Chapter 6 Resource Masters - Math Problem Solving - Homejaeproblemsolving.weebly.com/uploads/5/1/9/6/... · ©Glencoe/McGraw-Hill iv Glencoe Geometry Teacher’s Guide to Using the

Chapter 6Resource Masters

Geometry

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3

ANSWERS FOR WORKBOOKS The answers for Chapter 6 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-860183-5 GeometryChapter 6 Resource Masters

1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03

© Glencoe/McGraw-Hill iii Glencoe Geometry

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Proof Builder . . . . . . . . . . . . . . . . . . . . . . ix

Lesson 6-1Study Guide and Intervention . . . . . . . . 295–296Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 297Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 298Reading to Learn Mathematics . . . . . . . . . . 299Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 300






Chapter 6 AssessmentChapter 6 Test, Form 1 . . . . . . . . . . . . 331–332Chapter 6 Test, Form 2A . . . . . . . . . . . 333–334Chapter 6 Test, Form 2B . . . . . . . . . . . 335–336Chapter 6 Test, Form 2C . . . . . . . . . . . 337–338Chapter 6 Test, Form 2D . . . . . . . . . . . 339–340Chapter 6 Test, Form 3 . . . . . . . . . . . . 341–342Chapter 6 Open-Ended Assessment . . . . . . 343Chapter 6 Vocabulary Test/Review . . . . . . . 344Chapter 6 Quizzes 1 & 2 . . . . . . . . . . . . . . . 345Chapter 6 Quizzes 3 & 4 . . . . . . . . . . . . . . . 346Chapter 6 Mid-Chapter Test . . . . . . . . . . . . 347Chapter 6 Cumulative Review . . . . . . . . . . . 348Chapter 6 Standardized Test Practice . 349–350

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A29

© Glencoe/McGraw-Hill iv Glencoe Geometry

Teacher’s Guide to Using theChapter 6 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 6 Resource Masters includes the core materials neededfor Chapter 6. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theGeometry TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 6-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.

Vocabulary Builder Pages ix–xinclude another student study tool thatpresents up to fourteen of the key theoremsand postulates from the chapter. Studentsare to write each theorem or postulate intheir own words, including illustrations ifthey choose to do so. You may suggest thatstudents highlight or star the theorems orpostulates with which they are not familiar.

WHEN TO USE Give these pages tostudents before beginning Lesson 6-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toupdate it as they complete each lesson.

Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

© Glencoe/McGraw-Hill v Glencoe Geometry

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

Assessment OptionsThe assessment masters in the Chapter 6Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 338–339. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

66

© Glencoe/McGraw-Hill vii Glencoe Geometry

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 6.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

cross products

extremes

fractal

iteration

ID·uh·RAY·shuhn

means

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

midsegment

proportion

ratio

scale factor

self-similar

similar polygons

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

66

Learning to Read MathematicsProof Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

66

© Glencoe/McGraw-Hill ix Glencoe Geometry

Proo

f Bu

ilderThis is a list of key theorems and postulates you will learn in Chapter 6. As you

study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 6.1Side-Side-Side (SSS) Similarity

Theorem 6.2Side-Angle-Side (SAS) Similarity

Theorem 6.3

Theorem 6.4Triangle Proportionality Theorem

Theorem 6.5Converse of the TriangleProportionality Theorem

Theorem 6.6Triangle Midsegment Theorem


© Glencoe/McGraw-Hill x Glencoe Geometry

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 6.7Proportional Perimeters Theorem

Theorem 6.8

Theorem 6.9

Theorem 6.10

Theorem 6.11Angle Bisector Theorem

Postulate 6.1Angle-Angle (AA) Similarity

Learning to Read MathematicsProof Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

66

Study Guide and InterventionProportions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1

© Glencoe/McGraw-Hill 295 Glencoe Geometry

Less

on

6-1

Write Ratios A ratio is a comparison of two quantities. The ratio a to b, where b is not zero, can be written as �

ab� or a:b. The ratio of two quantities is sometimes called a scale

factor. For a scale factor, the units for each quantity are the same.

In 2002, the Chicago Cubs baseball team won 67 games out of 162.Write a ratio for the number of games won to the total number of games played.To find the ratio, divide the number of games won by the total number of games played. The

result is �16672�, which is about 0.41. The Chicago Cubs won about 41% of their games in 2002.

A doll house that is 15 inches tall is a scale model of a real housewith a height of 20 feet. What is the ratio of the height of the doll house to theheight of the real house?To start, convert the height of the real house to inches.20 feet � 12 inches per foot � 240 inchesTo find the ratio or scale factor of the heights, divide the height of the doll house by theheight of the real house. The ratio is 15 inches:240 inches or 1:16. The height of the doll

house is �116� the height of the real house.

1. In the 2002 Major League baseball season, Sammy Sosa hit 49 home runs and was atbat 556 times. Find the ratio of home runs to the number of times he was at bat.

2. There are 182 girls in the sophomore class of 305 students. Find the ratio of girls to total students.

3. The length of a rectangle is 8 inches and its width is 5 inches. Find the ratio of length to width.

4. The sides of a triangle are 3 inches, 4 inches, and 5 inches. Find the scale factor betweenthe longest and the shortest sides.

5. The length of a model train is 18 inches. It is a scale model of a train that is 48 feet long.Find the scale factor.

Example 1Example 1

Example 2Example 2

ExercisesExercises


Use Properties of Proportions A statement that two ratios are equal is called aproportion. In the proportion �

ab� � �d

c�, where b and d are not zero, the values a and d are

the extremes and the values b and c are the means. In a proportion, the product of themeans is equal to the product of the extremes, so ad � bc.

�ab� � �d

c�

a � d � b � c↑ ↑

extremes means

Solve �196� � �

2x7�.

�196� � �

2x7�

9 � x � 16 � 27 Cross products

9x � 432 Multiply.

x � 48 Divide each side by 9.

A room is 49 centimeters by 28 centimeters on a scale drawing of ahouse. For the actual room, the larger dimension is 14 feet. Find the shorterdimension of the actual room.If x is the room’s shorter dimension, then

�2489� � �1

x4�

49x � 392 Cross products

x � 8 Divide each side by 49.

The shorter side of the room is 8 feet.

Solve each proportion.

1. �12� � �

2x8� 2. �

38� � �2

y4� 3. �

xx

��

222

� � �3100�

4. �183.2� � �

9y� 5. �

2x8� 3� � �

54� 6. �

xx

��

11� � �

34�

Use a proportion to solve each problem.

7. If 3 cassettes cost $44.85, find the cost of one cassette.

8. The ratio of the sides of a triangle are 8:15:17. If the perimeter of the triangle is 480 inches, find the length of each side of the triangle.

9. The scale on a map indicates that one inch equals 4 miles. If two towns are 3.5 inchesapart on the map, what is the actual distance between the towns?

shorter dimension��longer dimension

Study Guide and Intervention (continued)

Proportions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1

Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticeProportions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1


Less

on

6-1

1. FOOTBALL A tight end scored 6 touchdowns in 14 games. Find the ratio of touchdownsper game.

2. EDUCATION In a schedule of 6 classes, Marta has 2 elective classes. What is the ratioof elective to non-elective classes in Marta’s schedule?

3. BIOLOGY Out of 274 listed species of birds in the United States, 78 species made theendangered list. Find the ratio of endangered species of birds to listed species in theUnited States.

4. ART An artist in Portland, Oregon, makes bronze sculptures of dogs. The ratio of theheight of a sculpture to the actual height of the dog is 2:3. If the height of the sculptureis 14 inches, find the height of the dog.

5. SCHOOL The ratio of male students to female students in the drama club at CampbellHigh School is 3:4. If the number of male students in the club is 18, what is the numberof female students?


6. �25� � �4

x0� 7. �1

70� � �

2x1� 8. �

250� � �

46x�

9. �54x� � �

385� 10. �

x �3

1� � �

72� 11. �

135� � �

x �5

3�

Find the measures of the sides of each triangle.

12. The ratio of the measures of the sides of a triangle is 3:5:7, and its perimeter is 450 centimeters.

13. The ratio of the measures of the sides of a triangle is 5:6:9, and its perimeter is 220 meters.

14. The ratio of the measures of the sides of a triangle is 4:6:8, and its perimeter is 126 feet.

15. The ratio of the measures of the sides of a triangle is 5:7:8, and its perimeter is 40 inches.


1. NUTRITION One ounce of cheddar cheese contains 9 grams of fat. Six of the grams of fatare saturated fats. Find the ratio of saturated fats to total fat in an ounce of cheese.

2. FARMING The ratio of goats to sheep at a university research farm is 4:7. The numberof sheep at the farm is 28. What is the number of goats?

3. ART Edward Hopper’s oil on canvas painting Nighthawks has a length of 60 inches anda width of 30 inches. A print of the original has a length of 2.5 inches. What is the widthof the print?


4. �58� � �1

x2� 5. �1.

x12� � �

15� 6. �2

67x� � �

43�

7. �x �

32

� � �89� 8. �

3x4� 5� � �

�75� 9. �

x �4

2� � �

x �2

4�

Find the measures of the sides of each triangle.

10. The ratio of the measures of the sides of a triangle is 3:4:6, and its perimeter is 104 feet.

11. The ratio of the measures of the sides of a triangle is 7:9:12, and its perimeter is 84 inches.

12. The ratio of the measures of the sides of a triangle is 6:7:9, and its perimeter is 77 centimeters.

Find the measures of the angles in each triangle.

13. The ratio of the measures of the angles is 4:5:6.

14. The ratio of the measures of the angles is 5:7:8.

15. BRIDGES The span of the Benjamin Franklin suspension bridge in Philadelphia,Pennsylvania, is 1750 feet. A model of the bridge has a span of 42 inches. What is theratio of the span of the model to the span of the actual Benjamin Franklin Bridge?

Practice Proportions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1

Reading to Learn MathematicsProportions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1


Less

on

6-1

Pre-Activity How do artists use ratios?

Read the introduction to Lesson 6-1 at the top of page 282 in your textbook.

Estimate the ratio of length to width for the background rectangles inTiffany’s Clematis Skylight.

Reading the Lesson

1. Match each description in the first column with a word or phrase from the secondcolumn.

a. The ratio of two corresponding quantities i. proportion

b. r and u in the equation �rs� � �u

t� ii. cross products

c. a comparison of two quantities iii. means

d. ru and st in the equation �rs� � �u

t� iv. scale factor

e. an equation stating that two ratios are equal v. extremes

f. s and t in the equation �rs� � �u

t� vi. ratio

2. If m, n, p, and q are nonzero numbers such that �mn� � �

pq�, which of the following

statements could be false?

A. np � mq B. �np

� � �mq�

C. mp � nq D. qm � pn

E. �mn� � �p

q� F. �p

q� � �m

n�

G. m:p � n:q H. m:n � p:q

3. Write two proportions that match each description.

a. Means are 5 and 8; extremes are 4 and 10.

b. Means are 5 and 4; extremes are positive integers that are different from means.

Helping You Remember

4. Sometimes it is easier to remember a mathematical idea if you put it into words withoutusing any mathematical symbols. How can you use this approach to remember theconcept of equality of cross products?


Golden Rectangles

Use a straightedge, compass, and the instructions belowto construct a golden rectangle.

1. Construct square ABCD with sides of 2 cm.

2. Construct the midpoint of A�B�. Call the midpoint M.

3. Draw AB��. Set your compass at an opening equal to MC.Use M as the center to draw an arc that intersects AB��. Call the point of intersection P.

4. Construct a line through P that is perpendicular to AB��.

5. Draw DC�� so that it intersects the perpendicular line in step 4.Call the intersection point Q. APQD is a golden rectanglebecause the ratio of its length to its width is 1.618. Check this

conclusion by finding the value of �QAPP�. Rectangles whose sides

have this ratio are, it is said, the most pleasing to the human eye.

A figure consisting of similar golden rectangles is shown below. Use a compassand the instructions below to draw quarter-circle arcs that form a spiral like thatfound in the shell of a chambered nautilus.

6. Using A as a center, draw an arc that passes through B and C.

7. Using D as a center, draw an arc that passes through C and E.

8. Using F as a center, draw an arc that passes through E and G.

9. Using H as a center, draw an arc that passes through G and J.

10. Using K as a center, draw an arc that passes through J and L.

11. Using M as a center, draw an arc that passes through L and N.

M NH

L FD

JK

B A

C

G

E

A M B P

D C Q

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1

Study Guide and InterventionSimilar Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2


Less

on

6-2

Identify Similar Figures

Determine whether the triangles are similar.Two polygons are similar if and only if their corresponding angles are congruent and their corresponding sides are proportional.

�C � �Z because they are right angles, and �B � �X.By the Third Angle Theorem, �A � �Y.

For the sides, �BX

CZ� � �

2203�, �

BX

AY� � � �

2203�, and �

AY

CZ� � �

2203�.

The side lengths are proportional. So �BCA � �XZY.

Is polygon WXYZ � polygon PQRS?

For the sides, �WPQ

X� � �

182� � �

32�, �Q

XYR� � �

1182� � �

32�, �R

YZS� � �

1150� � �

32�,

and �ZSWP� � �

96� � �

32�. So corresponding sides are proportional.

Also, �W � �P, �X � �Q, �Y � �R, and �Z � �S, so corresponding angles are congruent. We can conclude that polygon WXYZ � polygon PQRS.

Determine whether each pair of figures is similar. If they are similar, give theratio of corresponding sides.

1. 2.

3. 4. 10y

8y

37�

37�116�

27�

5x

4x

15z12z

22

11

1610

equilateral triangles

15

9

12

18

XW

Z Y10

6

8

12

QP

S R

20�2��23�2�

20 23

2320 20��2 23��245�

45�C A X Z

B Y

ExercisesExercises

Example 1Example 1

Example 2Example 2


Scale Factors When two polygons are similar, the ratio of the lengths of corresponding sides is called the scale factor. At the right, �ABC � �XYZ. The scale factor of �ABC to �XYZ is 2 and the scale factor of

�XYZ to �ABC is �12�.

3 cm

6 cm

8 cm

10 cm5 cm4 cm

Z

Y

XC

A

B


Similar Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2

The two polygons aresimilar. Find x and y.

Use the congruent angles to write thecorresponding vertices in order.�RST � �MNPWrite proportions to find x and y.

�3126� � �1

x3� �

3y8� � �

3126�

16x � 32(13) 32y � 38(16)x � 26 y � 19

x

y38

3216 13

T

S

R P

N

M

�ABC � �CDE. Find the scale factor and find the lengths of C�D�and D�E�.

AC � 3 � 0 � 3 and CE � 9 � 3 � 6. Thescale factor of �CDE to �ABC is 6:3 or 2:1.Using the Distance Formula,AB � �1 � 9� � �10� and BC � �4 � 9� � �13�. The lengths of thesides of �CDE are twice those of �ABC,so DC � 2(BA) or 2�10� and DE � 2(BC) or 2�13�.

x

y

A(0, 0)

B(1, 3)

C(3, 0)

E(9, 0)

D

Example 1Example 1 Example 2Example 2

ExercisesExercises

Each pair of polygons is similar. Find x and y.

1. 2.

3. 4.

5. In Example 2 above, point D has coordinates (5, 6). Use the Distance Formula to verifythe lengths of C�D� and D�E�.

40

36x

30

18 y12

y � 1

18

24

x18

10

y

4.5x

2.5

5

9

12

y8

10x

Skills PracticeSimilar Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2


Less

on

6-2

Determine whether each pair of figures is similar. Justify your answer.

1. 2.

Each pair of polygons is similar. Write a similarity statement, and find x, themeasure(s) of the indicated side(s), and the scale factor.

3. G�H� 4. S�T� and S�U�

5. W�T� 6. T�S� and S�P�

10

8

x � 2

x � 1

SP

M N

L

T

9

103

x � 1

5

P

S

RU V

WT

Q

x � 5

x � 5

9

3

44 S

W

XY

T

U26

14

12 13

B C

DA

13

76 x

G

E H

F

7.5

7.5

7.57.5

Z Y

XW

3

3

33S R

QP

10.5

6 9

59� 35�A C

B

7

4 659� 35�D F

E


Determine whether each pair of figures is similar. Justify your answer.

1. 2.

Each pair of polygons is similar. Write a similarity statement, and find x, themeasure(s) of the indicated side(s), and the scale factor.

3. L�M� and M�N� 4. D�E� and D�F�

5. COORDINATE GEOMETRY Triangle ABC has vertices A(0, 0), B(�4, 0), and C(�2, 4).The coordinates of each vertex are multiplied by 3 to create �AEF. Show that �AEF issimilar to �ABC.

6. INTERIOR DESIGN Graham used the scale drawing of his living room to decide where to place furniture. Find the dimensions of theliving room if the scale in the drawing is 1 inch � 4.5 feet.

4 in.

21–2 in.

6 12

40�

40�

x � 1

x � 3AF

E

D

B C14

10

x � 6

x � 9

C N MD

A B LP

2412

14 162118

VUAC

B T

25 12

9

14.4

1524

20

15

JM Q R

SP

KL

Practice Similar Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2

Reading to Learn MathematicsSimilar Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2


Less

on

6-2

Pre-Activity How do artists use geometric patterns?

Read the introduction to Lesson 6-2 at the top of page 289 in your textbook.• Describe the figures that have similar shapes.

• What happens to the figures as your eyes move from the center to theouter edge?

Reading the Lesson1. Complete each sentence.

a. Two polygons that have exactly the same shape, but not necessarily the same size,are .

b. Two polygons are congruent if they have exactly the same shape and the same .

c. Two polygons are similar if their corresponding angles are and their corresponding sides are .

d. Two polygons are congruent if their corresponding angles are and their corresponding sides are .

e. The ratio of the lengths of corresponding sides of two similar figures is called the .

f. Multiplying the coordinates of all points of a figure in the coordinate plane by a scale factor to get a similar figure is called a .

g. If two polygons are similar with a scale factor of 1, then the polygons are .

2. Determine whether each statement is always, sometimes, or never true.a. Two similar triangles are congruent.b. Two equilateral triangles are congruent.c. An equilateral triangle is similar to a scalene triangle.d. Two rectangles are similar.e. Two isosceles right triangles are congruent.f. Two isosceles right triangles are similar.g. A square is similar to an equilateral triangle.h. Two acute triangles are similar.i. Two rectangles in which the length is twice the width are similar.j. Two congruent polygons are similar.

Helping You Remember3. 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 phrasesthat include the word scale in a way that is related to proportions.


Constructing Similar PolygonsHere are four steps for constructing a polygon that is similar to and withsides twice as long as those of an existing polygon.

Step 1 Choose any point either inside or outside the polygon and label it O.Step 2 Draw rays from O through each vertex of the polygon.Step 3 For vertex V, set the compass to length OV. Then locate a new point

V� on ray OV such that VV� � OV. Thus, OV� � 2(OV ).Step 4 Repeat Step 3 for each vertex. Connect points V�, W�, X� and Y� to

form the new polygon.

Two constructions of polygons similar to and with sides twice those of VWXYare shown below. Notice that the placement of point O does not affect the sizeor shape of V�W�X�Y�, only its location.

