.7 cm3

cone heighth = 19 cm

9 ft11 ft

radiusr = 6.6 cm


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Surface Areas and Volumes of Spheres12-6

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What You’ll Learn Scan the text in Lesson 12-6. Write two facts you learned about surface areas and volumes of spheres.

Sample answer: what it means to be a great circle1. ____________________________________________________

____________________________________________________

____________________________________________________

Sample answer: the formula for fi nding the 2. ____________________________________________________

volume of a sphere ____________________________________________________

____________________________________________________

New Vocabulary Label the parts of the sphere pictured below.

Fill in each blank with the correct term or phrase.

One of the two congruent halves of a sphere separated by the great circle is known as a(n) ___________________.

When a plane intersects a sphere to form a circle which contains the center of the sphere, this forms a(n) ___________________.

A(n) ___________________ is one of the endpoints of a diameter of a great circle.

great circle

pole

hemisphere

great circle

pole

Active Vocabulary

hemisphere

radius

tangent

diameter

chord


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Main Idea Details

Follow the steps of the flowchart to find the surface area of the hemisphere.

Use the formula V = 4 − 3 πr3 (the formula for finding the

volume of a sphere) to answer each question. Round your answers to the nearest tenth.

1. Find the volume of a sphere with a diameter of 1.6 feet. 2.1 ft3

2. Find the volume of a sphere with a great circle circumference of 19 inches. 115.8 in3

3. Find the volume of a hemisphere with a radius of 7 centimeters. 718.4 cm3

Surface Area of Spherespp. 864–865

Volume of Spherespp. 866–867

The hemisphere’s surface area is comprised of half of the surface area of the sphere, plus the area of the great circle.

Numerically, that is

S = 1 − 2 (4π r2) + π r2.

Using r = 5 cm, find the value, S, of the surface area of this hemisphere.S = 235.5 cm2

r = 5 cm

Helping You Remember Many students have trouble remembering all of the formulas they have learned in this chapter. What is an easy way to remember the formula for the surface area of a sphere?

Sample answer: A sphere does not have any lateral faces or bases, so the —————————————————————————————————————————

expression in the formula for its surface area has just one term, 4π r2, rather —————————————————————————————————————————

than being the sum of the expressions for the lateral area and the area of the —————————————————————————————————————————

bases like the others.—————————————————————————————————————————


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Spherical Geometry12-7

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Sample answer: Geometry on a Sphere1. ____________________________________________________

____________________________________________________

Sample answer: Comparing Euclidean and 2. ____________________________________________________

Spherical Geometries ____________________________________________________

Review Vocabulary Match each term with its definition. (Lessons 12-1 and 12-3)

the intersection of a solid and a plane

a figure with a base that is a regular polygon and the altitude has an endpoint at the center of the base

The axis of a cone is also its altitude.

a corner view of a three-dimensional geometric solid on two-dimensional paper

the height of each lateral face of a regular pyramid

The axis of a cone is not its altitude.


is geometry where a plane is the surface of a sphere.

A geometry where a plane is a flat surface made up of points that extend infinitely in all directions is known as

.

A type of geometry in which at least one of the postulates is from Euclidean geometry is known as

.

oblique cone

right cone

cross section

isometric view

regular pyramid

slant height

Active Vocabulary

Spherical geometry

Euclidean geometry

non-Euclidean geometry


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Main Idea Details

Use the diagram of the sphere below to answer each question.Sample answers:

1. Name two lines containing point P. � ⎯ � FC and � ⎯ � BE

2. Name a segment containing point Q.

−− BP

3. Name a triangle. �BAF

Complete the table below which compares different facts about Euclidean geometry and Spherical geometry.

Euclidean Spherical

A plane is a flat surface. A plane is a spherical surface.

The shortest distance between two points is a line segment.

The shortest distance between two points is a(n) arc of a great circle.

A line is infinite. A line is fi nite.

Through any two given points, there is exactly one line passing through them.

Through any two given points, there is one great circle passing through them.

A unit used to measure length is a(n) centimeter.

Degrees are the unit used to measure length.

Two points can be any distance apart. There is no greatest distance.

The greatest distance between two points is 180°.

Geometry on a Spherepp. 873–874

Compare Plane Euclidean and Spherical Geometriesp. 874

AB

C

DE

F

P

Q


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Congruent and Similar Solids12-8

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Sample answer: how to identify congruent or 1. ____________________________________________________

similar solids ____________________________________________________

____________________________________________________


similar solids ____________________________________________________

____________________________________________________

New Vocabulary Define the following terms in your own words.

Sample answer: solids that have exactly the same ————————————————————————————

shape, but not necessarily the same size————————————————————————————

Sample answer: solids that have exactly the same ————————————————————————————

shape and the same size————————————————————————————

In your own words, what is meant by scale factor?

Sample answer: Scale factor is the common ratio ————————————————————————————

between the corresponding linear measures of ————————————————————————————

similar solids.————————————————————————————

————————————————————————————

————————————————————————————

Active Vocabulary

similar solids

congruent solids


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Main Idea Details

Fill in the missing parts to the organizer below.

They have the same shape.

Ratios of corresponding linear measures are equal.

Apply what you learned with Theorem 12.1 in the Student Edition to answer the following:

Two similar pyramids have volumes of 125 cubic inches and 64 cubic inches. What is the ratio of the slant height of the large pyramid to the slant height of the small pyramid?

5:4

Identify Congruent or Similar Solidspp. 880–881

Properties of Congruent and Similar Solidspp. 881–882

How do you know if solids are similar?

How to Identify Congruent

Solids

Corresponding � are �.

Corresponding faces

are �.

Volumes are equal.Corresponding edges

are �.

Helping You Remember A good way to remember a new mathematical concept is to relate it to something you already know. How can what you know about the units used to measure lengths, areas, and volumes help you to remember the theorem about the ratios of surface areas and volumes of similar solids?

Sample answer: Lengths are measured in linear units, surface areas in square —————————————————————————————————————————

units, and volumes in cubic units. Take the scale factor, which is the ratio of—————————————————————————————————————————

linear measurements in the solids, and square it to get the ratio of their —————————————————————————————————————————

surface areas or cube it to get the ratio of their volumes.—————————————————————————————————————————


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Tie It Together

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Complete the graphic organizer with a formula from the chapter.


Shape Lateral Area Surface Area Volume

Prism L = Ph S = L + 2B V = Bh

Cylinder L = 2πrh S = L + 2πr2 V = Bh

Pyramid L = 1 − 2 Pℓ S = L + B V = 1 −

3 Bh

Cone L = πrℓ S = L + πr2 V = 1 − 3 πr2h

Sphere S = 4πr2 V = 4 − 3 πr3


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Before the Test

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Study TipsStudy Tips • Use flash cards to study for tests by writing the concept on one side of the card and its

definition on the other.

K

What I know…

W


L

What I learned…




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Before You Read

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Note Taking TipsNote Taking Tips • When taking notes, use a table to make comparisons about the new material.

Determine what will be compared, decide what standards will be used, and then use what is known to find similarities and differences.

• When taking notes on statistics, include your own statistical examples as you write down concepts and definitions.

This will help you to better understand statistics.

Probability and Measurement

Before You Read Probability and Measurement

See students’ work. • There are many different ways to

represent the sample space of an experiment.

• The Fundamental Counting Principle uses addition to count the total number of outcomes.

• In a permutation, the order is not important.

• Compound events can be independent or dependent.

• When one event affects another event, the events are mutually exclusive.


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Key Points

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Scan the pages in the chapter and write at least one specific fact concerning each lesson. For example, in the lesson on geometric probability, one fact might be probability that involves a geometric measure such as length or area is called geometric probability. After completing the chapter, you can use this table to review for your chapter test.


Lesson Fact

13-1 Representing Sample Spaces See students’ work.

13-2 Probabilities with Permutations and Combinations

13-3 Geometric Probability

13-4 Simulations

13-5 Probabilities of Independent and Dependent Events

13-6 Probabilities of Mutually Exclusive Events


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Representing Sample Spaces13-1

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Sample answer: how to use lists, tables, and tree1. ____________________________________________________

diagrams to represent sample spaces ____________________________________________________

Sample answer: how to use the Fundamental2. ____________________________________________________

Counting Principle to count outcomes ____________________________________________________


one way of representing a sample space, using line segments known as branches, to display possible outcomes

experiments with more than two stages

the set of all possible outcomes of an experiment

a means used to find the entire sample space from an experiment that does not require listing all the possible outcomes

an experiment with two stages or events

Vocabulary Link Experiment is a word that is used in everyday English. Find the definition of experiment using a dictionary. Explain how its English definition can help you understand the meaning of experiment in mathematics.

Sample answer: In everyday English, the most————————————————————————————

common definition of experiment is a test or trial. In————————————————————————————

mathematics, an experiment involves testing a————————————————————————————

specific situation when chance, likelihood, or odds————————————————————————————

are involved.————————————————————————————

sample space

tree diagram

two-stage experiment

multi-stage experiment

Fundamental Counting Principle

Active Vocabulary


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Main Idea Details

A number cube is rolled twice. Complete the sample space for this experiment.