Trace each polygon. Then construct a similar polygon with sides twice as long asthose of the given polygon.

1. 2. D E F

GC

A

B

VW

Y X

VW

Y X

V�

W�

X�Y�

O

VW

YX

V�

W�

X�Y�

O

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2

3. Explain how to construct a similarpolygon with sides three times the lengthof those of polygon HIJKL. Then do theconstruction.

4. Explain how to construct a similar

polygon 1�12� times the length of those of

polygon MNPQRS. Then do theconstruction.

M

S

R

N

Q

P

H I

LK

J

Study Guide and InterventionSimilar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3


Less

on

6-3

Identify Similar Triangles Here are three ways to show that two triangles are similar.

AA Similarity Two angles of one triangle are congruent to two angles of another triangle.

SSS Similarity The measures of the corresponding sides of two triangles are proportional.

SAS SimilarityThe measures of two sides of one triangle are proportional to the measures of two corresponding sides of another triangle, and the included angles are congruent.

Determine whether thetriangles are similar.

�DAC

F� � �69� � �

23�

�BE

CF� � �1

82� � �

23�

�DAB

E� � �1105� � �

23�

�ABC � �DEF by SSS Similarity.

15

129

D E

F

10

86

A B

C

Determine whether thetriangles are similar.

�34� � �

68�, so �MQR

N� � �

NRS

P�.

m�N � m�R, so �N � �R.�NMP � �RQS by SAS Similarity.

8

4

70�

Q

R S6

370�

M

N P


ExercisesExercises

Determine whether each pair of triangles is similar. Justify your answer.

1. 2.

3. 4.

5. 6. 32

20

154025

24

3926

1624

18

8

65�9

465�

36

2018

24


Use Similar Triangles Similar triangles can be used to find measurements.


Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3

�ABC � �DEF.Find x and y.

�DAC

F� � �BE

CF� �D

ABE� � �

BE

CF�

� �198� �1

y8� � �

198�

18x � 9(18�3�) 9y � 324x � 9�3� y � 36

18�3��x

18��3

B

CA

18 18 9y

x

E

FD

A person 6 feet tall castsa 1.5-foot-long shadow at the same timethat a flagpole casts a 7-foot-longshadow. How tall is the flagpole?

The sun’s rays form similar triangles.Using x for the height of the pole, �

6x� � �

17.5�,

so 1.5x � 42 and x � 28.The flagpole is 28 feet tall.

7 ft

6 ft?

1.5 ft


ExercisesExercises

Each pair of triangles is similar. Find x and y.

1. 2.

3. 4.

5. 6.

7. The heights of two vertical posts are 2 meters and 0.45 meter. When the shorter postcasts a shadow that is 0.85 meter long, what is the length of the longer post’s shadow to the nearest hundredth?

y

x32

10

2230yx

9

8

7.2

16

yx

6038.6

23

3024

36��2yx

3636

20

13

26y

x � 235

13

10 20y

x

Skills PracticeSimilar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3


Less

on

6-3


1. 2.

3. 4.

ALGEBRA Identify the similar triangles, and find x and the measures of theindicated sides.

5. A�C� and E�D� 6. J�L� and L�M�

7. E�H� and E�F� 8. U�T� and R�T�

6

14

S

U

V

R T

x � 4

x � 612

9

6

9

D

E

FH

Gx � 5

16

4M

NL

J

K

x � 18

x � 3

1512

D

E CB

Ax � 1

x � 5

30�60�RQ

M

S

K

T

2170�

15

MJ

P

1470�10

U

S

T

12

128

9

9 6

C P

QR

A

B

2014

13 1221

9

S W X

Y

R

T



1. 2.

ALGEBRA Identify the similar triangles, and find x and the measures of theindicated sides.

3. L�M� and Q�P� 4. N�L� and M�L�

Use the given information to find each measure.

5. If T�S� || Q�R�, TS � 6, PS � x � 7, 6. If E�F� || H�I�, EF � 3, EG � x � 1,QR � 8, and SR � x � 1, HI � 4, and HG � x � 3,find PS and PR. find EG and HG.

INDIRECT MEASUREMENT For Exercises 7 and 8, use the following information.A lighthouse casts a 128-foot shadow. A nearby lamppost that measures 5 feet 3 inches castsan 8-foot shadow.

7. Write a proportion that can be used to determine the height of the lighthouse.

8. What is the height of the lighthouse?

F

GE

H

I

S RP

QT

8

N

J LK

Mx � 5

6x � 2

24

12

18N

P

QL

M

x � 3x � 1

18

16 12.516

14

11

M

L N

S R

T42�

42�

24

1812

16J

A K

Y S

W

Practice Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3

Reading to Learn MathematicsSimilar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3


Less

on

6-3

Pre-Activity How do engineers use geometry?

Read the introduction to Lesson 6-3 at the top of page 298 in your textbook.• What does it mean to say that triangular shapes result in rigid

construction?

• What would happen if the shapes used in the construction werequadrilaterals?

Reading the Lesson1. State whether each condition guarantees that two triangles are congruent or similar.

If the condition guarantees that the triangles are both similar and congruent, writecongruent. If there is not enough information to guarantee that the triangles will becongruent or similar, write neither.a. Two sides and the included angle of one triangle are congruent to two sides and the

included angle of the other triangle.b. The measures of all three pairs of corresponding sides are proportional.c. Two angles of one triangle are congruent to two angles of the other triangle.d. Two angles and a nonincluded side of one triangle are congruent to two angles and

the corresponding nonincluded side of the other triangle.e. The measures of two sides of a triangle are proportional to the measures of two

corresponding sides of another triangle, and the included angles are congruent.

f. The three sides of one triangle are congruent to the three sides of the other triangle.

g. The three angles of one triangle are congruent to the three angles of the other triangle.

h. One acute angle of a right triangle is congruent to one acute angle of another righttriangle.

i. The measures of two sides of a triangle are proportional to the measures of two sidesof another triangle.

2. Identify each of the following as an example of a reflexive, symmetric, or transitive property.a. If �RST � �UVW, then �UVW � �RST.b. If �RST � �UVW and �UVW � �OPQ, then �RST � �OPQ.c. �RST � �RST

Helping You Remember3. A good way to remember something is to explain it to someone else. Suppose one of your

classmates is having trouble understanding the difference between SAS for congruenttriangles and SAS for similar triangles. How can you explain the difference to him?


Ratio Puzzles with TrianglesIf you know the perimeter of a triangle and the ratios of the sides, you canfind the lengths of the sides.

The perimeter of a triangle is 84 units. The sides havelengths r, s, and t. The ratio of s to r is 5:3, and the ratio of t to r is2:1. Find the length of each side.

Since both ratios contain r, rewrite one or both ratios to make r the same. Youcan write the ratio of t to r as 6:3. Now you can write a three-part ratio.r : s : t � 3: 5: 6

There is a number x such that r � 3x, s � 5x, and t � 6x. Since you know theperimeter, 84, you can use algebra to find the lengths of the sides.

r � s � t � 843x � 5x � 6x � 84

14x � 84x � 6

3x � 18, 5x � 30, 6x � 36

So r � 18, s � 30, and t � 36.

Find the lengths of the sides of each triangle.

1. The perimeter of a triangle is 75 units. The sides have lengths a, b, and c. The ratio of b to a is 3:5, and the ratio of c to a is 7:5. Find the length of each side.

2. The perimeter of a triangle is 88 units. The sides have lengths d, e, and f. The ratio of e to d is 3:1, and the ratio of f to e is 10:9. Find the length of each side.

3. The perimeter of a triangle is 91 units. The sides have lengths p, q, and r. The ratio of p to r is 3:1, and the ratio of q to r is 5:2. Find the length of each side.

4. The perimeter of a triangle is 68 units. The sides have lengths g, h, and j. The ratio of j to g is 2:1, and the ratio of h to g is 5:4. Find the length of each side.

5. Write a problem similar to those above involving ratios in triangles.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3

ExampleExample

Study Guide and InterventionParallel Lines and Proportional Parts

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4


Less

on

6-4

Proportional Parts of Triangles In any triangle, a line parallel to one side of a triangle separates the other two sides proportionally. The converse is also true.

If X and Y are the midpoints of R�T� and S�T�, then X�Y� is amidsegment of the triangle. The Triangle Midsegment Theoremstates that a midsegment is parallel to the third side and is half its length.

If XY�� || RS��, then �RXTX� � �Y

SYT�.

If �RXTX� � �Y

SYT�, then XY�� || RS��.

If X�Y� is a midsegment, then XY�� || RS�� and XY � �12�RS.

Y ST

R

X

In �ABC, E�F� || C�B�. Find x.

Since E�F� || C�B�, �FA

BF� � �E

AEC�.

�xx

��

222

� � �168�

6x � 132 � 18x � 3696 � 12x8 � x

F

C

BA

18

x � 22 x � 2

6E

A triangle has verticesD(3, 6), E(�3, �2), and F(7, �2).Midsegment G�H� is parallel to E�F�. Find the length of G�H�.G�H� is a midsegment, so its length is one-half that of E�F�. Points E and F have thesame y-coordinate, so EF � 7 � (�3) � 10.The length of midsegment G�H� is 5.


ExercisesExercises

Find x.

1. 2. 3.

4. 5. 6.

7. In Example 2, find the slope of E�F� and show that E�F� || G�H�.

30 10x � 10

x11x � 12

x33

30 10

24x

35

x9

20

x

185

5 x

7


Divide Segments Proportionally When three or more parallel lines cut two transversals,they separate the transversals into proportional parts. If the ratio of the parts is 1, then the parallellines separate the transversals into congruent parts.

If �1 || �2 || �3, If �4 || �5 || �6 and

then �ab� � �d

c�. �

uv� � 1, then �

wx� � 1.

Refer to lines �1, �2 ,and �3 above. If a � 3, b � 8, and c � 5, find d.

�1 || �2 || �3 so �38� � �d

5�. Then 3d � 40 and d � 13�

13�.

Find x and y.

1. 2.

3. 4.

5. 6.16 32

y � 3y

3 4

x � 4x

5

8 y � 2

y2x � 4

3x � 1 2y � 2

3y

2x � 6

x � 3

12

x � 12 12

3x5x

a

bd

u v

xw

ctn

m

s �1

�4 �5 �6

�2

�3


Parallel Lines and Proportional Parts

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4

ExampleExample

ExercisesExercises

Skills PracticeParallel Lines and Proportional Parts

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4


Less

on

6-4

1. If JK � 7, KH � 21, and JL � 6, 2. Find x and TV if RU � 8, US � 14,find LI. TV � x � 1 and VS � 17.5.

Determine whether B�C� || D�E�.

3. AD � 15, DB � 12, AE � 10, and EC � 8

4. BD � 9, BA � 27, and CE is one third of EA

5. AE � 30, AC � 45, and AD is twice DB

COORDINATE GEOMETRY For Exercises 6–8, use the following information.Triangle ABC has vertices A(�5, 2), B(1, 8), and C(4, 2). Point Dis the midpoint of A�B� and E is the midpoint of A�C�.

6. Identify the coordinates of D and E.

7. Show that B�C� is parallel to D�E�.

8. Show that DE � �12�BC.

9. Find x and y. 10. Find x and y.

3–2x � 2

x � 3

2y � 1

3y � 52x � 1 3y � 8

y � 5x � 7

x

y

O

A C

B

D

E

C

B

E

D

A

R

U V

T

S

HL

K J

I


1. If AD � 24, DB � 27, and EB � 18, 2. Find x, QT, and TR if QT � x � 6,find CE. SR � 12, PS � 27, and TR � x � 4.

Determine whether J�K� || N�M�.

3. JN � 18, JL � 30, KM � 21, and ML � 35

4. KM � 24, KL � 44, and NL � �56�JN

COORDINATE GEOMETRY For Exercises 5 and 6, use the following information.Triangle EFG has vertices E(�4, �1), F(2, 5), and G(2, �1). Point Kis the midpoint of E�G� and H is the midpoint of F�G�.

5. Show that E�F� is parallel to K�H�.

6. Show that KH � �12�EF.

7. Find x and y. 8. Find x and y.

9. MAPS The distance from Wilmington to Ash Grove along Kendall is 820 feet and along Magnolia, 660 feet. If the distance between Beech and Ash Grove along Magnolia is 280 feet, what is the distance between the two streets alongKendall?

Ash Grove

Beech

Magnolia Kendall

Wilmington

4–3y � 2

3x � 4 x � 1

4y � 43x � 4

2–3y � 3

1–3y � 65–

4x � 3

x

y

O

E G

F

H

K

N LJ

KM

S

Q

TRP

D

E

C

BA

Practice Parallel Lines and Proportional Parts

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4

Reading to Learn MathematicsParallel Lines and Proportional Parts

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4


Less

on

6-4

Pre-Activity How do city planners use geometry?


Use a geometric idea to explain why the distance between Chicago Avenueand Ontario Street is shorter along Michigan Avenue than along LakeShore Drive.

Reading the Lesson

1. Provide the missing words to complete the statement of each theorem. Then state thename of the theorem.

a. If a line intersects two sides of a triangle and separates the sides into corresponding

segments of lengths, then the line is to the

third side.

b. A midsegment of a triangle is to one side of the triangle and its

length is the length of that side.

c. If a line is to one side of a triangle and intersects the other two sides

in distinct points, then it separates these sides into

of proportional length.

2. Refer to the figure at the right.

a. Name the three midsegments of �RST.

b. If RS � 8, RU � 3, and TW � 5, find the length of each of themidsegments.

c. What is the perimeter of �RST?

d. What is the perimeter of �UVW?

e. What are the perimeters of �RUV, �SVW, and �TUW?

f. How are the perimeters of each of the four small triangles related to the perimeter ofthe large triangle?

g. Would the relationship that you found in part f apply to any triangle in which themidpoints of the three sides are connected?


3. A good way to remember a new mathematical term is to relate it to other mathematicalvocabulary that you already know. What is an easy way to remember the definition ofmidsegment using other geometric terms?

R

U

V

W

T

S


Parallel Lines and Congruent PartsThere is a theorem stating that if three parallel lines cut off congruent segments on onetransversal, then they cut off congruent segments on any transversal. This can be shown forany number of parallel lines. The following drafting technique uses this fact to divide asegment into congruent parts.

A�B� is to be separated into five congruent parts. This can be done very accurately without using a ruler. All that is needed is a compass and a piece of notebook paper.

Step 1 Hold the corner of a piece of notebook paper at point A.

Step 2 From point A, draw a segment along the paper that is five spaces long. Mark where the lines of the notebook paper meet the segment. Label the fifth point, P.

Step 3 Draw P�B�. Through each of the other marks on A�P�, construct a line parallel to B�P�. The points where these lines intersect A�B� will divide A�B� into five congruent segments.

Use a compass and a piece of notebook paper to divide each segment into the given number of congruent parts.

1. six congruent parts

2. seven congruent parts A B

A B

A

P

B

A B

P

A B

A B

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4

Study Guide and InterventionParts of Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5


Less

on

6-5

Perimeters If two triangles are similar, their perimeters have the same proportion as the corresponding sides.

If �ABC � �RST, then

�AR

BS

��

BST

C��

RA

TC

� � �AR

BS� � �

BST

C� � �R

ACT�.

Use the diagram above with �ABC � �RST. If AB � 24 and RS � 15,find the ratio of their perimeters.Since �ABC � �RST, the ratio of the perimeters of �ABC and �RST is the same as theratio of corresponding sides.

Therefore � �2145�

� �85�

Each pair of triangles is similar. Find the perimeter of the indicated triangle.

1. �XYZ 2. �BDE

3. �ABC 4. �XYZ

5. �ABC 6. �RST

25

20

10

P T

SM

R

95

13

12

A D

E

C

B

5 6

8.7

13

10

X Y

Z T

R S18

20

25 15 17

8

NB

A C PM

36

9

11C

DE

A

B

44

24

20 22 10

TRX Z

YS

perimeter of �ABC��perimeter of �RST

A C

BS

R T

ExampleExample

ExercisesExercises


Special Segments of Similar Triangles When two triangles are similar,corresponding altitudes, angle bisectors, and medians are proportional to the correspondingsides. Also, in any triangle an angle bisector separates the opposite side into segments thathave the same ratio as the other two sides of the triangle.


Parts of Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5

In the figure,�ABC � �XYZ, with angle bisectors asshown. Find x.

Since �ABC � �XYZ, the measures of theangle bisectors are proportional to themeasures of a pair of corresponding sides.

�AX

BY� � �Y

BWD�

�2x4� � �

180�

10x � 24(8)10x � 192

x � 19.2

2410

DW

Y

ZXC

B

A

8x

S�U� bisects �RST. Find x.

Since S�U� is an angle bisector, �RTU

U� � �

RTS

S�.

�2x0� � �

1350�

30x � 20(15)30x � 300

x � 10

x 20

3015

UT

S

R


ExercisesExercises

Find x for each pair of similar triangles.

1. 2.

3. 4.

5. 6. 2x � 2x � 1

281617

11

x � 7

x

10

10x

8

873

3

4x

69

12x

x

36

1820

Skills PracticeParts of Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5


Less

on

6-5

Find the perimeter of the given triangle.

1. �JKL, if �JKL � �RST, RS � 14, 2. �DEF, if �DEF � �ABC, AB � 27,ST � 12, TR � 10, and LJ � 14 BC � 16, CA � 25, and FD � 15

3. �PQR, if �PQR � �LMN, LM � 16, 4. �KLM, if �KLM � �FGH, FG � 30,MN � 14, NL � 27, and RP � 18 GH � 38, HF � 38, and KL � 24


5. Find FG if �RST � �EFG, S�H� is an 6. Find MN if �ABC � �MNP, A�D� is an altitude of �RST, F�J� is an altitude of altitude of �ABC, M�Q� is an altitude of�EFG, ST � 6, SH � 5, and FJ � 7. �MNP, AB � 24, AD � 14, and MQ � 10.5.

Find x.

7. �HKL � �XYZ 8.

LH

K

18

10

Y

15

x

ZX7

20

24

x

C

A

BD P NQ

M

JGE

F

HTR

S

GL

K

M

F

H

L MP Q

RN

A

B

C

D

E

F

R S

T

J K

L


Find the perimeter of the given triangle.

1. �DEF, if �ABC � �DEF, AB � 36, 2. �STU, if �STU � �KLM, KL � 12,BC � 20, CA � 40, and DE � 35 LM � 31, MK � 32, and US � 28


3. Find PR if �JKL � �NPR, K�M� is an 4. Find ZY if �STU � �XYZ, U�A� is an altitude of �JKL, P�T� is an altitude of altitude of �STU, Z�B� is an altitude of �NPR, KL � 28, KM � 18, and �XYZ, UT � 8.5, UA � 6, and PT � 15.75. ZB � 11.4.

Find x.

5. 6.

PHOTOGRAPHY For Exercises 7 and 8, use the following information.Francine has a camera in which the distance from the lens to the film is 24 millimeters.

7. If Francine takes a full-length photograph of her friend from a distance of 3 meters andthe height of her friend is 140 centimeters, what will be the height of the image on thefilm? (Hint: Convert to the same unit of measure.)