How many possible outcomes are there? ____________

Sara takes orders at a restaurant where each sandwich is customized. Sara’s manager requires her to ask the following series of questions when placing the orders.

Use the Fundamental Counting Principle to find the number of different types of sandwiches that represent the sample space for chicken sandwich orders?

16 different types of sandwiches

Represent a Sample Spacepp. 899–900

Fundamental Counting Principlep. 901

36

Sandwich Shop1. Would you like your chicken fried

or grilled?2. Would you like bbq, honey mustard,

mayo, or no sauce?3. Would you like your sandwich

deluxe or regular?


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Probability with Permutations and Combinations

13-2

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Sample answer: Probability Using Permutations1. ____________________________________________________

____________________________________________________

Sample answer: Probability Using Combinations2. ____________________________________________________

____________________________________________________

Review Vocabulary Define each term in your own words. (Lesson 12-7)

a type of geometry in which at least one————————————————————————————

of the postulates from Euclidean geometry fails————————————————————————————

a geometry where a plane is a flat surface made up————————————————————————————

of points that extend infinitely in all directions————————————————————————————

a geometry where a plane is the surface of a sphere————————————————————————————

————————————————————————————


given a positive integer n, it is the product of the integers less than or equal to n

an arrangement of objects in which order is important

an arrangement of objects in which order is not important

an arrangement of objects in the form of a circle or loop

Non-Euclidean geometry

Euclidean geometry

Spherical geometry

________________________

________________________

________________________

________________________

Active Vocabulary

factorial

permutation

combination

circular permutation


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Main Idea Details

Follow and complete the steps in the flowchart below.A marching band is divided into squads of 8 musicians. Each squad is required to select leaders: a head squad leader and an assistant squad leader. Anders and Matthew are in one of the squads. If the positions are decided at random, what is the probability that Anders and Matthew are selected as leaders?

Leah is packing for a trip to Florida. She decides to pack 3 of her 8 travel games to take on the trip. If she chooses to select games at random, what is the probability that the games chosen are Leah’s 3 favorites?Use the formula for combinations, nCr = n! −

(n - r)!r! , when you

have n objects taken r at a time.

8C

3 = 8!

− (8 - 3)!3!

= 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 −−

(5!)3! =

8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 −−

(5 · 4 · 3 · 2 · 1)(3 · 2 · 1) = 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1

−− (5 · 4 · 3 · 2 · 1)(3 · 2 · 1)

= 56

There is only one favorable outcome that the games————————————————————————————

chosen are Leah’s favorites. So, the probability of————————————————————————————

these 3 games being chosen is 1 − 56

.————————————————————————————

Probability Using Permutationspp. 906–909

Probability Using Combinationspp. 909–910

Step 1 The number of possible outcomes in the number of permutations of 8 people taken 2 at a time, 8P2.

8P2 = 8! − (8 - 2)!

= 8 · 7 · 6! − 6!

= 56

Step 2 The number of favorable outcomes in the number of permutations of these 2 students in each of the 2-leader positions is 2!, which is 2 · 1 or 2.

Step 3 So, the probability of Anders and Matthew being

selected as the 2 squad leaders is 2 − 56

or 1 − 28

.


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Geometric Probability13-3

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What You’ll Learn


base of a triangle



Scan the text in Lesson 13-3. Write two facts you learned about geometric probability as you scanned the text.

Sample answer: There are many real-world1. ____________________________________________________

applications of probability. ____________________________________________________

____________________________________________________

Sample answer: Probabilities can be found with2. ____________________________________________________

length and with area. ____________________________________________________

____________________________________________________


The height of a triangle is the length of an altitude drawn to a given base.

Any side of a triangle is known as the base of a triangle .

Any side of a parallelogram is known as the base of a parallelogram .

The height of a parallelogram is the perpendicular distance between any two parallel bases.

New Vocabulary Define geometric probability in your own words.

Sample answer: Geometric probability is probability————————————————————————————

that involves a geometric measure such as length or————————————————————————————

area.————————————————————————————

————————————————————————————

————————————————————————————

Active Vocabulary


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Main Idea Details

The schedule for Meghan’s school is shown on the number line. The local fire department is going to randomly have a fire drill today. Find the probability the fire drill will be during Meghan’s lunch.

Darcy has a magnetic dartboard. What is the probability that Darcy’s magnetic darts will land in the central circle?

Sample answer: The probability is found by dividing————————————————————————————

the area of the center circle by the area of the entire————————————————————————————

dartboard: π · 52

− π · 92

= 25π

− 81π

≈ 31%.————————————————————————————

————————————————————————————

Probability with Lengthpp. 915–916

Probability with Areapp. 916–917

Find the total time of the school day. (Include time between classes.) 390 minutes

Find the total time of the lunch period. 35 minutes

probability of fire drill during lunch

Find the probability the drill will be during lunch.

35 minutes

− 390 minutes

≈ 9%

1st bell7:20-8:10

2nd bell8:15-9:05

Homeroom7:00-7:15

3rd bell9:10-10:00

4th bell10:05-10:55

lunch10:55-11:30

dismissal1:30

5th bell11:35-1:30

9 in.5 in.


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Simulations13-4

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Sample answer: how to design a simulation and1. ____________________________________________________

how to perform all the steps it involves ____________________________________________________

Sample answer: how to use different methods for2. ____________________________________________________

summarizing and displaying simulation data ____________________________________________________


the average value of a random variable that one anticipates after repeating an experiment or simulation a theoretically infinite number of times

a variable that can assume a set of values, each with fixed probabilities

a mathematical model used to match a random phenomenon

as the number of trials of a random process increases, the average will approach the expected value

the use of a probability model to recreate a situation again and again so that the likelihood of various outcomes can be estimated

Vocabulary Link Simulation is a word that is used in everyday English. Explain how the English definition can help you remember how simulation is used in mathematics.

Sample answer: something anticipated or in testing; ————————————————————————————

This is primarily the same as its mathematical ————————————————————————————

definition as it is applied in the study of probability.————————————————————————————

————————————————————————————

probability model

simulation

random variable

expected value

Law of Large Numbers

Active Vocabulary


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Main Idea Details

Miranda bowled a strike in 75% of her frames over the last 9 months. Design a simulation that can be used to estimate the probability that she will bowl a strike in the next frame.Complete the missing pieces to the steps below. Step 1 Possible Outcomes

Step 2 Assume that Miranda will have the opportunity to bowl 50 more frames.

Step 3 Divide the spinner provided into two sectors to represent the two possible outcomes listed in Step 1.

bowl a strike not bowl a strike 75%(360°) 25%(360°)

Step 4 A trial, one spin of a spinner, will represent bowling 1 frame.

What will a successful trial represent? strike

What will a failed trial represent? not a strike

The simulation will consist of how many spins? 50

Conduct the simulation above and record your results in the table provided below. When your simulation is complete, find the experimental probability of Miranda bowling a strike on her next frame.

Outcome Tally Frequency

strike

11

not a strike

39

Total 50

number of strikes −− number of non-strikes

=

Design a Simulationpp. 923–924

Summarize Data from a Simulationpp. 924–926

She bowls a strike. She does not bowl a strike.

theoretical probability: 75% theoretical probability: 25%

11 −

39


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Probabilities of Independent and Dependent Events

13-5

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What You’ll Learn Skim the Examples in Lesson 13-5. Predict two things you think you will learn about this lesson.

Sample answer: how to identify independent and1. ____________________________________________________

dependent events ____________________________________________________

Sample answer: how to find the probability of2. ____________________________________________________

independent events ____________________________________________________


a specific type of tree diagram that includes probabilities

events where the probability of one event occurring does not affect the probability of the other event occurring

consists of two or more simple events

a probability that uses the notation P(B�A)

events where the probability of one event occurring in some way changes the probability that the other event occurs

Vocabulary Link Independent and dependent are words used in everyday English. How do their English definitions apply to their definitions in probability?

Sample answer: Dependent means relying on————————————————————————————

something. In the case of probabilities, dependent————————————————————————————

means one event’s probability does rely on another————————————————————————————

event’s probability. Independent means not influenced————————————————————————————

by others. In the case of probabilities, independent————————————————————————————

events have no influence on each other’s probabilities.————————————————————————————

________________________

________________________

________________________

________________________

________________________

Active Vocabulary

probability tree

independent events

compound event

conditional probability

dependent events


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Main Idea Details

The probability that two independents occur follows the multiplication rule for probability: P(A and B) = P(A) · P(B). Extend the use of this multiplication rule for probability by creating a model of another set of independent events that can be used to demonstrate this rule. Sample answer:

Suppose a number cube is rolled and a coin is

tossed.

The sample space for this experiment is: {(1, H), (2, H),

(3, H), (4, H), (5, H), (6, H), (1, T), (2, T), (3, T), (4, T),

(5, T), (6, T)}.