8. Suppose the height of the image on the film of her friend is 15 millimeters. If Francinetook a full-length shot, what was the distance between the camera and her friend?

x28

20 30402x � 1 25x � 4

YX B

Z

TS A

U

MJ L

K

TN R

P

KS

LT

UMBE

CFD

A

Practice Parts of Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5

Reading to Learn MathematicsParts of Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5


Less

on

6-5

Pre-Activity How is geometry related to photography?


• How is similarity involved in the process of making a photographic printfrom a negative?

• Why do photographers place their cameras on tripods?

Reading the Lesson

1. In the figure, �RST � �UVW. Complete each proportion involving the lengths of segments inthis figure by replacing the question mark. Thenidentify the definition or theorem from the listbelow that the completed proportion illustrates.

i. Definition of congruent polygons ii. Definition of similar polygons

iii. Proportional Perimeters Theorem iv. Angle Bisectors Theorem

v. Similar triangles have corresponding altitudes proportional to corresponding sides.

vi. Similar triangles have corresponding medians proportional to corresponding sides.

vii. Similar triangles have corresponding angle bisectors proportional to corresponding sides.

a. �RS � S?T � TR� � �U

RSV� b. �U

RWT� � �

S?X�

c. �RU

MN� � �V

?W� d. �U

RSV� � �

S?T�

e. �RPS

P� � �S

?T� f. �

U?N� � �

UR

WT�

g. �WTP

Q� � �R?T� h. �

UVW

W� � �Q

?V�


2. A good way to remember a large amount of information is to remember key words. Whatkey words will help you remember the features of similar triangles that are proportionalto the lengths of the corresponding sides?

W

Q N

YU

V

T

P M

XR

S


Proportions for Similar TrianglesRecall that if a line crosses two sides of a triangle and is parallel to the thirdside, then the line separates the two sides that it crosses into segments ofproportional lengths.

You can write many proportions by identifying similar triangles in thefollowing diagram. In the diagram, AM�� BN��, AE�� FL�� MR��, and DQ�� ER��.

Answer each question. Use the diagram above.

1. Name a triangle similar to �GNP. 2. Name a triangle similar to �CJH.

3. Name two triangles similar to �JKS. 4. Name a triangle similar to �ACP.

Complete each proportion.

5. �AA

GP� � �

A?F� 6. �C

CHP� � �

C?R� 7. �J

JRS� � �J

?L�

8. �PP

HC� � �

P?G� 9. �

ELR

R� � �J

?R� 10. �

MM

NP� � �A

?P�

Solve.

11. If CJ � 16, JR � 48, and LR � 30, find EL.

12. If DK � 5, KS � 7, and CJ � 8, find JS.

13. If MN � 12, NP � 32, and AP � 48, find AG. Round to the nearest tenth.

14. If CH � 18, HP � 82, and CR � 130, find CJ.

15. Write three more problems that can be solved using the diagram above.

A B C D E

FG H J K L

M N P RQ

S

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

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Study Guide and InterventionFractals and Self-Similarity

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6-6

Characteristics of Fractals The act of repeating a process over and over, such asfinding a third of a segment, then a third of the new segment, and so on, is called iteration.When the process of iteration is applied to some geometric figures, the results are calledfractals. For objects such as fractals, when a portion of the object has the same shape orcharacteristics as the entire object, the object can be called self-similar.

In the diagram at the right, notice that the details at each stage are similar to the details at Stage 1.

1. Follow the iteration process below to produce a fractal.

Stage 1• Draw a square.• Draw an isosceles right triangle on the top side of the square.

Use the side of the square as the hypotenuse of the triangle.• Draw a square on each leg of the right triangle.

Stage 2Repeat the steps in Stage 1, drawing an isosceles triangle and two small squares for each of the small squares from Stage 1.

Stage 3Repeat the steps in Stage 1 for each of the smallest squares in Stage 2.

2. Is the figure produced in Stage 3 self-similar?

Stage 1 Stage 2 Stage 3

ExercisesExercises

ExampleExample


Nongeometric Iteration An iterative process can be applied to an algebraicexpression or equation. The result is called a recursive formula.

Find the value of x3, where the initial value of x is 2. Repeat theprocess three times and describe the pattern.

Initial value: 2First time: 23 � 8Second time: 83 � 512Third time: 5123 � 134,217,728

The result of each step of the iteration is used for the next step. For this example, the xvalues are greater with each iteration. There is no maximum value, so the values aredescribed as approaching infinity.

For Exercises 1–5, find the value of each expression. Then use that value as thenext x in the expression. Repeat the process three times, and describe yourobservations.

1. �x�, where x initially equals 5

2. �1x�, where x initially equals 2

3. 3x, where x initially equals 1

4. x � 5 where x initially equals 10

5. x2 � 4, where x initially equals 16

6. Harpesh paid $1000 for a savings certificate. It earns interest at an annual rate of 2.8%,and interest is added to the certificate each year. What will the certificate be worth afterfour years?


Fractals and Self-Similarity

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6

ExercisesExercises

ExampleExample

Skills PracticeFractals and Self-Similarity

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6-6

Stages 1 and 2 of a fractal known as the Cantor set are shown. To get Stage 2, thesegment in Stage 1 is trisected, and the interior of the middle segment is removed.(The interior of a segment is the segment with its endpoints removed.) Theprocess is repeated for subsequent stages.

1. Draw stages 3 and 4 of the Cantor set.

2. How many segments are there in Stage 3? Stage 4?

3. What happens to the length of the line segments in each stage?

4. The Cantor set is the set of points after infinitely many iterations. Is the Cantor set self-similar?

Find the value of each expression. Then, use that value as the next x in theexpression. Repeat the process three times, and describe your observations.

5. 2x � 3, where x initially equals 5

6. �2x

�, where x initially equals 1

Find the first three iterates of each expression.

7. x2 � 1, where x initially equals 2 8. �1x�, where x initially equals 8

Stage 3

Stage 4

Stage 1

Stage 2


An artist is designing a book cover and wants to show a copy of the book cover inthe lower left corner of the cover. After Stages 1 and 2 of the design, he realizesthat the design is developing into a fractal!

1. Draw Stages 3 and 4 of the book cover fractal.

2. After infinitely many iterations, will the result be a self similar fractal?

3. On what part of the book cover should you focus your attention to be sure you can find acopy of the entire figure?

Find the value of each expression. Then, use that value as the next x in theexpression. Repeat the process three times and describe your observations.

4. 2(x � 3), where x initially equals 0 5. �2x

� � 3, where x initially equals 10

Find the first three iterates of each expression.

6. �2x

� � 1, where x initially equals 1 7. 2x2, where x initially equals 2

8. HOUSING The Andrews purchased a house for $96,000. The real estate agent who soldthe house said that comparable houses in the area appreciate at a rate of 4.5% per year.If this pattern continues, what will be the value of the house in three years? Round tothe nearest whole number.

Stage 3

MATHMATH

MATH

Stage 4

MATHMATH

MATHMATH

Stage 1

MATHStage 2

MATHMATH

Practice Fractals and Self-Similarity

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6

Reading to Learn MathematicsFractals and Self-Similarity

NAME ______________________________________________ DATE ____________ PERIOD _____

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Less

on

6-6

Pre-Activity How is mathematics found in nature?


Name two objects from nature other than broccoli in which a small pieceresembles the whole.

Reading the Lesson1. Match each definition from the first column with a term from the second column. (Some

words of phrases in the second column may be used more than once or not at all.)

Phrase Terma. a geometric figure that is created using iteration i. similar b. a pattern in which smaller and smaller details of a shape have ii. iteration

the same geometric characteristics as the original shape iii. fractalc. the result of translating an iterative process into a formula or iv. self-similar

algebraic equation v. recursiond. the process of repeating the same procedure over and over formula

again vi. congruent

2. Refer to the three patterns below.

i.

ii.

iii.

a. Give the special name for each of the fractals you obtain by continuing the patternwithout end.

b. Which of these fractals are self-similar?

c. Which of these fractals are strictly self-similar?

Helping You Remember3. A good way to remember a new mathematical term is to relate it to everyday English

words. The word fractal is related to fraction and fragment. Use your dictionary to findat least one definition for each of these words that you think is related to the meaning ofthe word fractal and explain the connection.


The Möbius StripA Möbius strip is a special surface with only one side. It was discovered byAugust Ferdinand Möbius, a German astronomer and mathematician.

1. To make a Möbius strip, cut a strip of paper about 16 inches long and 1inch wide. Mark the ends with the letters A, B, C, and D as shown below.

Twist the paper once, connecting A to D and B to C. Tape the endstogether on both sides.

2. Use a crayon or pencil to shade one side of the paper. Shade around thestrip until you get back to where you started. What happens?

3. What do you think will happen if you cut the Möbius strip down themiddle? Try it.

4. Make another Möbius strip. Starting a third of the way in from one edge,cut around the strip, staying always the same distance in from the edge.What happens?

5. Start with another long strip of paper. Twist the paper twice and connectthe ends. What happens when you cut down the center of this strip?

6. Start with another long strip of paper. Twist the paper three times andconnect the ends. What happens when you cut down the center of this strip?

A

B

C

D

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6

Chapter 6 Test, Form 166


Ass

essm

ents

Write the letter for the correct answer in the blank at the right of eachquestion.

1. There are 15 plums and 9 apples in a fruit bowl. Find the ratio of apples toplums.A. 3:5 B. 3:8 C. 5:3 D. 8:3

2. The scale drawing of a porch is 8 inches wide by 12 inches long. If the actualporch is 12 feet wide, find the length of the porch.A. 8 ft B. 10 ft C. 16 ft D. 18 ft

3. Solve �56� � �

4x�.

A. 4.6 B. 4.8 C. 5 D. 7

4. A quality control technician checked a sample of 30 bulbs. Two of the bulbswere defective. If the sample was representative, find the number of bulbsexpected to be defective in a case of 450.A. 24 B. 30 C. 36 D. 45

5. Find the triangle similar to �ABC at the right.A. B.

C. D.

6. Find x if �ABC � �JKL.A. 10 B. 14C. 25 D. 29

7. Quadrilateral ABCD � quadrilateral PQRS. If AB � 10, BC � 6, PS � 12,and QR � 4, find the scale factor of ABCD to PQRS.

A. �12� B. �

32� C. �

53� D. �

56�

8. If quadrilateral ABCD � quadrilateralEFGH, find x.A. 15 B. 20C. 25 D. 30

9. Which theorem or postulate can be used to prove that these two triangles are similar?A. AA B. SAS C. SSA D. SSS

10. Find MN.

A. 5�13� B. 6 �

34� C. 7 D. 12

35�8

96

35�

Q

PM

L

N

6

84

27�

63�

B

C

D

A

F

E

H

G 60� 120�80�

100�4x �

A C

B J L

K186 9

x � 2

4

352210

24

15

36

39

5

12

35

12

13

CB

A

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 6 Test, Form 1 (continued)66

11.

12.

13.

14.

15.

16.

17.

18.

19.

11. A 5-foot tall student cast a 4-foot shadow. If the tree next to her cast a 44-footshadow, what is the height of the tree?

A. 35�15� ft B. 45 ft C. 51�

12� ft D. 55 ft

12. In �ABC, D�E� || A�C�. If AD � 12, BD � 3, and CE � 10,find BE.A. 1 B. 1 �

12�

C. 2 D. 2 �12�

13. Find x so that A�C� || M�N� in �ABC.A. 8 B. 10C. 25 D. 29

14. Find x.A. 14 B. 15C. 16 D. 18

15. If �FGH � �PQR, FG � 6, PQ � 10, and the perimeter of �PQR is 35, findthe perimeter of �FGH.A. 21 B. 27 C. 31 D. 58�

13�

16. �LMN � �XYZ with altitudes K�L�and W�X�. Find KL.A. 6 B. 7C. 9 D. 19

17. Find x.A. 5 B. 6

C. 6�12� D. 7�

12�

18. Find x.A. 16 B. 18C. 20 D. 21

19. Each arrangement of dots forms a box number. Find the number of dots in the eighth box number.A. 32 B. 48C. 64 D. 512

Bonus In �ABC, AB � 10, BC � 16, D�E� ⊥ A�C�,and DE � 6. Find CD.

106

B C

D

E

A

16

4 8 12

15

1824

x

1610

129x

28 123

K

L X

M YWZN

1610

24x

C B

A

N

M12

18

163x

C B

A

E

D12

10

3

B:

NAME DATE PERIOD

Chapter 6 Test, Form 2A66


Ass

essm

ents

Write the letter for the correct answer in the blank at the right of eachquestion.1. Of the 240 students eating lunch, 96 purchased their lunch and the rest

brought a bag lunch. Find the ratio of students purchasing lunch to studentsbringing a bag lunch.A. 2:3 B. 2:5 C. 3:2 D. 5:2

2. In a rectangle, the ratio of the width to the length is 4:5. If the rectangle is 40 centimeters long, find its width.A. 32 cm B. 36 cm C. 44 cm D. 50 cm

3. A postage stamp 25 millimeters wide and 40 millimeter tall is enlarged tomake a poster. The poster is 4 feet wide. Find the height of the poster.A. 2.5 ft B. 5.25 ft C. 5.8 ft D. 6.4 ft

4. Find the polygon that is similar to ABCD.A. B.

C. D.

5. If �PQR � �STU, find x.A. 4.4 B. 7C. 24.6 D. 35

6. If ABCD � EFGH, find x.A. 18.75 B. 20C. 22.75 D. 28

7. If �ABC � �LMN, AB � 18, BC � 12, LN � 9, and LM � 6, find the scalefactor of �ABC to �LMN.

A. �92� B. �

32� C. �

31� D. �

21�

8. Name the theorem or postulate that can be used to prove that these triangles are similar.A. AA B. SSSC. SAS D. SSA

For Questions 9 and 10, refer to the figure at the right.9. Identify the true statement.

A. �PQR � �RST B. �PQR � �STRC. �PQR � �TSR D. �PQR � �TRS

10. Find x.

A. 2�12� B. 3 C. 3�

12� D. 4

15

18

4x2

21–2

3

T

R

P

Q

S

14

43

7

16

30

10

x � 4

A

BC

DE H

GF

22�123�

5x�12 8

4.5

R U

S

TQ

P

882 6

2–3

218

6

4

12

A D

B C

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 6 Test, Form 2A (continued)66

11.

12.

13.

14.

15.

16.

17.

18.

19.

11. A 24-foot flagpole cast a 20-foot shadow. The building next to it cast an 85-footshadow. Find the height of the building.

A. 70�56� ft B. 89 ft C. 96�

16� ft D. 102 ft

12. Find QT.A. 15 B. 17C. 19 D. 21

13. Find x so that S�T� || P�R�.A. 4 B. 4�

12�

C. 6 D. 6�12�

14. Find y.

A. �43� B. 2

C. �73� D. 3

15. If �KLM � �XYZ, find the perimeter of �XYZ.A. 40 B. 42C. 45 D. 48

16. �ABC � �JKL with altitudes B�X� and K�Y�.Find BX.A. 19.2 B. 21C. 24.6 D. 28

17. Find x.A. 4 B. 5C. 6 D. 8

18. Find y.

A. 2�14� B. 2�

34�

C. 3�12� D. 4�

12�

19. Find the third iterate of the expression 2(3x � 1), where x initially equals 1.A. 24 B. 64 C. 302 D. 474

Bonus Find x.12

4 x

6

8

y

DA C

B

6S

Q

P R

12 8

x � 2 x

32

X CA

B10 6

Y LJ

K

9

X Z

Y4

K M

L7

9

69

2y

1–3y � 2

25

5

34x � 3 QR

P

S

T

24

40

35 RQ

PS

T

B:

NAME DATE PERIOD

Chapter 6 Test, Form 2B66


Ass

essm

ents

Write the letter for the correct answer in the blank at the right of eachquestion.

1. Given the choice between doing an oral and a written report, 18 of the 28 students chose to do an oral report. Find the ratio of written to oral reports.A. 5:9 B. 9:5 C. 9:14 D. 14:9

2. A model of a lighthouse has diameter 8 inches and height 18 inches. If theactual diameter of the lighthouse is 20 feet, find its actual height.A. 30 ft B. 35 ft C. 45 ft D. 50 ft

3. The three sides of a triangle are in the ratio 2:4:5. If the shortest side of thetriangle is 4 meters long, find the perimeter.A. 17 m B. 22 m C. 32 m D. 40 m

4. Find the polygon that may be similar to ABCD.A. B.

C. D.

5. If �ABC � �DEF, find m�C.A. 54° B. 59°C. 67° D. 69°

6. If quadrilateral JKLM � quadrilateral WXYZ, JK � 15, LM � 10, XY � 6,and WX � 9, find KL.A. 8 B. 10 C. 11 D. 12

7. If �LMN � �RST, LN � 21, MN � 28, and the scale factor of �RST to �LMNis �

43�, find ST.

A. 15�34� B. 21 C. 28 D. 37�

13�

8. Name the theorem or postulate that can be used to prove that these triangles are similar.A. AA B. SASC. SSA D. SSS

For Questions 9 and 10, refer to the figure at the right.9. Identify the similar triangles.

A. �LMN � �MPQ B. �LMN � �QMPC. �LMN � �QPM D. �LMN � �PQM

10. Find LM.A. 16 B. 17 C. 18 D. 20

1510 12

L

N

MP

Q

20

8

11.228

54�67�67�

DECA

FB

30

24

8

22

18

6

17

14

6

12

15

5

12

15

54

A B

D C

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 6 Test, Form 2B (continued)66

11.

12.

13.

14.

15.

16.

17.

18.

19.

11. A 6-foot tall fence post cast a 2�12�-foot shadow. A nearby clock tower cast a

35-foot shadow. Find the height of the tower.A. 37�

12� ft B. 71 ft C. 78 ft D. 84 ft

12. Find CE.A. 25 B. 26C. 27 D. 28

13. Find FH so that G�H� || D�E�.A. 4 B. 5C. 6 D. 7

14. Find x.A. 4 B. 6C. 9 D. 12

15. If �ABC � �PQR, find the perimeter of �PQR.A. 12 B. 13�

12�

C. 14�12� D. 16

16. �ABC � �XYZ with altitudes A�P�and X�Q�. Find AP.

A. 11�14� B. 14

C. 15�14� D. 20

17. Find x.A. 7 B. 8C. 9 D. 10

18. Find y.

A. 3�34� B. 4

C. 6 D. 7�12�

19. Find the third iterate of the expression 3(2x � 1), where x initially equals 2.A. 27 B. 63 C. 174 D. 303

Bonus Find CE.6

5

4

39

A

B

C

D

E

6

21–4

16

y

R T

SU

40

16

x � 2

3x

125

Q

X

Y Z27

P

A

B C

6 Q

R

P8

7 3

B

C

A

58

2x

x � 3

9

10

15D

G

FE H

8

20

EB C

DA

35

B:

NAME DATE PERIOD

Chapter 6 Test, Form 2C66


Ass

essm

ents

1. Of the 300 television sets sold at an electronics store lastmonth, 90 were flat-screen TVs. Find the ratio of flat-screenTVs to other TVs sold last month.

2. Determine whether �ABC � �DEF. Justify your answer.

3. When a 5-foot vertical pole casts a 3-foot, 4-inch shadow, anoak tree casts a 20-foot shadow. Find the height of the tree.

4. If quadrilateral ABCD � quadrilateral WXYZ, AB � 15,BC � 27, and the scale factor of WXYZ to ABCD is �

23�, find XY.

5. The blueprint for a swimming pool is 8 inches by 2�12� inches.

The actual pool is 136 feet long. Find the width of the pool.

6. Find CD.

7. If quadrilateral ABCD � quadrilateral PQRS, find BC.

8. Determine whether �ABC � �DEF. Justify your answer.

9. �ABC � �XYZ, AB � 12, AC � 16, BC� 20, and XZ � 24.Find the perimeter of �XYZ.

For Questions 10 and 11, use the figure.

10. Identify the similar triangles.

11. Find x.

15 2x � 4

x6

P

Q

R S

T

9

8

6

42

5

B

EF

DC

A

28

10 6.5R

S P

Q B

A

C

D

1.8

2.46C EG

FD

22.5

15 1510

A

B

C DE

F

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 6 Test, Form 2C (continued)66

12. If �ABC � �PQR and B�M� and Q�N� are medians, find BM.

13. The ratio of the measures of the three sides of a triangle is 3:4:6.If the perimeter is 91, find the measure of the longest side.

14. If �RST � �UVW, find m�W.

15. In �ABC, A�X� bisects �BAC. Find x.

16. Find y so that M�N� || B�C�.

17. �ABC � �LMN, and A�D� and L�P� are altitudes. Find AD.

18. Find x.

19. Find the third iterate of the expression x2 � 2, where xinitially equals 1.

Bonus Find EG.18

10

12

B

F

D EG

C

A

28

818

2x � 2

13.5

3620

8D

A

C B M P

L

N

18

8y4

y

B

CAN

M

4x � 5

2x

8

6

A C

X

B

85�

48�R T

S

U W

V

11

4.43.8

A C P R

Q

B

M N

12.

13.

14.

15.

16.

17.

18.

19.

B:

NAME DATE PERIOD

Chapter 6 Test, Form 2D66


Ass

essm

ents

1. Of the 112 students in the marching band, 35 were in thedrum section. Find the ratio of drummers to other musiciansin the band.

2. Determine whether quadrilateral ABCD � quadrilateralEFGH. Justify your answer.

3. When a 9-foot tall garden shed cast a 5-foot, 3-inch shadow, ahouse cast a 28-foot shadow. Find the height of the house.

4. �ABC � �FGH, AB � 24, AC � 16, GH � 9, and FH � 12.Find the scale factor of �ABC to �FGH.

5. The model of a suspension bridge is 18 inches long and 2 inchestall. If the length of the actual bridge is 1650 feet, find its height.

6. Find GP.

7. If �JKL � �PQR, find x.

8. Determine whether �ABC � �PQR. Justify your answer.

9. �ABC � �PQR, AB � 18, BC � 20, AC � 22, and QR � 25.Find the perimeter of �PQR.

For Questions 10 and 11, use the figure.

10. Identify the similar triangles.

11. Find MN.20

x

6

Y Z

X

M

N

16

20

22.5

3012

158

P

BC

A

Q

R

35

14

6

x

J L

PQ

R

K

2.73.3

4.5

F H

Q

G

P

A

9

7

14 17

9 8.5

13.512

B

C

D H G

FE

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 6 Test, Form 2D (continued)66

12. If �FGH � �LMN and A�F� and B�L� are medians, find BL.

13. The ratio of the measures of the three angles of a triangle is3:4:8. Find the measure of the largest angle.

14. If quadrilateral DEFG � quadrilateral WXYZ, find m�Y.

15. In �PQR, Q�S� bisects �PQR. Find x.

16. Find x so that P�Q� || F�H�.

17. If �FGH � �JKL, find GX.

18. Find y.

19. Find the third iterate of the expression 4(x � 2), where xinitially equals 10.

Bonus Find FG.

2.8 3.2 4.2

8 7

A

B F

GEC

D

42 2y

12 16

X Y

K

F H JL

27 12 21–2

G

63

3x2x � 1

F H

Q

G

P

P Q

R

S3x � 1

32

40

2x � 3

E F Z W

XY

GD

68� 85�

82�125�

16.8 4.8

7.7

F H L

MB

N

G

A

12.

13.

14.

15.

16.

17.

18.

19.

B:

NAME DATE PERIOD

Chapter 6 Test, Form 366


Ass

essm

ents

1. In an orchard of apple and peach trees, �37� of the trees are

peaches. Find the ratio of apple trees to peach trees.

2. Determine whether trapezoid ABCD � trapezoid PQRS.Justify your answer.

3. In �ABC, m�A � 51, AB � 14, and AC � 20. In �DEF,m�D � 51, DE � 16.8, and DF � 24. Determine whether�ABC � �DEF. Justify your answer.

4. A painting that is 48 inches by 12 inches is reduced to fit onan area that is 30 centimeters by 10 centimeters. Find themaximum dimensions of the reduced painting.

5. Find x.

6. If �ABC � �XYZ, find y.

7. The ratio of the measures of the sides of a triangle is 2:5:6. Ifthe length of the longest side is 48 inches, find the perimeter.

8. Find m�I.

9. �ABC � �DEF, AB � 8, BC � 13, AC � 15, and DF � 20.Find the perimeter of �DEF.

10. �ABC � �JKL, AB � 12, BC � 18.4, KL � 6.9, and JL � 5.6.Find the scale factor of �ABC to �JKL.

11. Find y.y � 8

4y � 7

F

G

HI

J45�

53�

17.5A C

B

2y

14X Z

Yy � 3

6

4.8

6.5K L

J

N

M

2x

9.6

15.6

12.47.8

9

15

6

23.4D

R S

PQ

C

A

B

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 6 Test, Form 3 (continued)66

12. Find SR.

For Questions 13 and 14, �FGH � �JKL.

13. Find x.

14. Find the ratio of the perimeter of �FGH to the perimeter of �JKL.

15. Find AD so that D�E� || A�B�.

16. Find the fourth iterate of the expression 3x � 1, where xinitially equals 0.8.

17. When a 15-foot tall climbing wall cast a 20-foot shadow, abuilding cast a 32-foot shadow. Find the height of the building.

18. If �ABC � �EFG and B�D� and F�H� are medians, find BD.

19. Stage 1 is repeated in the upper left half of the unshaded square of each stage. Find the perimeter of thesmallest square in Stage 3.

Bonus The ratio of the lengths of the sides of �ABC is 4:5:2.5. The scale factor of �ABC to �DECis 5:2, and the scale factor of�DEC to �FEG is 1:4. B�C� is the shortest side and A�B� is thelongest side of �ABC. Find FG.

10

A

B C

D E

G

F

Stage 1

10 in.

Stage 2

HE G

F2x � 5

751–4

x � 5

DA C

B

A B

ED

C

2x � 4 3x

8x5x

12x

F H

G7.5

4.5

J L

K

30 12Q

RPS

40

12.

13.

14.

15.

16.

17.

18.

19.

B:

NAME DATE PERIOD

Chapter 6 Open-Ended Assessment66


Ass

essm

ents

Demonstrate your knowledge by giving a clear, concise solution toeach problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate beyond the requirements of the problem.

1. Write a possible proportion if the extremes are 3 and 10.

2. �ABC and �WXY are isosceles triangles.

a. Write a possible ratio for the sides of �ABC if its perimeter is 42 inches.

b. Name possible measures for the sides of �ABC using your answer toPart a.

c. If �WXY has a perimeter of 28 and �ABC has sides with the measuresyou gave in Part b, what must be the measure of the sides of �WXY sothat �WXY � �ABC?

3. Write as many triangle similarity statements as possible for the figurebelow. How do you know that these triangles are similar?

4. Sketch two triangles that are not similar, but have one pair of correspondingangles congruent and two pairs of corresponding sides proportional. Labelthe corresponding angles and the proportional sides.

5.

a. Draw �XYZ inside �PQR with half the perimeter of �PQR. Explainyour process and why it works.

b. Draw the figure that results when your process in Part a is iterated 3 times.

c. Explain why your figure in Part b is a fractal.

d. The perimeter of �PQR is �12�x. If x equals 96, what is the perimeter of

the smallest triangle in your figure in Part b?

P R

Q

A B C D

I E

F

G

H

NAME DATE PERIOD

SCORE


Chapter 6 Vocabulary Test/Review66

Choose from the terms above to complete each sentence.

1. If there are 15 girls and 9 boys in an art class, the ofgirls to boys in the class is 5:3.

2. If �ABC � �DEF, AB � 10, and DE � 2.5, then the of�ABC to �DEF is 4:1.

3. In �LMN, P lies on L�M� and Q on L�N�. If PQ � �12�MN, P�Q� is

called a(n) .

4. The product of the in the equation �3x� � �

2340� is 90.

5. The product of the in the equation �3x� � �

2340� is 24x.

6. A geometric figure created by using iteration is called a(n).

7. If quadrilaterals ABCD and WXYZ have corresponding anglescongruent and corresponding sides proportional, they arecalled .

8. In a recursive formula, an algebraic expression is evaluatedrepeatedly in a process called .

9. The equation �3x� � �

2340� is called a(n) .

10. If some parts of a figure resemble the figure as a whole, thefigure is called .

In your own words—

11. Explain what is meant by equality of cross products.

12. Describe a self-similar figure.

?

?

?

?

?

?

?

?

?

? 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

cross productsextremesfractal

iterationmeansmidsegment

proportionratioscale factor

self-similarsimilar polygons

NAME DATE PERIOD

SCORE

Chapter 6 Quiz (Lessons 6–1 and 6–2)

66


Ass

essm

ents

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

1. GARDENS The model of a circular garden is 8 inches indiameter. The actual garden will be 20 feet in diameter. Whatis the ratio of the diameter of the model to the diameter of theactual garden?

2. PHOTOS A 4-inch by 6-inch photograph, set vertically, isenlarged to make a poster 22 inches wide. How tall is the poster?

3. Are any of the three trianglessimilar? If so, write theappropriate similaritystatement.

4. If �ABC � �DEC, find x and the scale factor of �ABC to �DEC.

5. STANDARDIZED TEST PRACTICE The perimeter of arectangle is 126 centimeters. The ratio of the length to thewidth is 5:2. Find the width of the rectangle.A. 9 cm B. 18 cm C. 45 cm D. 50.4 cm

D

E

CB

A

20

253x � 2

2x

BP X Y

Z

Q

R

A

C

5

4 8 8 8

10

4

Chapter 6 Quiz (Lesson 6–3)

66

1.

2.

3.

4.

5.

For Questions 1 and 2, determine whether each pair oftriangles is similar. Justify your answer.

1. 2.

3. Identify the similar triangles in the figure,then find x.

4. SHADOWS A person who is 5 feet tall casts a shadow that is4 feet long. At the same time, a flagpole casts a shadow that is18 feet long. How tall is the flagpole?

5. Find x.

46

9 x

8

A

B D

EF

C 4

6x

20

15

98

10 1241�

52�41�

87�

NAME DATE PERIOD

SCORE


Chapter 6 Quiz (Lessons 6–4 and 6–5)

66

1.

2.

3.

4.

5.

1. If �PQR � � XYZ, find the perimeter of �XYZ.

2. Find x so that L�M� || A�B�.

3. In �ABC, D�E� is parallel to A�C� and DE � 10. What is thelength of A�C� if D�E� is the midsegment of �ABC?

4. Find DE.

5. In �RST, TU�� bisects �T. If U is a point on R�S�, RU � 6,RT � 9, and ST � 12, find RS.

20

12

15 DE

B

A

C

6 8

4A B

C

L Mx

16

128

P R

Q

5

X Z

Y

NAME DATE PERIOD

SCORE

Chapter 6 Quiz (Lesson 6–6)

66

1.

2.

3.

4.

1. DIAMOND NUMBERS The dots in each arrangement form a diamondnumber. How many dots are in the ninth diamond number?

2. In the fractal, Stage 1 is repeated in the upper right and lowerleft unshaded squares of each stage. Draw Stage 3.

For Questions 3 and 4, find the first 3 iterates of the givenexpression.

3. 4x � 1, where x initially equals 2

4. x2 � x, where x initially equals 3

Stage 1 Stage 2

Stage 3

NAME DATE PERIOD

SCORE

Chapter 6 Mid-Chapter Test (Lessons 6–1 through 6–3)

66


Ass

essm

ents

1. If quadrilateral ABCD � quadrilateral PQRS, which proportion must be true?

A. �AA

DC� � �

PP

QS� B. �C

BDC� � �

QR

RS� C. �B

ADB� � �Q

PQR� D. �

CA

DB� � �

PR

QS�

2. This fall 126 students participated in the soccer program, while 54 playedvolleyball. What was the ratio of soccer players to volleyball players?

A. �34� B. �

37� C. �

43� D. �

73�

3. The ratio of the measures of the angles of a triangle is 2:3:10. What is theleast angle measure?A. 12 B. 15 C. 24 D. 36

4. Find x.A. 2 B. 4.8C. 6 D. 6.4

5. If rectangle ABCD � rectangle EFGH, the perimeter of ABCD is 54 centimeters, and the perimeter of EFGH is 36 centimeters, what is thescale factor of ABCD to EFGH?

A. �23� B. �

32� C. �

35� D. �

53�

10 8

x

D

E

AC B18

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

Part II

For Questions 6 and 7, determine whether each pair oftriangles is similar. Justify your answer.

6. 7.

8. If �ABE � �BCD, find DE and the scale factor of �ABE to �BCD.

9. Quadrilateral ABCD � quadrilateral RSUV, m�ABC � 120,and the scale factor of ABCD to RSUV is �

85�. What is m�RSU?

10. MODELS Sasha made a model of a clipper ship. If her modelhas a length of 18 inches, and the original ship had a length of160 feet and a width of 32 feet, what should be the width ofher model?

24 32

9

A

B ED

C

45251810 85�85�6

415

10

Part I Write the letter for the correct answer in the blank at the right of each question.


Chapter 6 Cumulative Review(Chapters 1–6)

66

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

1. Find the coordinates of the midpoint of A�B� for A(�24, 15) andB(13, �31). (Lesson 1-3)

For Questions 2–4, use the figure at the right.2. The perimeter of rectangle DEFG is 176,

EF � h, and DE � 7h. Find h. (Lesson 1-6)

3. If p: DEFG is a rectangle and q: DE � EF, find the truth valueof p � q. (Lesson 2-2)

4. How many line segments can be drawn connecting D, E, F,and G? (Lesson 2-5)

5. For S(�5, 7), T(1, 9), P(12, �1), and R(3, 26), determinewhether ST�� and PR�� are parallel, perpendicular, or neither.(Lesson 3-3)

6. Write an equation in slope-intercept form for the line y � 7 � 4(x �10). (Lesson 3-4)

7. Given T(3, �1), U(1, �7), V(8, �5), W(2, 6), X(�4, 8), andY(�2, 1) determine whether �TUV � �WXY. Explain.(Lesson 4-4)

8. Name a pair of congruent angles and a pair of congruent sides that are not marked in the figure. (Lesson 4-6)

For Questions 9 and 10,use the figure at the right.9. Points D, E, and F are

the midpoints of A�B�, B�C�,and C�A�, respectively.Find x, y, and z. (Lesson 5-1)

10. Compare m�BEG to m�CEG. (Lesson 5-5)

For Questions 11 and 12, use the figure at the right.11. Determine whether �HJL � �HKM.

Justify your answer. (Lesson 6-3)

12. If JK � 7, HJ � 8, HL � 12, and LM � 2x � 7.5, find x.(Lesson 6-4)

13. Find a and b. (Lesson 6-4)

L3a � 2

3b

b � 46 � a

M

N

R

Q

P

H ML

K

J

B

CAF 3y � 1

8 � 5z

2.57

1.5

x � 3D EG

L N

M

P

QR

D E

FG

NAME DATE PERIOD

SCORE

Standardized Test Practice (Chapters 1–6)


1. Find the length of Y�Z�. (Lesson 1-2)

A. 1.9 in. B. 5.3 in.C. 7.2 in. D. 12.5 in.

2. Given: 3b � 4 � 16 Conjecture: b � 0Which of the following would be a counterexample? (Lesson 2-1)

E. b � �1 F. b � 0 G. b � 3.5 H. b � 4

3. Find the plane that is parallel to plane PTU. (Lesson 3-1)

A. plane QRU B. plane QRSC. plane PQS D. plane SPU

4. Find m�1. (Lesson 3-2)

E. 5 F. 12G. 41 H. 44

5. Which statement must be true in order to prove �ABC � �DCB by SAS? (Lesson 4-4)

A. C�B� bisects �ABD.B. �BCA � �CBDC. �BDC � �CABD. A�B� � B�C�

6. In an indirect proof, you assume that the conclusion is false andthen find a(n) . (Lesson 5-3)

E. assumption F. contradictionG. truth value H. conditional statement

7. Demont and Tony are competing to see whose house is the tallest.Early in the afternoon, Tony, who is 4 feet tall, measured his shadowto be 9.6 inches and the shadow of his house to be 62.4 inches. Laterin the day, Demont, who is 5 feet tall, measured his shadow to be15.6 inches and the shadow of his house to be 62.4 inches. Who livesin the taller house? (Lesson 6-3)

A. Demont B. Both houses are the same height.C. Tony D. There is not enough information.

8. Which geometric figure is created using iteration? (Lesson 6-6)

E. scalene triangle F. squareG. fractal H. midsegment

?

D

A C

B

(3x � 5)�

(125 � 7x)�

1

R

P

Q

STU

X Z5.5 in.

3.6 in. Y

NAME DATE PERIOD

SCORE

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.

1.

2.

3.

4.

5.

6.

7.

8. E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

Ass

essm

ents

66


Standardized Test Practice (continued)

9. What is the distance between �76 and 43 on anumber line? (Lesson 1-3)

10. If 28 � 4x � �12, then 5 � 2x � .(Lesson 2-6)

11. Find m�2. (Lesson 3-2)

12. �LMN is equilateral, LM is one more thanthree times a number, MN is nine less than fivetimes the number, and NL is eleven more thanthe number. Find LM. (Lesson 4-1)

52�

12 34

?

NAME DATE PERIOD

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

Part 3: Short Response

Instructions: Show your work or explain in words how you found your answer.

9. 10.

11. 12.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

13. Solve ��4106

� � �4x �

510

�. (Lesson 6-1)

14. X�A� is an altitude of �XYB. Find y.(Lesson 5-1)

15. Two sides of a triangle measure 21 inches and 32 inches, andthe third side measures x inches. Find the range for x. (Lesson 5-4)

16. If �ABC � �HIJ, find the perimeter of �HIJ. (Lesson 6-5) 32 cm

80 cm

60 cm

A B

C 12 cmH I

J

(2y � 81)� BA

X

Y

13.

14.

15.

16.

1 1 9 2 5

1 2 8 1 6

66

Standardized Test PracticeStudent Record Sheet (Use with pages 338–339 of the Student Edition.)