Using the sample space, the probability of the

compound event of rolling a 6 and the coin landing on

heads is P(6 and H) = 1 −

12 .

The same probability can be found by multiplying the

probabilities of each simple event (hence, using the

multiplication rule for probability). P(6) = 1 − 6 ; P(H) = 1 −

2 ;

P(6 and H) = 1 − 6 · 1 −

2 or 1 −

12

Ms. Alexander’s class of 16 students sits at four different tables, with four students at each table. Ms. Alexander chose tables A and B to go to the lab, while those students at tables C and D remained seated. If Carly has to go to the lab, then what is the probability that she sits at Table A?

Sample answer: Because Carly is going to the lab, the————————————————————————————

sample space is reduced from the entire class, to just ————————————————————————————

the 8 students seated at tables A and B. So, the————————————————————————————

probability that Carly is at Table A is P(A| lab) = 4 −

8 or

1 −

2 .

———————————————————————————

Independent and Dependent Eventspp. 931–933

Conditional Probabilitiesp. 934


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Probabilities of Mutually Exclusive Events13-6

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What You’ll Learn Skim the lesson. Write two things you already know about mutually exclusive events.

Sample answer: Some events affect each other, 1. ____________________________________________________

while others do not affect each other. ____________________________________________________

Sample answer: The probability that an event will2. ____________________________________________________

not occur is called the complement. ____________________________________________________

Review Vocabulary Label the part of each diagram to which the arrow is pointing. (Lessons 11-1 and 11-2)


trapezoid

3. base of a parallelogram

4. height of a parallelogram

New Vocabulary Define the following terms in your own words.

Sample answer: two events that cannot happen at the————————————————————————————

same time and have no common outcomes————————————————————————————

Sample answer: the probability that an event will not————————————————————————————

occur; This is found by subtracting the probability————————————————————————————

that the event will occur from 1.————————————————————————————

————————————————————————————

trapezoid




mutually exclusive events

complement

Active Vocabulary


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Main Idea Details

Model the following mutually exclusive events by representing their relationship in a Venn diagram below.When a die is rolled, find the probability of rolling a 1 or 6.

Is the following event mutually exclusive or not mutually exclusive?

when a die is rolled, the probability of rolling a number less than 5 or an odd number not mutually exclusive

At the local carnival, the probability of a dart to land on one of the playing cards on the board is 33%.

Name the complement of landing on one of the cards and the probability of the complement occurring.

Sample answer: The complement is the dart not ————————————————————————————

landing on a playing card. The probability of this ————————————————————————————

happening is 67%.————————————————————————————

————————————————————————————

Mutually Exclusive Eventspp. 938–940

Probabilities of Complementspp. 941–942


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Tie It Together

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Complete each graphic organizer with a definition or formula from the chapter.

Sample Space

Experiment a situation involving chance that leads to results

Outcome the result of a single trial of an experiment

Event one or more outcomes of an experiment

Calculating Probability

Fundamental Counting Principle

The total possible outcomes in an experiment with k stages is n

1 • n

2 • n

3 • … • nk.

PermutationsnP

r = n!

− (n - r)!

CombinationsnC

r = n!

− (n - r)!r!

Circular Permutations n! −

n

Simulations the use of a probability model to recreate a situation and estimate the probability of various outcomes

Complement P(not A) = 1 - P(A)

Types of Probability

Geometric Probability probability that involves a geometric measure like length or area

Independent Events P(A and B) = P(A) • P(B)

Dependent Events P(A and B) = P(A) • P(B|A)

Conditional Probability P(B|A) = P(A and B)

− P(A)

Mutually Exclusive Events P(A and B) = P(A) + P(B)


001-026_GEOSNEM_890858.indd Sec12:227001-026_GEOSNEM_890858.indd Sec12:227 6/17/08 8:52:11 PM6/17/08 8:52:11 PM

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Before the Test

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Study TipsStudy Tips • You will do better on a test if you are relaxed. If you feel anxious, try some deep

breathing exercises. Don’t worry about how quickly others are finishing; do your best and use all the time that is available to you.


Probability and Measurement After You Read

• There are many different ways to represent the sample space of an experiment.

A

• The Fundamental Counting Principle uses addition to count the total number of outcomes.

D

• In a permutation, the order is not important. D

• Compound events can be independent or dependent. A

• When one event affects another event the events are mutually exclusive.

D


Documents

Teacher Annotated Edition - SecondaryMathsecondarymath.cmswiki.wikispaces.net/file/view/geosnbte.pdf7-2 Similar Polygons ... 13-2 Permutations and Combinations ... already know about