66

© Glencoe/McGraw-Hill A1 Glencoe Geometry

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7

2 5 8

3 6 DCBADCBA

DCBADCBADCBA

DCBADCBADCBA

NAME DATE PERIOD

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 3 Open-EndedPart 3 Open-Ended

Solve the problem and write your answer in the blank.

For Questions 11 and 12, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.

9 11 12

10

11 (grid in)

12 (grid in)

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

Record your answers for Questions 13–14 on the back of this paper.


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fee

t.F

ind

th

e sh

orte

rd

imen

sion

of

the

actu

al r

oom

.If

xis

th

e ro

om’s

sh

orte

r di

men

sion

,th

en

�2 48 9��

� 1x 4�

49x

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ross

pro

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s

x�

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ivid

e ea

ch s

ide

by 4

9.

Th

e sh

orte

r si

de o

f th

e ro

om i

s 8

feet

.

Sol

ve e

ach

pro

por

tion

.

1.�1 2�

��2 x8 �

562.

�3 8��

� 2y 4�9

3.�x x� �

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8

4.� 18

3 .2��

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.65.

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3.5

6.�x x

� �1 1

��

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7

Use

a p

rop

orti

on t

o so

lve

each

pro

ble

m.

7.If

3 c

asse

ttes

cos

t $4

4.85

,fin

d th

e co

st o

f on

e ca

sset

te.

$14.

95

8.T

he

rati

o of

th

e si

des

of a

tri

angl

e ar

e 8:

15:1

7.If

th

e pe

rim

eter

of

the

tria

ngl

e is

48

0 in

ches

,fin

d th

e le

ngt

h o

f ea

ch s

ide

of t

he

tria

ngl

e.96

in.,

180

in.,

204

in.

9.T

he

scal

e on

a m

ap i

ndi

cate

s th

at o

ne

inch

equ

als

4 m

iles

.If

two

tow

ns

are

3.5

inch

esap

art

on t

he

map

,wh

at i

s th

e ac

tual

dis

tan

ce b

etw

een

th

e to

wn

s?14

mi

shor

ter

dim

ensi

on�

��

long

er d

imen

sion

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Pro

po

rtio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-1

6-1

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Answers (Lesson 6-1)


An

swer

s

Skil

ls P

ract

ice

Pro

po

rtio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-1

6-1

©G

lenc

oe/M

cGra

w-H

ill29

7G

lenc

oe G

eom

etry

Lesson 6-1

1.FO

OTB

ALL

A t

igh

t en

d sc

ored

6 t

ouch

dow

ns

in 1

4 ga

mes

.Fin

d th

e ra

tio

of t

ouch

dow

ns

per

gam

e.

0.43

:1

2.E

DU

CA

TIO

NIn

a s

ched

ule

of

6 cl

asse

s,M

arta

has

2 e

lect

ive

clas

ses.

Wh

at i

s th

e ra

tio

of e

lect

ive

to n

on-e

lect

ive

clas

ses

in M

arta

’s s

ched

ule

?

1:2

3.B

IOLO

GY

Ou

t of

274

lis

ted

spec

ies

of b

irds

in

th

e U

nit

ed S

tate

s,78

spe

cies

mad

e th

een

dan

gere

d li

st.F

ind

the

rati

o of

en

dan

gere

d sp

ecie

s of

bir

ds t

o li

sted

spe

cies

in

th

eU

nit

ed S

tate

s.

� 13 39 7�

4.A

RT

An

art

ist

in P

ortl

and,

Ore

gon

,mak

es b

ron

ze s

culp

ture

s of

dog

s.T

he

rati

o of

th

eh

eigh

t of

a s

culp

ture

to

the

actu

al h

eigh

t of

th

e do

g is

2:3

.If

the

hei

ght

of t

he

scu

lptu

reis

14

inch

es,f

ind

the

hei

ght

of t

he

dog.

21 in

.

5.SC

HO

OL

Th

e ra

tio

of m

ale

stu

den

ts t

o fe

mal

e st

ude

nts

in

th

e dr

ama

clu

b at

Cam

pbel

lH

igh

Sch

ool

is 3

:4.I

f th

e n

um

ber

of m

ale

stu

den

ts i

n t

he

clu

b is

18,

wh

at i

s th

e n

um

ber

of f

emal

e st

ude

nts

?

24

Sol

ve e

ach

pro

por

tion

.

6.�2 5�

�� 4x 0�

167.

� 17 0��

�2 x1 �30

8.�2 50 �

��4 6x �

6

9.�5 4x �

��3 85 �

3.5

10.�

x� 3

1�

��7 2�

9.5

11.�

1 35 ��

�x� 5

3�

28

Fin

d t

he

mea

sure

s of

th

e si

des

of

each

tri

angl

e.

12.T

he

rati

o of

th

e m

easu

res

of t

he

side

s of

a t

rian

gle

is 3

:5:7

,an

d it

s pe

rim

eter

is

450

cen

tim

eter

s.90

cm

,150

cm

,210

cm

13.T

he r

atio

of

the

mea

sure

s of

the

sid

es o

f a

tria

ngle

is 5

:6:9

,and

its

peri

met

er is

220

met

ers.

55 m

,66

m,9

9 m

14.T

he

rati

o of

th

e m

easu

res

of t

he

side

s of

a t

rian

gle

is 4

:6:8

,an

d it

s pe

rim

eter

is

126

feet

.28

ft,

42 f

t,56

ft

15.T

he r

atio

of

the

mea

sure

s of

the

sid

es o

f a

tria

ngle

is

5:7:

8,an

d it

s pe

rim

eter

is

40 i

nche

s.10

in.,

14 in

.,16

in.

©G

lenc

oe/M

cGra

w-H

ill29

8G

lenc

oe G

eom

etry

1.N

UTR

ITIO

NO

ne o

unce

of

ched

dar

chee

se c

onta

ins

9 gr

ams

of f

at.S

ix o

f th

e gr

ams

of f

atar

e sa

tura

ted

fats

.Fin

d th

e ra

tio

of s

atur

ated

fat

s to

tot

al f

at i

n an

oun

ce o

f ch

eese

.

2:3

2.FA

RM

ING

Th

e ra

tio

of g

oats

to

shee

p at

a u

niv

ersi

ty r

esea

rch

far

m i

s 4:

7.T

he

nu

mbe

rof

sh

eep

at t

he

farm

is

28.W

hat

is

the

nu

mbe

r of

goa

ts?

16

3.A

RT

Edw

ard

Hop

per’

s oi

l on

can

vas

pain

tin

g N

igh

thaw

ksh

as a

len

gth

of

60 i

nch

es a

nd

a w

idth

of

30 i

nch

es.A

pri

nt

of t

he

orig

inal

has

a l

engt

h o

f 2.

5 in

ches

.Wh

at i

s th

e w

idth

of t

he

prin

t?

1.25

in.

Sol

ve e

ach

pro

por

tion

.

4.�5 8�

�� 1x 2�

7.5

5.� 1.

x 12��

�1 5�0.

224

6.� 26 7x �

��4 3�

6

7.�x

� 32

��

�8 9��2 3�

8.�3x

4�5

��

�� 75 ��5 7�

9.�x

� 42

��

�x� 2

4�

�10

Fin

d t

he

mea

sure

s of

th

e si

des

of

each

tri

angl

e.

10.T

he

rati

o of

th

e m

easu

res

of t

he

side

s of

a t

rian

gle

is 3

:4:6

,an

d it

s pe

rim

eter

is

104

feet

.

24 f

t,32

ft,

48 f

t

11.T

he r

atio

of

the

mea

sure

s of

the

sid

es o

f a

tria

ngle

is 7

:9:1

2,an

d it

s pe

rim

eter

is 8

4 in

ches

.

21 in

.,27

in.,

36 in

.

12.T

he

rati

o of

th

e m

easu

res

of t

he

side

s of

a t

rian

gle

is 6

:7:9

,an

d it

s pe

rim

eter

is

77 c

enti

met

ers.

21 c

m,2

4.5

cm,3

1.5

cm

Fin

d t

he

mea

sure

s of

th

e an

gles

in

eac

h t

rian

gle.

13.T

he

rati

o of

th

e m

easu

res

of t

he

angl

es i

s 4:

5:6.

48,6

0,72

14.T

he

rati

o of

th

e m

easu

res

of t

he

angl

es i

s 5:

7:8.

45,6

3,72

15.B

RID

GES

Th

e sp

an o

f th

e B

enja

min

Fra

nkl

in s

usp

ensi

on b

ridg

e in

Ph

ilad

elph

ia,

Pen

nsy

lvan

ia,i

s 17

50 f

eet.

A m

odel

of

the

brid

ge h

as a

spa

n o

f 42

in

ches

.Wh

at i

s th

era

tio

of t

he

span

of

the

mod

el t

o th

e sp

an o

f th

e ac

tual

Ben

jam

in F

ran

klin

Bri

dge?

� 51 00�

Pra

ctic

e (

Ave

rag

e)

Pro

po

rtio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-1

6-1



Readin

g t

o L

earn

Math

em

ati

csP

rop

ort

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-1

6-1

©G

lenc

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w-H

ill29

9G

lenc

oe G

eom

etry

Lesson 6-1

Pre-

Act

ivit

yH

ow d

o ar

tist

s u

se r

atio

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

6-1

at

the

top

of p

age

282

in y

our

text

book

.

Est

imat

e th

e ra

tio

of l

engt

h t

o w

idth

for

th

e ba

ckgr

oun

d re

ctan

gles

in

Tif

fan

y’s

Cle

mat

is S

kyli

ght.

Sam

ple

an

swer

:ab

ou

t 5

to 2

Rea

din

g t

he

Less

on

1.M

atch

eac

h d

escr

ipti

on i

n t

he

firs

t co

lum

n w

ith

a w

ord

or p

hra

se f

rom

th

e se

con

dco

lum

n.

a.T

he

rati

o of

tw

o co

rres

pon

din

g qu

anti

ties

ivi.

prop

orti

on

b.

ran

d u

in t

he

equ

atio

n �r s�

�� ut �

vii

.cro

ss p

rodu

cts

c.a

com

pari

son

of

two

quan

titi

esvi

iii.

mea

ns

d.

ruan

d st

in t

he

equ

atio

n �r s�

�� ut �

iiiv

.sc

ale

fact

or

e.an

equ

atio

n s

tati

ng

that

tw

o ra

tios

are

equ

ali

v.ex

trem

es

f.s

and

tin

th

e eq

uat

ion

�r s��

� ut �iii

vi.r

atio

2.If

m,n

,p,a

nd

qar

e n

onze

ro n

um

bers

su

ch t

hat

�m n��

�p q� ,w

hic

h o

f th

e fo

llow

ing

stat

emen

ts c

ould

be

fals

e?B

,C,E

A.

np

�m

qB

.�np �

�� mq �

C.

mp

�n

qD

.qm

�pn

E.

�m n��

� pq �F.

� pq ��

� mn �

G.m

:p�

n:q

H.m

:n�

p:q

3.W

rite

tw

o pr

opor

tion

s th

at m

atch

eac

h d

escr

ipti

on.

a.M

ean

s ar

e 5

and

8;ex

trem

es a

re 4

an

d 10

.

any

two

of

�4 5��

� 18 0�,�

4 8��

� 15 0�,�

1 50 ��

�8 4� ,�1 80 �

��5 4�

b.

Mea

ns

are

5 an

d 4;

extr

emes

are

pos

itiv

e in

tege

rs t

hat

are

dif

fere

nt

from

mea

ns.

any

two

of

�2 5��

� 14 0�,�

2 4��

� 15 0�,�

1 4��

� 25 0�,�

1 5��

� 24 0�,�

1 50 ��

�4 2� ,�1 40 �

��5 2� ,

�2 40 ��

�5 1� ,

or

�2 50 ��

�4 1�

Hel

pin

g Y

ou

Rem

emb

er

4.S

omet

imes

it

is e

asie

r to

rem

embe

r a

mat

hem

atic

al i

dea

if y

ou p

ut

it i

nto

wor

ds w

ith

out

usi

ng

any

mat

hem

atic

al s

ymbo

ls.H

ow c

an y

ou u

se t

his

app

roac

h t

o re

mem

ber

the

con

cept

of

equ

alit

y of

cro

ss p

rodu

cts?

Sam

ple

an

swer

:Th

e p

rod

uct

of

the

mea

ns

equ

als

the

pro

du

ct o

f th

e ex

trem

es.

©G

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w-H

ill30

0G

lenc

oe G

eom

etry

Go

lden

Rec

tan

gle

s

Use

a s

trai

ghte

dge

,com

pas

s,an

d t

he

inst

ruct

ion

s b

elow

to c

onst

ruct

a g

old

en r

ecta

ngl

e.

1.C

onst

ruct

squ

are

AB

CD

wit

h s

ides

of

2 cm

.

2.C

onst

ruct

th

e m

idpo

int

of A �

B�.C

all

the

mid

poin

t M

.

3.D

raw

AB

� �� .

Set

you

r co

mpa

ss a

t an

ope

nin

g eq

ual

to

MC

.U

se M

as t

he

cen

ter

to d

raw

an

arc

th

at i

nte

rsec

ts A

B� �

� .C

all

the

poin

t of

in

ters

ecti

on P

.

4.C

onst

ruct

a l

ine

thro

ugh

Pth

at i

s pe

rpen

dicu

lar

to A

B� �

� .

5.D

raw

DC

� ��

so t

hat

it

inte

rsec

ts t

he

perp

endi

cula

r li

ne

in s

tep

4.C

all

the

inte

rsec

tion

poi

nt

Q.A

PQ

Dis

a g

old

en r

ecta

ngl

ebe

cau

se t

he

rati

o of

its

len

gth

to

its

wid

th i

s 1.

618.

Ch

eck

this

con

clu

sion

by

fin

din

g th

e va

lue

of �Q A

PP �.R

ecta

ngl

es w

hos

e si

des

hav

e th

is r

atio

are

,it

is s

aid,

the

mos

t pl

easi

ng

to t

he

hu

man

eye

.

A f

igu

re c

onsi

stin

g of

sim

ilar

gol

den

rec

tan

gles

is

show

n b

elow

.Use

a c

omp

ass

and

th

e in

stru

ctio

ns

bel

ow t

o d

raw

qu

arte

r-ci

rcle

arc

s th

at f

orm

a s

pir

al l

ike

that

fou

nd

in

th

e sh

ell

of a

ch

amb

ered

nau

tilu

s.

6.U

sin

g A

as a

cen

ter,

draw

an

arc

th

at p

asse

s th

rou

gh

Ban

d C

.

7.U

sin

g D

as a

cen

ter,

draw

an

arc

th

at p

asse

s th

rou

gh

Can

d E

.

8.U

sin

g F

as a

cen

ter,

draw

an

arc

th

at p

asse

s th

rou

gh

Ean

d G

.

9.U

sin

g H

as a

cen

ter,

draw

an

arc

th

at p

asse

s th

rou

gh

Gan

d J

.

10.U

sin

g K

as a

cen

ter,

draw

an

arc

th

at p

asse

s th

rou

gh

Jan

d L

.

11.U

sin

g M

as a

cen

ter,

draw

an

arc

th

at p

asse

s th

rou

gh

Lan

d N

.

MN H

LF

D JK

BAC

G

E

AM

BP

DC

Q

Ch

eck

stu

den

ts’

con

stru

ctio

nm

arks

.Th

e fi

nal

fig

ure

sh

ou

ld lo

ok

like

the

on

e b

elo

w.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-1

6-1



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Sim

ilar

Po

lyg

on

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-2

6-2

©G

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ill30

1G

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eom

etry

Lesson 6-2

Iden

tify

Sim

ilar

Fig

ure

s

Det

erm

ine

wh

eth

er t

he

tria

ngl

es

are

sim

ilar

.T

wo

poly

gon

s ar

e si

mil

ar i

f an

d on

ly i

f th

eir

corr

espo

ndi

ng

angl

es a

re c

ongr

uen

t an

d th

eir

corr

espo

ndi

ng

side

s ar

e pr

opor

tion

al.

�C

��

Zbe

cau

se t

hey

are

rig

ht

angl

es,a

nd

�B

��

X.

By

the

Th

ird

An

gle

Th

eore

m,�

A�

�Y

.

For

th

e si

des,

�B XC Z�

��2 20 3�

,�B X

A Y��

��2 20 3�

,an

d �A Y

C Z��

�2 20 3�.

Th

e si

de l

engt

hs

are

prop

orti

onal

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�B

CA

��

XZ

Y.

Is p

olyg

on W

XY

Z�

pol

ygon

PQ

RS

?

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th

e si

des,

�W PQX �

��1 82 �

��3 2� ,

� QXY R�

��1 18 2�

��3 2� ,

� RYZ S�

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��3 2� ,

and

�Z SW P��

�9 6��

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o co

rres

pon

din

g si

des

are

prop

orti

onal

.

Als

o,�

W�

�P

,�X

��

Q,�

Y�

�R

,an

d �

Z�

�S

,so

corr

espo

ndi

ng

angl

es a

re c

ongr

uen

t.W

e ca

n c

oncl

ude

th

at

poly

gon

WX

YZ

�po

lygo

n P

QR

S.

Det

erm

ine

wh

eth

er e

ach

pai

r of

fig

ure

s is

sim

ilar

.If

they

are

sim

ilar

,giv

e th

era

tio

of c

orre

spon

din

g si

des

.

1.2.

yes;

�5 8�n

o

3.4.

yes;

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s;�5 4�

10y

8y

37�

37�

116�

27�

5x

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z12

z

2211

1610 equi

late

ral t

riang

les

15

9

12

18XW

ZY

10

6

8

12

QP S

R

20�

2�� 23

�2�

2023

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20�

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45�

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CA

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BY

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cises

Exer

cises

Exam

ple1

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ple1

Exam

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lenc

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eom

etry

Scal

e Fa

cto

rsW

hen

tw

o po

lygo

ns

are

sim

ilar

,th

e ra

tio

of t

he

len

gth

s of

cor

resp

ondi

ng

side

s is

cal

led

the

scal

e fa

ctor

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the

righ

t,�

AB

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8 cm

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uid

e a

nd I

nte

rven

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(con

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)

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s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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____

____

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ER

IOD

____

_

6-2

6-2

Th

e tw

o p

olyg

ons

are

sim

ilar

.Fin

d x

and

y.

Use

th

e co

ngr

uen

t an

gles

to

wri

te t

he

corr

espo

ndi

ng

vert

ices

in

ord

er.

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ST

��

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rite

pro

port

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s to

fin

d x

and

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x

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RP

N

M

�A

BC

��

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ind

th

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fact

or a

nd

fin

d t

he

len

gth

s of

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and

D�E�

.

AC

�3

�0

�3

and

CE

�9

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esc

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fact

or o

f �

CD

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BC

is 6

:3 o

r 2:

1.U

sin

g th

e D

ista

nce

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mu

la,

AB

��

1 �

9�

��

10�an

d B

C�

�4

�9

��

�13�

.Th

e le

ngt

hs

of t

he

side

s of

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DE

are

twic

e th

ose

of �

AB

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so D

C�

2(B

A)

or 2

�10�

and

DE

�2(

BC

) or

2�

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x

y

A( 0

, 0)

B( 1

, 3) C( 3

, 0)

E( 9

, 0)

D

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Eac

h p

air

of p

olyg

ons

is s

imil

ar.F

ind

xan

d y

.

1.2.

x�

6;y

�15

x�

2.25

;y

�5

3.4.

x�

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y�

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�24

;y

�27

5.In

Exa

mpl

e 2

abov

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int

Dh

as c

oord

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es (

5,6)

.Use

th

e D

ista

nce

For

mu

la t

o ve

rify

the

len

gth

s of

C �D�

and

D�E�

.

CD

�2�

10�,D

E�

2�13�

40

36x

30

18y

12

y �

1

18

24

x18

10y

4.5

x2.

5 5

9

12y8 10

x



Skil

ls P

ract

ice

Sim

ilar

Po

lyg

on

s

NA

ME

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____

____

____

____

____

____

____

____

____

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AT

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____

____

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6-2

6-2

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Lesson 6-2

Det

erm

ine

wh

eth

er e

ach

pai

r of

fig

ure

s is

sim

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.Ju

stif

y yo

ur

answ

er.

1.2.

�A

BC

��

DE

F;

�A

��

D,

rho

mbu

s P

QR

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rho

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s W

XY

Z;�

C�

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,an

d �

B�

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��

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,�R

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Y,

bec

ause

if t

wo

an

gle

s o

f o

ne

�S

��

Z;

tria

ng

le a

re c

on

gru

ent

to t

wo

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S Z��

� ZS WP ��

� 73 .5��

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ang

les

of

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con

d t

rian

gle

,th

en

the

thir

d a

ng

les

are

con

gru

ent;

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��B E

C F��

�C FDA �

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h p

air

of p

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imil

ar.W

rite

a s

imil

arit

y st

atem

ent,

and

fin

d x

,th

em

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re(s

) of

in

dic

ated

sid

e(s)

,an

d t

he

scal

e fa

ctor

.

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S�T�

and

S�U�

AB

CD

�E

FG

H;

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��

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r �

SU

T)

6�1 2� ;

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27;

12;

12;

�1 3�

5.W�

T�6.

T�S�

and

S�P�

PQ

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SP

MN

L

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9

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5

P

S

RU

V

WT

Q

x �

5

x �

5

9

3

44

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W

XY

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1213

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DA

1376

x G

EH

F

7.5

7.5

7.5

7.5

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SR

QP

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69

59�

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AC

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7

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Det

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wh

eth

er e

ach

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fig

ure

s is

sim

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.Ju

stif

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ur

answ

er.

1.2.

JKL

M�

QR

SP

;�

AB

C�

�U

VT

;�

A�

�U

,�

J�

�Q

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��

R,

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nd

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th

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hir

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ng

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heo

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an

d

� QJKR�

�� RK

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�L SM P��

� PMQJ �

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or

1.67

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C T��

�C TUA �

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Eac

h p

air

of p

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imil

ar.W

rite

a s

imil

arit

y st

atem

ent,

and

fin

d x

,th

em

easu

re(s

) of

in

dic

ated

sid

e(s)

,an

d t

he

scal

e fa

ctor

.

3.L �

M�an

d M�

N�4.

D�E�

and

D�F�

AB

CD

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MN

P;

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DE

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RD

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TE G

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ETRY

Tri

angl

e A

BC

has

ver

tice

s A

(0,0

),B

(�4,

0),a

nd

C(�

2,4)

.T

he

coor

din

ates

of

each

ver

tex

are

mu

ltip

lied

by

3 to

cre

ate

�A

EF

.Sh

ow t

hat

�A

EF

issi

mil

ar t

o �

AB

C.

AB

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BC

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2�5�

and

AE

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A ��

�1 3� ;�

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and

sin

ce B�

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F�,�

B�

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and

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F.S

ince

th

e co

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po

nd

ing

an

gle

s ar

e co

ng

ruen

t an

d t

he

corr

esp

on

din

g s

ides

are

pro

po

rtio

nal

,�A

BC

��

AE

F.

6.IN

TER

IOR

DES

IGN

Gra

ham

use

d th

e sc

ale

draw

ing

of h

is l

ivin

g ro

om t

o de

cide

wh

ere

to p

lace

fu

rnit

ure

.Fin

d th

e di

men

sion

s of

th

eli

vin

g ro

om i

f th

e sc

ale

in t

he

draw

ing

is 1

in

ch �

4.5

feet

.18

ft

by 1

1 ft

3 in

.4

in.

21 – 2 in.

612

40�

40�

x �

1

x �

3A

F

E

D

BC

14

10

x �

6

x �

9

CN

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Sim

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Po

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NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-2

6-2



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csS

imila

r P

oly

go

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

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____

____

__P

ER

IOD

____

_

6-2

6-2

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etry

Lesson 6-2

Pre-

Act

ivit

yH

ow d

o ar

tist

s u

se g

eom

etri

c p

atte

rns?

Rea

d th

e in

trod

uct

ion

to

Les

son

6-2

at

the

top

of p

age

289

in y

our

text

book

.•

Des

crib

e th

e fi

gure

s th

at h

ave

sim

ilar

sh

apes

. Sam

ple

an

swer

:L

igh

t an

d d

ark

fig

ure

s ar

e ar

ran

ged

in a

lter

nat

ing

pat

tern

s.F

igu

res

that

hav

e th

eir

feet

po

inti

ng

to

war

d t

he

cen

ter

of

the

circ

le h

ave

the

sam

e sh

ape

but

dif

fere

nt

size

s.•

Wh

at h

appe

ns

to t

he

figu

res

as y

our

eyes

mov

e fr

om t

he

cen

ter

to t

he

oute

r ed

ge?

Sam

ple

an

swer

:Th

e fi

gu

res

bec

om

e sm

alle

r.

Rea

din

g t

he

Less

on

1.C

ompl

ete

each

sen

ten

ce.

a.T

wo

poly

gon

s th

at h

ave

exac

tly

the

sam

e sh

ape,

but

not

nec

essa

rily

th

e sa

me

size

,ar

e .

b.

Tw

o po

lygo

ns

are

con

gru

ent

if t

hey

hav

e ex

actl

y th

e sa

me

shap

e an

d th

e sa

me

.c.

Tw

o po

lygo

ns

are

sim

ilar

if

thei

r co

rres

pon

din

g an

gles

are

an

d th

eir

corr

espo

ndi

ng

side

s ar

e .

d.

Tw

o po

lygo

ns

are

con

gru

ent

if t

hei

r co

rres

pon

din

g an

gles

are

an

d th

eir

corr

espo

ndi

ng

side

s ar

e .

e.T

he

rati

o of

th

e le

ngt

hs

of c

orre

spon

din

g si

des

of t

wo

sim

ilar

fig

ure

s is

cal

led

the

.f.

Mu

ltip

lyin

g th

e co

ordi

nat

es o

f al

l po

ints

of

a fi

gure

in

th

e co

ordi

nat

e pl

ane

by a

sca

le

fact

or t

o ge

t a

sim

ilar

fig

ure

is

call

ed a

.

g.If

tw

o po

lygo

ns a

re s

imila

r w

ith

a sc

ale

fact

or o

f 1,

then

the

pol

ygon

s ar

e .

2.D

eter

min

e w

het

her

eac

h s

tate

men

t is

alw

ays,

som

etim

es,o

r n

ever

tru

e.a.

Tw

o si

mil

ar t

rian

gles

are

con

gru

ent.

som

etim

esb

.T

wo

equ

ilat

eral

tri

angl

es a

re c

ongr

uen

t.so

met

imes

c.A

n e

quil

ater

al t

rian

gle

is s

imil

ar t

o a

scal

ene

tria

ngl

e.n

ever

d.

Tw

o re

ctan

gles

are

sim

ilar

.so

met

imes

e.T

wo

isos

cele

s ri

ght

tria

ngl

es a

re c

ongr

uen

t.so

met

imes

f.T

wo

isos

cele

s ri

ght

tria

ngl

es a

re s

imil

ar.

alw

ays

g.A

squ

are

is s

imil

ar t

o an

equ

ilat

eral

tri

angl

e.n

ever

h.

Tw

o ac

ute

tri

angl

es a

re s

imil

ar.

som

etim

esi.

Tw

o re

ctan

gles

in

wh

ich

th

e le

ngt

h i

s tw

ice

the

wid

th a

re s

imil

ar.

alw

ays

j.T

wo

con

gru

ent

poly

gon

s ar

e si

mil

ar.

alw

ays

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al v

ocab

ula

ry t

erm

is

to r

elat

e it

to

wor

dsu

sed

in e

very

day

life

.Th

e w

ord

scal

eh

as m

any

mea

nin

gs i

n E

ngl

ish

.Giv

e th

ree

phra

ses

that

in

clu

de t

he

wor

d sc

ale

in a

way

th

at i

s re

late

d to

pro

port

ion

s.S

amp

le a

nsw

er:

scal

e d

raw

ing

,sca

le m

od

el,s

cale

of

mile

s (o

n a

map

)

con

gru

ent

dila

tio

n

scal

e fa

cto

r

con

gru

ent

con

gru

ent

pro

po

rtio

nal

con

gru

ent

sizesi

mila

r

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eom

etry

Co

nst

ruct

ing

Sim

ilar

Po

lyg

on

sH

ere

are

fou

r st

eps

for

con

stru

ctin

g a

poly

gon

th

at i

s si

mil

ar t

o an

d w

ith

side

s tw

ice

as l

ong

as t

hos

e of

an

exi

stin

g po

lygo

n.

Ste

p 1

Ch

oose

an

y po

int

eith

er i

nsi

de o

r ou

tsid

e th

e po

lygo

n a

nd

labe

l it

O.

Ste

p 2

Dra

w r

ays

from

Oth

rou

gh e

ach

ver

tex

of t

he

poly

gon

.S

tep

3F

or v

erte

x V

,set

th

e co

mpa

ss t

o le

ngt

h O

V.T

hen

loc

ate

a n

ew p

oin

tV

�on

ray

OV

such

th

at V

V�

�O

V.T

hu

s,O

V�

�2(

OV

).S

tep

4R

epea

t S

tep

3 fo

r ea

ch v

erte

x.C

onn

ect

poin

ts V

�,W

�,X

�an

d Y

�to

form

th

e n

ew p

olyg

on.

Tw

o co

nst

ruct

ion

s of

pol

ygon

s si

mil

ar t

o an

d w

ith

sid

es t

wic

e th

ose

of V

WX

Yar

e sh

own

bel

ow.N

otic

e th

at t

he

plac

emen

t of

poi

nt

Odo

es n

ot a

ffec

t th

e si

zeor

sh

ape

of V

�W�X

�Y�,

only

its

loc

atio

n.

Tra

ce e

ach

pol

ygon

.Th

en c

onst

ruct

a s

imil

ar p

olyg

on w

ith

sid

es t

wic

e as

lon

g as

thos

e of

th

e gi

ven

pol

ygon

.S

ee s

tud

ents

’co

nst

ruct

ion

s.

1.2.

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6-2

6-2

3.E

xpla

in h

ow t

o co

nst

ruct

a s

imil

arpo

lygo

n w

ith

side

s th

ree

tim

es t

he l

engt

hof

th

ose

of p

olyg

on H

IJK

L.T

hen

do

the

con

stru

ctio

n.

See

stu

den

ts’w

ork

.

4.E

xpla

in h

ow t

o co

nst

ruct

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imil

ar

poly

gon

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tim

es t

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len

gth

of

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en d

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ee s

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6-3

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tify

Sim

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Tria

ng

les

Her

e ar

e th

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way

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sh

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tw

o tr

ian

gles

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ilari

tyTw

o an

gles

of

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ngle

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con

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o tw

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gles

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anot

her

tria

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imila

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The

mea

sure

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cor

resp

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ides

of

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tria

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e pr

opor

tiona

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S S

imila

rity

The

mea

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corr

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des

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noth

er t

riang

le,

and

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incl

uded

ang

les

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cong

ruen

t.

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erm

ine

wh

eth

er t

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tria

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es a

re s

imil

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32

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angl

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an b

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ts.

Stu

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nte

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ME

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t th

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me

tim

eth

at a

fla

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sts

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foot

-lon

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tal

l is

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e fl

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e su

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rm s

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7.W

rite

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hat

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use

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hei

ght

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ple

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6-3

6-3



Readin

g t

o L

earn

Math

em

ati

csS

imila

r Tri

ang

les

NA

ME

____

____

____

____

____

____

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6-3

6-3

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Pre-

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o en

gin

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use

geo

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ry?

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d th

e in

trod

uct

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to

Les

son

6-3

at

the

top

of p

age

298

in y

our

text

book

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Wh

at d

oes

it m

ean

to

say

that

tri

angu

lar

shap

es r

esu

lt i

n r

igid

con

stru

ctio

n?

Sam

ple

an

swer

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tri

ang

ula

r st

ruct

ure

will

no

tch

ang

e it

s sh

ape

or

size

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ou

pu

sh o

n it

s si

des

wit

ho

ut

bre

akin

g o

r b

end

ing

th

em.

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hat

wou

ld h

appe

n i

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e sh

apes

use

d in

th

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nst

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wer

equ

adri

late

rals

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amp

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nsw

er:T

he

stru

ctu

re m

igh

t co

llap

se,

bec

ause

qu

adri

late

rals

are

no

t ri

gid

.

Rea

din

g t

he

Less

on

1.S

tate

wh

eth

er e

ach

con

diti

on g

uar

ante

es t

hat

tw

o tr

ian

gles

are

con

gru

ent

or s

imil

ar.

If t

he

con

diti

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uar

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es t

hat

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ian

gles

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bot

h s

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th

ere

is n

ot e

nou

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rmat

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to

guar

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at t

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ill

beco

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uen

t or

sim

ilar

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te n

eith

er.

a.T

wo

side

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d th

e in

clu

ded

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e of

on

e tr

ian

gle

are

con

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ent

to t

wo

side

s an

d th

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clu

ded

angl

e of

th

e ot

her

tri

angl

e.co

ng

ruen

tb

.T

he

mea

sure

s of

all

th

ree

pair

s of

cor

resp

ondi

ng

side

s ar

e pr

opor

tion

al.

sim

ilar

c.T

wo

angl

es o

f on

e tr

ian

gle

are

con

gru

ent

to t

wo

angl

es o

f th

e ot

her

tri

angl

e.si

mila

rd

.T

wo

angl

es a

nd

a n

onin

clu

ded

side

of

one

tria

ngl

e ar

e co

ngr

uen

t to

tw

o an

gles

an

dth

e co

rres

pon

din

g n

onin

clu

ded

side

of

the

oth

er t

rian

gle.

con

gru

ent

e.T

he

mea

sure

s of

tw

o si

des

of a

tri

angl

e ar

e pr

opor

tion

al t

o th

e m

easu

res

of t

wo

corr

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ng

side

s of

an

oth

er t

rian

gle,

and

the

incl

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d an

gles

are

con

gru

ent.

sim

ilar

f.T

he

thre

e si

des

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ne

tria

ngl

e ar

e co

ngr

uen

t to

th

e th

ree

side

s of

th

e ot

her

tri

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ng

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The

thr

ee a

ngle

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one

tri

angl

e ar

e co

ngru

ent

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he t

hree

ang

les

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he o

ther

tri

angl

e.si

mila

rh

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ne

acu

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e of

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igh

t tr

ian

gle

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t to

on

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ute

an

gle

of a

not

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ri.

Th

e m

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the

mea

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not

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tri

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fol

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as a

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wri

te t

he

rati

o of

tto

ras

6:3

.Now

you

can

wri

te a

th

ree-

part

rat

io.

r:s:

t�

3:5

:6

Th

ere

is a

nu

mbe

r x

such

th

at r

�3

x,s

�5

x,an

d t

�6

x.S

ince

you

kn

ow t

he

peri

met

er,8

4,yo

u c

an u

se a

lgeb

ra t

o fi

nd

the

len

gth

s of

th

e si

des.

r�

s�

t�

843

x�

5x�

6x�

8414

x�

84x

�6

3x

�18

,5x

�30

,6x

�36

So

r�

18,s

�30

,an

d t

�36

.

Fin

d t

he

len

gth

s of

th

e si

des

of

each

tri

angl

e.

1.T

he

peri

met

er o

f a

tria

ngl

e is

75

un

its.

Th

e si

des

hav

e le

ngt

hs

a,b,

and

c.T

he

rati

o of

bto

ais

3:5

,an

d th

e ra

tio

of c

to a

is 7

:5.F

ind

the

len

gth

of

each

sid

e.

a�

25,b

�15

,c�

35

2.T

he

peri

met

er o

f a

tria

ngl

e is

88

un

its.

Th

e si

des

hav

e le

ngt

hs

d,e

,an

d f.

Th

e ra

tio

of e

to d

is 3

:1,a

nd

the

rati

o of

fto

eis

10:

9.F

ind

the

len

gth

of

each

sid

e.

d�

12,e

�36

,f�

40

3.T

he

peri

met

er o

f a

tria

ngl

e is

91

un

its.

Th

e si

des

hav

e le

ngt

hs

p,q,

and

r.T

he

rati

o of

pto

ris

3:1

,an

d th

e ra

tio

of q

to r

is 5

:2.F

ind

the

len

gth

of

each

sid

e.

p�

42,q

�35

,r�

14

4.T

he

peri

met

er o

f a

tria

ngl

e is

68

un

its.

Th

e si

des

hav

e le

ngt

hs

g,h

,an

d j.

Th

e ra

tio

of j

to g

is 2

:1,a

nd

the

rati

o of

hto

gis

5:4

.Fin

d th

e le

ngt

h o

f ea

ch s

ide.

g�

16,h

�20

,j�

32

5.W

rite

a p

robl

em s

imil

ar t

o th

ose

abov

e in

volv

ing

rati

os i

n t

rian

gles

.

See

stu

den

ts’w

ork

.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-3

6-3

Exam

ple

Exam

ple



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Par

alle

l Lin

es a

nd

Pro

po

rtio

nal

Par

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-4

6-4

©G

lenc

oe/M

cGra

w-H

ill31

3G

lenc

oe G

eom

etry

Lesson 6-4

Pro

po

rtio

nal

Par

ts o

f Tr

ian

gle

sIn

an

y tr

ian

gle,

a li

ne

para

llel

to

one

side

of

a tr

ian

gle

sepa

rate

s th

e ot

her

tw

o si

des

prop

orti

onal

ly.T

he

con

vers

e is

als

o tr

ue.

If X

and

Yar

e th

e m

idpo

ints

of

R �T�

and

S�T�

,th

en X�

Y�is

am

idse

gmen

tof

th

e tr

ian

gle.

Th

e T

rian

gle

Mid

segm

ent

Th

eore

mst

ates

th

at a

mid

segm

ent

is p

aral

lel

to t

he

thir

d si

de a

nd

is h

alf

its

len

gth

.

If X

Y� �

�|| R

S��

� ,th

en �R X

TX ��

� YSY T�

.

If �R X

TX ��

� YSY T�

,th

en X

Y� �

�|| R

S� �

� .

If X�

Y�is

a m

idse

gmen

t,th

en X

Y� �

�|| R

S� �

�an

d X

Y�

�1 2� RS

.

YS

T

R

X

In �

AB

C,E�

F�|| C�

B�.F

ind

x.

Sin

ce E�

F�|| C�

B�,�

FABF �

�� EA

E C�.

�x x� �2 22

��

�1 68 �

6x�

132

�18

x�

3696

�12

x8

�x

F

C

BA

18

x �

22

x �

2

6E

A t

rian

gle

has

ver

tice

sD

(3,6

),E

(�3,

�2)

,an

d F

(7,�

2).

Mid

segm

ent

G �H�

is p

aral

lel

to E�

F�.F

ind

th

e le

ngt

h o

f G �

H�.

G �H�

is a

mid

segm

ent,

so i

ts l

engt

h i

s on

e-h

alf

that

of

E �F�

.Poi

nts

Ean

d F

hav

e th

esa

me

y-co

ordi

nat

e,so

EF

�7

�(�

3) �

10.

Th

e le

ngt

h o

f m

idse

gmen

t G �

H�is

5.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d x

.

1.7

2.10

3.17

.5

4.8

5.12

6.5

7.In

Exa

mpl

e 2,

fin

d th

e sl

ope

of E�

F�an

d sh

ow t

hat

E�F�

|| G�H�

.

Th

e en

dp

oin

ts o

f G�

H�ar

e at

(0,

2) a

nd

(5,

2).T

he

slo

pe

of

G�H�

is 0

an

d t

he

slo

pe

of

E�F�

is 0

.G�H�

and

E�F�

are

par

alle

l.

3010

x �

10

x11

x �

12

x33

3010

24x

35x9

20

x

185

5x

7

©G

lenc

oe/M

cGra

w-H

ill31

4G

lenc

oe G

eom

etry

Div

ide

Seg

men

ts P

rop

ort

ion

ally

Wh

en

thre

e or

mor

e pa

rall

el l

ines

cu

t tw

o tr

ansv

ersa

ls,

they

sep

arat

e th

e tr

ansv

ersa

ls i

nto

pro

port

ion

al

part

s.If

th

e ra

tio

of t

he

part

s is

1,t

hen

th

e pa

rall

elli

nes

sep

arat

e th

e tr

ansv

ersa

ls i

nto

con

gru

ent

part

s.

If �

1|| �

2|| �

3,If

�4

|| �5

|| �6

and

then

�a b��

� dc � .�u v�

�1,

then

�w x��

1.

Ref

er t

o li

nes

�1,

� 2,a

nd

�3

abov

e.If

a�

3,b

�8,

and

c�

5,fi

nd

d.

� 1|| �

2|| �

3so

�3 8��

� d5 � .T

hen

3d

�40

an

d d

�13

�1 3� .

Fin

d x

and

y.

1.2.

x�

8x

�9

3.4.

x�

5;y

�2

y�

3�1 3�

5.6.

x�

12y

�3

1632

y �

3y

34

x �

4x

5

8y

� 2

y2x

� 4

3 x �

12y

� 2

3 y

2x �

6

x �

3

12

x �

12

123x5x

a

bd

uv

xw

ct

n m

s� 1

� 4� 5

� 6

� 2

� 3

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Par

alle

l Lin

es a

nd

Pro

po

rtio

nal

Par

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-4

6-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Par

alle

l Lin

es a

nd

Pro

po

rtio

nal

Par

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-4

6-4

©G

lenc

oe/M

cGra

w-H

ill31

5G

lenc

oe G

eom

etry

Lesson 6-4

1.If

JK

�7,

KH

�21

,an

d J

L�

6,2.

Fin

d x

and

TV

if R

U�

8,U

S�

14,

fin

d L

I.T

V�

x�

1 an

d V

S�

17.5

.

1811

;10

Det

erm

ine

wh

eth

er B�

C�|| D�

E�.

3.A

D�

15,D

B�

12,A

E�

10,a

nd

EC

�8

yes

4.B

D�

9,B

A�

27,a

nd

CE

is o

ne

thir

d of

EA

no

5.A

E�

30,A

C�

45,a

nd

AD

is t

wic

e D

B

yes

CO

OR

DIN

ATE

GEO

MET

RYF

or E

xerc

ises

6–8

,use

th

e fo

llow

ing

info

rmat

ion

.T

rian

gle

AB

Ch

as v

erti

ces

A(�

5,2)

,B(1

,8),

and

C(4

,2).

Poi

nt

Dis

th

e m

idpo

int

of A �

B�an

d E

is t

he

mid

poin

t of

A�C�

.

6.Id

enti

fy t

he

coor

din

ates

of

Dan

d E

.

D(�

2,5)

;E

(�0.

5,2)

7.S

how

th

at B�

C�is

par

alle

l to

D�E�

.

Sin

ce t

he

slo

pe

of

B�C�

is �

2 an

d t

he

slo

pe

of

D�E�

is �

2,B�

C�is

par

alle

l to

D�E�.

8.S

how

th

at D

E�

�1 2� BC

.

Th

e le

ng

th o

f D�

E�is

�11

.25

�o

r an

d t

he

len

gth

of

B�C�

is �

45�.

9.F

ind

xan

d y.

10.F

ind

xan

d y.

x�

6,y

�6.

5x

�2,

y�

4

3 – 2x �

2

x �

3

2 y �

1

3y �

52x

� 1

3 y �

8y

� 5

x �

7

�45�

�2

x

y

O

AC

B

D

E

CB

ED

A

R

UV

T

S

HL

KJ

I

©G

lenc

oe/M

cGra

w-H

ill31

6G

lenc

oe G

eom

etry

1.If

AD

�24

,DB

�27

,an

d E

B�

18,

2.F

ind

x,Q

T,a

nd

TR

if Q

T�

x�

6,fi

nd

CE

.S

R�

12,P

S�

27,a

nd

TR

�x

�4.

1612

;18

;8

Det

erm

ine

wh

eth

er J�

K�|| N�

M�.

3.J

N�

18,J

L�

30,K

M�

21,a

nd

ML

�35

no

4.K

M�

24,K

L�

44,a

nd

NL

��5 6� J

N

yes

CO

OR

DIN

ATE

GEO

MET

RYF

or E

xerc

ises

5 a

nd

6,u

se t

he

foll

owin

g in

form

atio

n.

Tri

angl

e E

FG

has

ver

tice

s E

(�4,

�1)

,F(2

,5),

and

G(2

,�1)

.Poi

nt

Kis

th

e m

idpo

int

of E �

G�an

d H

is t

he

mid

poin

t of

F�G�

.

5.S

how

th

at E�

F�is

par

alle

l to

K�H�

.

Th

e sl

op

e o

f E�

F�is

1 a

nd

th

e sl

op

e o

f K�

H�is

1,

so E�

F�is

par

alle

l to

K�H�

.

6.S

how

th

at K

H�

�1 2� EF

.

Th

e le

ng

th o

f K�

H�is

3�

2�,an

d t

he

len

gth

of

E�F�

is 6

�2�.

7.F

ind

xan

d y.

8.F

ind

xan

d y.

x�

4,y

�9

x�

�5 2� ,y

��9 4�

9.M

APS

Th

e di

stan

ce f

rom

Wil

min

gton

to

Ash

Gro

ve a

lon

g K

enda

ll i

s 82

0 fe

et a

nd

alon

g M

agn

olia

,660

fee

t.If

th

e di

stan

ce b

etw

een

Bee

ch a

nd

Ash

Gro

ve a

lon

g M

agn

olia

is

280

feet

,wh

at i

s th

e di

stan

ce b

etw

een

th

e tw

o st

reet

s al

ong

Ken

dall

?

abo

ut

348

ft

Ash

Grov

e

Beec

h

Mag

nolia

Kend

all

Wilm

ingt

on

4 – 3y �

2

3 x �

4x

� 1

4 y �

43x

� 4

2 – 3y �

31 – 3y

� 6

5 – 4x �

3

x

y

O

EGF H

K

NL

J

KM

S

Q

TR

P

D

EC BA

Pra

ctic

e (

Ave

rag

e)

Par

alle

l Lin

es a

nd

Pro

po

rtio

nal

Par

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-4

6-4



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csP

aral

lel L

ines

an

d P

rop

ort

ion

al P

arts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-4

6-4

©G

lenc

oe/M

cGra

w-H

ill31

7G

lenc

oe G

eom

etry

Lesson 6-4

Pre-

Act

ivit

yH

ow d

o ci

ty p

lan

ner

s u

se g

eom

etry

?

Rea

d th

e in

trod

uct

ion

to

Les

son

6-4

at

the

top

of p

age

307

in y

our

text

book

.

Use

a g

eom

etri

c id

ea t

o ex

plai

n w

hy

the

dist

ance

bet

wee

n C

hic

ago

Ave

nu

ean

d O

nta

rio

Str

eet

is s

hor

ter

alon

g M

ich

igan

Ave

nu

e th

an a

lon

g L

ake

Sh

ore

Dri

ve.S

amp

le a

nsw

er:T

he

sho

rtes

t d

ista

nce

bet

wee

n t

wo

par

alle

l lin

es is

th

e p

erp

end

icu

lar

dis

tan

ce.

Rea

din

g t

he

Less

on

1.P

rovi

de t

he

mis

sin

g w

ords

to

com

plet

e th

e st

atem

ent

of e

ach

th

eore

m.T

hen

sta

te t

he

nam

e of

th

e th

eore

m.

a.If

a l

ine

inte

rsec

ts t

wo

side

s of

a t

rian

gle

and

sepa

rate

s th

e si

des

into

cor

resp

ondi

ng

segm

ents

of

len

gth

s,th

en t

he

lin

e is

to

th

e

thir

d si

de.

Co

nver

se o

f th

e Tr

ian

gle

Pro

po

rtio

nal

ity

Th

eore

mb

.A

mid

segm

ent

of a

tri

angl

e is

to

on

e si

de o

f th

e tr

ian

gle

and

its

leng

th is

th

e le

ngth

of

that

sid

e.Tr

ian

gle

Mid

seg

men

t Th

eore

mc.

If a

line

is

to o

ne s

ide

of a

tri

angl

e an

d in

ters

ects

the

oth

er t

wo

side

s

in

dist

inct

poi

nts

,th

en i

t se

para

tes

thes

e si

des

into

of p

ropo

rtio

nal

len

gth

.Tr

ian

gle

Pro

po

rtio

nal

ity

Th

eore

m

2.R

efer

to

the

figu

re a

t th

e ri

ght.

a.N

ame

the

thre

e m

idse

gmen

ts o

f �

RS

T.

U�V�

,V�W�

,U�W�

b.

If R

S�

8,R

U�

3,an

d T

W�

5,fi

nd

the

len

gth

of

each

of

the

mid

segm

ents

.U

V�

5,V

W�

3,U

W�

4c.

Wh

at i

s th

e pe

rim

eter

of

�R

ST

?24

d.

Wh

at i

s th

e pe

rim

eter

of

�U

VW

?12

e.W

hat

are

th

e pe

rim

eter

s of

�R

UV

,�S

VW

,an

d �

TU

W?

12;

12;

12f.

How

are

th

e pe

rim

eter

s of

eac

h o

f th

e fo

ur

smal

l tr

ian

gles

rel

ated

to

the

peri

met

er o

fth

e la

rge

tria

ngl

e?T

he

per

imet

er o

f ea

ch s

mal

l tri

ang

le is

on

e-h

alf

the

per

imet

er o

f th

e la

rge

tria

ng

le.

g.W

ould

th

e re

lati

onsh

ip t

hat

you

fou

nd

in p

art

f ap

ply

to a

ny

tria

ngl

e in

wh

ich

th

em

idpo

ints

of

the

thre

e si

des

are

con

nec

ted?

yes

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

a n

ew m

ath

emat

ical

ter

m i

s to

rel

ate

it t

o ot

her

mat

hem

atic

alvo

cabu

lary

th

at y

ou a

lrea

dy k

now

.Wh

at i

s an

eas

y w

ay t

o re

mem

ber

the

defi

nit

ion

of

mid

segm

ent

usi

ng

oth

er g

eom

etri

c te

rms?

Sam

ple

an

swer

:A

mid

seg

men

tis

ase

gm

ent

that

co

nn

ects

th

e m

idp

oin

tso

f tw

o s

ides

of

a tr

ian

gle

.

R U

V W

T

S

seg

men

tstw

op

aral

lel

on

e-h

alf

par

alle

l

par

alle

lp

rop

ort

ion

al

©G

lenc

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ill31

8G

lenc

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eom

etry

Par

alle

l Lin

es a

nd

Co

ng

ruen

t P

arts

Th

ere

is a

th

eore

m s

tati

ng

that

if

thre

e pa

rall

el l

ines

cu

t of

f co

ngr

uen

t se

gmen

ts o

n o

ne

tran

sver

sal,

then

th

ey c

ut

off

con

gru

ent

segm

ents

on

an

y tr

ansv

ersa

l.T

his

can

be

show

n f

oran

y n

um

ber

of p

aral

lel

lin

es.T

he

foll

owin

g dr

afti

ng

tech

niq

ue

use

s th

is f

act

to d

ivid

e a

segm

ent

into

con

gru

ent

part

s.

A �B�

is t

o be

sep

arat

ed i

nto

fiv

e co

ngr

uen

t pa

rts.

Th

is c

an b

e do

ne

very

acc

ura

tely

wit

hou

t u

sin

g a

rule

r.A

ll t

hat

is

nee

ded

is a

com

pass

an

d a

piec

e of

not

eboo

k pa

per.

Ste

p 1

Hol

d th

e co

rner

of

a pi

ece

of n

oteb

ook

pape

r at

po

int

A.

Ste

p 2

Fro

m p

oin

t A

,dra

w a

seg

men

t al

ong

the

pape

r th

at i

s fi

ve s

pace

s lo

ng.

Mar

k w

her

e th

e li

nes

of

th

e n

oteb

ook

pape

r m

eet

the

segm

ent.

Lab

el

the

fift

h p

oin

t,P.

Ste

p 3

Dra

w P�

B�.T

hro

ugh

eac

h o

f th

e ot

her

mar

ks

on A �

P�,c

onst

ruct

a l

ine

para

llel

to

B�P�

.Th

e po

ints

wh

ere

thes

e li

nes

in

ters

ect

A �B�

wil

l di

vide

A �B�

into

fiv

e co

ngr

uen

t se

gmen

ts.

Use

a c

omp

ass

and

a p

iece

of

not

eboo

k p

aper

to

div

ide

each

seg

men

t in

to t

he

give

n n

um

ber

of

con

gru

ent

par

ts.

See

stu

den

t’s w

ork

.

1.si

x co

ngr

uen

t pa

rts

2.se

ven

con

gru

ent

part

sA

B

AB

A

P

B

AB

P

AB

AB

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-4

6-4



Stu

dy G

uid

e a

nd I

nte

rven

tion

Par

ts o

f S

imila

r Tri

ang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-5

6-5

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eom

etry

Lesson 6-5

Peri

met

ers

If t

wo

tria

ngl

es a

re s

imil

ar,t

hei

r pe

rim

eter

s h

ave

the

sam

e pr

opor

tion

as

the

corr

espo

ndi

ng

side

s.

If �

AB

C�

�R

ST

,th

en

�A RB S

� �B S

TC��

RATC

��

�A RB S�

��B S

TC ��

� RAC T�

.

Use

th

e d

iagr

am a

bov

e w

ith

�A

BC

��

RS

T.I

f A

B�

24 a

nd

RS

�15

,fi

nd

th

e ra

tio

of t

hei

r p

erim

eter

s.S

ince

�A

BC

��

RS

T,t

he

rati

o of

th

e pe

rim

eter

s of

�A

BC

and

�R

ST

is t

he

sam

e as

th

era

tio

of c

orre

spon

din

g si

des.

Th

eref

ore

��2 14 5�

��8 5�

Eac

h p

air

of t

rian

gles

is

sim

ilar

.Fin

d t

he

per

imet

er o

f th

e in

dic

ated

tri

angl

e.

1.�

XY

Z2.

�B

DE

3345

3.�

AB

C4.

�X

YZ

8455

.4

5.�

AB

C6.

�R

ST

5411

0

25 20

10

PT

SM

R

95

13

12

AD

E

C

B

56

8.7

13

10

XY

ZT

RS

18

202515

17 8

NB

AC

PM

36

9 11 CDE

A

B

44

24

2022

10

TR

XZ

YS

peri

met

er o

f �

AB

C�

��

peri

met

er o

f �

RS

T

AC

BS

RT

Exam

ple

Exam

ple

Exer

cises

Exer

cises

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eom

etry

Spec

ial S

egm

ents

of

Sim

ilar

Tria

ng

les

Wh

en t

wo

tria

ngl

es a

re s

imil

ar,

corr

espo

ndi

ng

alti

tude

s,an

gle

bise

ctor

s,an

d m

edia

ns

are

prop

orti

onal

to

the

corr

espo

ndi

ng

side

s.A

lso,

in a

ny

tria

ngl

e an

an

gle

bise

ctor

sep

arat

es t

he

oppo

site

sid

e in

to s

egm

ents

th

ath

ave

the

sam

e ra

tio

as t

he

oth

er t

wo

side

s of

th

e tr

ian

gle.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Par

ts o

f S

imila

r Tri

ang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-5

6-5

In t

he

figu

re,

�A

BC

��

XY

Z,w

ith

an

gle

bis

ecto

rs a

ssh

own

.Fin

d x

.

Sin

ce �

AB

C�

�X

YZ

,th

e m

easu

res

of t

he

angl

e bi

sect

ors

are

prop

orti

onal

to

the

mea

sure

s of

a p

air

of c

orre

spon

din

g si

des.

�A XB Y�

�� YB

WD �

�2 x4 ��

�1 80 �

10x

�24

(8)

10x

�19

2x

�19

.2

2410 D

W

Y

ZX

C

B

A

8x

S�U�

bis

ects

�R

ST

.Fin

d x

.

Sin

ce S �

U�is

an

an

gle

bise

ctor

,�R T

UU ��

�R TSS �

.

� 2x 0��

�1 35 0�

30x

�20

(15)

30x

�30

0x

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x2030

15

UT

S

R

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d x

for

each

pai

r of

sim

ilar

tri

angl

es.

1.2.

108

3.4.

2.25

8.75

5.6.

12�5 6�

15

2x �

2x

� 1

2816

17

11

x �

7

x

10

10x

8

87

3

3

4x

69

12x

x

36

1820



An

swer

s

Skil

ls P

ract

ice

Par

ts o

f S

imila

r Tri

ang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-5

6-5

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eom

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Lesson 6-5

Fin

d t

he

per

imet

er o

f th

e gi

ven

tri

angl

e.

1.�

JK

L,i

f �

JK

L�

�R

ST

,RS

�14

,2.

�D

EF

,if

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EF

��

AB

C,A

B�

27,

ST

�12

,TR

�10

,an

d L

J�

14B

C�

16,C

A�

25,a

nd

FD

�15

50.4

40.8

3.�

PQ

R,i

f �

PQ

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�L

MN

,LM

�16

,4.

�K

LM

,if

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LM

��

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H,F

G�

30,

MN

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,NL

�27

,an

d R

P�

18G

H�

38,H

F�

38,a

nd

KL

�24

3884

.8

Use

th

e gi

ven

in

form

atio

n t

o fi

nd

eac

h m

easu

re.

5.F

ind

FG

if �

RS

T�

�E

FG

,S�H�

is a

n

6.F

ind

MN

if �

AB

C�

�M

NP,

A�D�

is a

n

alti

tude

of

�R

ST

,F �J�

is a

n a

ltit

ude

of

alti

tude

of

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BC

,M�Q�

is a

n a

ltit

ude

of

�E

FG

,ST

�6,

SH

�5,

and

FJ

�7.

�M

NP

,AB

�24

,AD

�14

,and

MQ

�10

.5.

8.4

18

Fin

d x

.

7.�

HK

L�

�X

YZ

8.

8�1 3�

8.4

LH

K

18

10

Y

15

x

ZX

7

20

24

x

C

A

BD

PN

QM

JG

E

F

HT

R

S

GL K

M

F

H

LM

PQ

RN

A

BC

D

EF

RS

T

JK

L

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lenc

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ill32

2G

lenc

oe G

eom

etry

Fin

d t

he

per

imet

er o

f th

e gi

ven

tri

angl

e.

1.�

DE

F,i

f �

AB

C�

�D

EF

,AB

�36

,2.

�S

TU

,if

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TU

��

KL

M,K

L�

12,

BC

�20

,CA

�40

,an

d D

E�

35L

M�

31,M

K�

32,a

nd

US

�28

93.3�

65.6

25

Use

th

e gi

ven

in

form

atio

n t

o fi

nd

eac

h m

easu

re.

3.F

ind

PR

if �

JK

L�

�N

PR

,K�M�

is a

n

4.F

ind

ZY

if �

ST

U�

�X

YZ

,U�A�

is a

n

alti

tude

of

�J

KL

,P �T�

is a

n a

ltit

ude

of

alti

tude

of

�S

TU

,Z�B�

is a

n a

ltit

ude

of

�N

PR

,KL

�28

,KM

�18

,an

d �

XY

Z,U

T�

8.5,

UA

�6,

and

PT

�15

.75.

ZB

�11

.4.

24.5

16.1

5

Fin

d x

.

5.6.

13.5

16.8

PHO

TOG

RA

PHY

For

Exe

rcis

es 7

an

d 8

,use

th

e fo

llow

ing

info

rmat

ion

.F

ran

cin

e h

as a

cam

era

in w

hic

h t

he

dist

ance

fro

m t

he

len

s to

th

e fi

lm i

s 24

mil

lim

eter

s.

7.If

Fra

nci

ne

take

s a

full

-len

gth

ph

otog

raph

of

her

fri

end

from

a d

ista

nce

of

3 m

eter

s an

dth

e h

eigh

t of

her

fri

end

is 1

40 c

enti

met

ers,

wh

at w

ill

be t

he

hei

ght

of t

he

imag

e on

th

efi

lm?

(Hin

t:C

onve

rt t

o th

e sa

me

un

it o

f m

easu

re.)

11.2

mm

8.S

upp

ose

the

hei

ght

of t

he

imag

e on

th

e fi

lm o

f h

er f

rien

d is

15

mil

lim

eter

s.If

Fra

nci

ne

took

a f

ull

-len

gth

sh

ot,w

hat

was

th

e di

stan

ce b

etw

een

th

e ca

mer

a an

d h

er f

rien

d?2.

24 m

x28

2030

402x

� 1

25x

� 4

YX

BZ

TS

AU

MJ

L

K

TN

R

P

KS

LT

UM

B E

CF

DA

Pra

ctic

e (

Ave

rag

e)

Par

ts o

f S

imila

r Tri

ang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-5

6-5



Readin

g t

o L

earn

Math

em

ati

csP

arts

of

Sim

ilar T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-5

6-5

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3G

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eom

etry

Lesson 6-5

Pre-

Act

ivit

yH

ow i

s ge

omet

ry r

elat

ed t

o p

hot

ogra

ph

y?

Rea

d th

e in

trod

uct

ion

to

Les

son

6-5

at

the

top

of p

age

316

in y

our

text

book

.

•H

ow i

s si

mil

arit

y in

volv

ed i

n t

he

proc

ess

of m

akin

g a

phot

ogra

phic

pri

nt

from

a n

egat

ive?

Sam

ple

an

swer

:Th

e p

rin

t is

an

en

larg

ed v

ersi

on

of

the

neg

ativ

e.T

he

imag

e o

n t

he

pri

nt

is s

imila

r to

th

e im

age

on

the

neg

ativ

e,so

wh

en y

ou

loo

k at

th

e p

rin

t,yo

u w

ill s

eeex

actl

y th

e sa

me

thin

gs

in t

he

sam

e p

rop

ort

ion

s as

on

th

en

egat

ive,

just

larg

er.

•W

hy

do p

hot

ogra

pher

s pl

ace

thei

r ca

mer

as o

n t

ripo

ds?

Sam

ple

an

swer

:Th

ree

po

ints

det

erm

ine

a p

lan

e,so

th

e th

ree

feet

of

the

trip

od

are

co

pla

nar

an

d t

he

trip

od

will

no

t w

ob

ble

.

Rea

din

g t

he

Less

on

1.In

th

e fi

gure

,�R

ST

��

UV

W.C

ompl

ete

each

pr

opor

tion

in

volv

ing

the

len

gth

s of

seg

men

ts i

nth

is f

igu

re b

y re

plac

ing

the

ques

tion

mar

k.T

hen

iden

tify

th

e de

fin

itio

n o

r th

eore

m f

rom

th

e li

stbe

low

th

at t

he

com

plet

ed p

ropo

rtio

n i

llu

stra

tes.

i.D

efin

itio

n o

f co

ngr

uen

t po

lygo

ns

ii.

Def

init

ion

of

sim

ilar

pol

ygon

s

iii.

Pro

port

ion

al P

erim

eter

s T

heo

rem

iv.

An

gle

Bis

ecto

rs T

heo

rem

v.S

imil

ar t

rian

gles

hav

e co

rres

pon

din

g al

titu

des

prop

orti

onal

to

corr

espo

ndi

ng

side

s.

vi.

Sim

ilar

tri

angl

es h

ave

corr

espo

ndi

ng

med

ian

s pr

opor

tion

al t

o co

rres

pon

din

g si

des.

vii.

Sim

ilar

tria

ngle

s ha

ve c

orre

spon

ding

ang

le b

isec

tors

pro

port

iona

l to

corr

espo

ndin

g si

des.

a.�R

S�

S ?T�

TR

��

� URS V�

UV

�V

W�

WU

;iii

b.

� URWT �

��S ?X �

VY

;v

c.�R U

M N��

� V? W�S

T;

vid

.� UR

S V��

�S ?T �V

W;

ii

e.�R P

SP ��

� S? T�R

T;

ivf.

�U ?N ��

�U RW T�

RM

;vi

g.� WT

P Q��

�R ?T �U

W;

vii

h.

�U VWW �

�� Q? V�

UQ

;iv

Hel

pin

g Y

ou

Rem

emb

er

2.A

goo

d w

ay t

o re

mem

ber

a la

rge

amou

nt

of i

nfo

rmat

ion

is

to r

emem

ber

key

wor

ds.W

hat

key

wor

ds w

ill

hel

p yo

u r

emem

ber

the

feat

ure

s of

sim

ilar

tri

angl

es t

hat

are

pro

port

ion

alto

th

e le

ngt

hs

of t

he

corr

espo

ndi

ng

side

s?

per

imet

ers,

alti

tud

es,a

ng

le b

isec

tors

,med

ian

s

W

QN

YU

V

T

PM

XR

S

©G

lenc

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ill32

4G

lenc

oe G

eom

etry

Pro

po

rtio

ns

for

Sim

ilar T

rian

gle

sR

ecal

l th

at i

f a

lin

e cr

osse

s tw

o si

des

of a

tri

angl

e an

d is

par

alle

l to

th

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6-5

6-5



An

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6-6

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Ch

arac

teri

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s o

f Fr

acta

lsT

he

act

of r

epea

tin

g a

proc

ess

over

an

d ov

er,s

uch

as

fin

din

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thir

d of

a s

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then

a t

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d of

th

e n

ew s

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and

so o

n,i

s ca

lled

ite

rati

on.

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en t

he

proc

ess

of i

tera

tion

is

appl

ied

to s

ome

geom

etri

c fi

gure

s,th

e re

sult

s ar

e ca

lled

frac

tals

.For

obj

ects

su

ch a

s fr

acta

ls,w

hen

a p

orti

on o

f th

e ob

ject

has

th

e sa

me

shap

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char

acte

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ics

as t

he

enti

re o

bjec

t,th

e ob

ject

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be

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elf-

sim

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dia

gram

at

the

righ

t,n

otic

e th

at t

he

det

ails

at

each

sta

ge a

re s

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ls a

t S

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1.

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ollo

w t

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iter

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elow

to

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use

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6-6

6-6

Exer

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Skil

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6-6

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Lesson 6-6

Sta

ges

1 an

d 2

of

a fr

acta

l k

now

n a

s th

e C

anto

r se

t ar

e sh

own

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get

Sta

ge 2

,th

ese

gmen

t in

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ge 1

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tris

ecte

d,a

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th

e in

teri

or o

f th

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idd

le s

egm

ent

is r

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ed.

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e in

teri

or o

f a

segm

ent

is t

he

segm

ent

wit

h i

ts e

nd

poi

nts

rem

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.) T

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pro

cess

is

rep

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or s

ub

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raw

sta

ges

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d 4

of t

he

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tor

set.

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ow m

any

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ents

are

th

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in S

tage

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ge 4

?

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gm

ents

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men

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hat

hap

pen

s to

th

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ngt

h o

f th

e li

ne

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ents

in

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h s

tage

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e se

gm

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dec

reas

e in

len

gth

by

two

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s at

eac

h s

tag

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fter

in

fin

itel

y m

any

iter

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ns.

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he

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tor

set

self

-sim

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?

yes

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d t

he

valu

e of

eac

h e

xpre

ssio

n.T

hen

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at v

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e as

th

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ext

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serv

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6-6

6-6



An

swer

s

Readin

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o L

earn

Math

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csF

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6-6

6-6

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Lesson 6-6

Pre-

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ath

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fou

nd

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d th

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trod

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Les

son

6-6

at

the

top

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age

325

in y

our

text

book

.

Nam

e tw

o ob

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s fr

om n

atu

re o

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th

an b

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ter

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Ter

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ula

or

iv.

self

-sim

ilar

alge

brai

c eq

uat

ion

vv.

recu

rsio

nd

.th

e pr

oces

s of

rep

eati

ng

the

sam

e pr

oced

ure

ove

r an

d ov

erfo

rmu

laag

ain

iivi

.con

gru

ent

2.R

efer

to

the

thre

e pa

tter

ns

belo

w.

i. ii.

iii.

a.G

ive

the

spec

ial

nam

e fo

r ea

ch o

f th

e fr

acta

ls y

ou o

btai

n b

y co

nti

nu

ing

the

patt

ern

wit

hou

t en

d.i.

Ko

ch s

no

wfl

ake;

ii.S

ierp

insk

i tri

ang

le;

iii.K

och

cu

rve

b.

Wh

ich

of

thes

e fr

acta

ls a

re s

elf-

sim

ilar

?S

ierp

insk

i tri

ang

le,K

och

cu

rve

c.W

hic

h o

f th

ese

frac

tals

are

str

ictl

y se

lf-s

imil

ar?

Sie

rpin

ski t

rian

gle

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al t

erm

is

to r

elat

e it

to

ever

yday

En

glis

hw

ords

.Th

e w

ord

frac

tal

is r

elat

ed t

o fr

acti

onan

d fr

agm

ent.

Use

you

r di

ctio

nar

y to

fin

dat

lea

st o

ne

defi

nit

ion

for

eac

h o

f th

ese

wor

ds t

hat

you

th

ink

is r

elat

ed t

o th

e m

ean

ing

ofth

e w

ord

frac

tal

and

expl

ain

th

e co

nn

ecti

on.

Sam

ple

an

swer

:F

ract

ion

:a

smal

lp

art;

a d

isco

nn

ecte

d p

iece

;fr

agm

ent:

a p

iece

det

ach

ed o

r b

roke

n o

ff;

som

eth

ing

inco

mp

lete

;in

a f

ract

al,t

he

smal

ler

pie

ces

of

the

shap

ere

sem

ble

th

e w

ho

le.

©G

lenc

oe/M

cGra

w-H

ill33

0G

lenc

oe G

eom

etry

Th

e M

öb

ius

Str

ipA

Möb

ius

stri

p is

a s

peci

al s

urf

ace

wit

h o

nly

on

e si

de.I

t w

as d

isco

vere

d by

Au

gust

Fer

din

and

Möb

ius,

a G

erm

an a

stro

nom

er a

nd

mat

hem

atic

ian

.

1.T

o m

ake

a M

öbiu

s st

rip,

cut

a st

rip

of p

aper

abo

ut

16 i

nch

es l

ong

and

1in

ch w

ide.

Mar

k th

e en

ds w

ith

th

e le

tter

s A

,B,C

,an

d D

as s

how

n b

elow

.

Tw

ist

the

pape

r on

ce,c

onn

ecti

ng

Ato

Dan

d B

to C

.Tap

e th

e en

dsto

geth

er o

n b

oth

sid

es.

See

stu

den

ts’w

ork

.

2.U

se a

cra

yon

or

pen

cil

to s

had

e on

e si

de o

f th

e pa

per.

Sh

ade

arou

nd

the

stri

p u

nti

l yo

u g

et b

ack

to w

her

e yo

u s

tart

ed.W

hat

hap

pen

s?

Th

e en

tire

str

ip is

sh

aded

on

bo

th s

ides

.

3.W

hat

do

you

th

ink

wil

l h

appe

n i

f yo

u c

ut

the

Möb

ius

stri

p do

wn

th

em

iddl

e? T

ry i

t.

Inst

ead

of

two

loo

ps,

you

get

on

e lo

op

th

at is

tw

ice

as lo

ng

as t

he

ori

gin

al o

ne.

4.M

ake

anot

her

Möb

ius

stri

p.S

tart

ing

a th

ird

of t

he

way

in

fro

m o

ne

edge

,cu

t ar

oun

d th

e st

rip,

stay

ing

alw

ays

the

sam

e di

stan

ce i

n f

rom

th

e ed

ge.

Wh

at h

appe

ns?

Two

inte

rlo

ckin

g lo

op

s ar

e fo

rmed

.

5.S

tart

wit

h a

not

her

lon

g st

rip

of p

aper

.Tw

ist

the

pape

r tw

ice

and

con

nec

tth

e en

ds.W

hat

hap

pen

s w

hen

you

cu

t do

wn

th

e ce

nte

r of

th

is s

trip

?

Two

inte

rlo

ckin

g lo

op

s ar

e fo

rmed

.

6.S

tart

wit

h an

othe

r lo

ng s

trip

of

pape

r.T

wis

t th

e pa

per

thre

e ti

mes

and

conn

ect

the

ends

.Wha

t ha

ppen

s w

hen

you

cut

dow

n th

e ce

nter

of

this

str

ip?

On

e tw

o-s

ided

loo

p w

ith

a k

no

t is

fo

rmed

.

A B

C D

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-6

6-6



Chapter 6 Assessment Answer Key Form 1 Form 2APage 331 Page 332 Page 333


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

A

D

B

B

B

D

B

C

A

B

11.

12.

13.

14.

15.

16.

17.

18.

19.

B:

D

D

A

B

A

B

D

C

A

9.6

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

A

A

D

C

B

C

C

A

C

B


Chapter 6 Assessment Answer KeyForm 2A (continued) Form 2BPage 334 Page 335 Page 336

An

swer

s

11.

12.

13.

14.

15.

16.

17.

18.

19.

B:

D

D

B

B

C

A

A

A

C

32

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

A

C

B

D

B

B

D

B

B

C

11.

12.

13.

14.

15.

16.

17.

18.

19.

B:

D

A

C

D

B

A

D

C

D

26


Chapter 6 Assessment Answer KeyForm 2CPage 337 Page 338

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

3:7

Yes; corres. �are �.

30 ft

18

42.5 ft

6.3

18.2

No; the sides arenot proportional.

72

�PQR � �STR

2�12

�

12.

13.

14.

15.

16.

17.

18.

19.

B:

9.5

42

47

2

3

14.4

4

123

7


Chapter 6 Assessment Answer KeyForm 2DPage 339 Page 340

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

5:11

No; thecorrespondingsides are notproportional.

48 ft

�43

�

183�13

� ft

5.5

15

yes; SSSSimilarity

75

�XYZ � �XNM

7.5

12.

13.

14.

15.

16.

17.

18.

19.

B:

2.2

96

85

9.5

2

5�58

�

28

472

4.8


Chapter 6 Assessment Answer KeyForm 3Page 341 Page 342

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

4:3

No; corr. sides arenot proportional.

yes; SAS

30 cm by 7.5 cm

5.85

5

104 in.

82

48

�83

�

5

12.

13.

14.

15.

16.

17.

18.

19.

B:

11�37

�

7.2

�85

�

60

24.8

24 ft

9

10 in.

25.6

Chapter 6 Assessment Answer KeyPage 343, Open-Ended Assessment

Scoring Rubric


Score General Description Specific Criteria

• Shows thorough understanding of the concepts of ratios,properties of proportions, similar figures, similar triangles,dividing segments into parts, proportional parts of triangles,corresponding perimeters and altitudes, fractals, anditeration.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Figures are accurate and appropriate.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of ratios,properties of proportions, similar figures, similar triangles,dividing segments into parts, proportional parts oftriangles, corresponding perimeters and altitudes, fractals,and iteration.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Figures are mostly accurate and appropriate.• Satisfies all requirements of problems.

• Shows an understanding of most of the concepts of ratios,properties of proportions, similar figures, similar triangles,dividing segments into parts, proportional parts of triangles,corresponding perimeters and altitudes, fractals, anditeration.

• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Figures are mostly accurate.• Satisfies the requirements of most of the problems.

• Final computation is correct.• No written explanations or work is shown to substantiate

the final computation.• Figures may be accurate but lack detail or explanation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of the concepts ofratios, properties of proportions, similar figures, similartriangles, dividing segments into parts, proportional partsof triangles, corresponding perimeters and altitudes,fractals, and iteration.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Figures are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer may be given.

0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation

4 SuperiorA correct solution that is supported by well-developed, accurateexplanations

An

swer

s


Chapter 6 Assessment Answer KeyPage 343, Open-Ended Assessment

Sample Answers

In addition to the scoring rubric found on page A25, the following sample answers may be used as guidance in evaluating open-ended assessment items.

1. �53� � �

160� or �

135� � �

120�

2. a. 3:2:2 (Note: Check to be sure that the sum of any two sides of thetriangle is greater than the third side. For example, a ratio of 1:1:2would not be acceptable.)

b. 18, 12, 12

c. 12, 8, 8

3. �ABH � �CDI � �GFI � �ADG � �GDE � �CFE � �AGE by the AAPostulate.

4.

5. a.

Mark the midpoint of each side of �PQR. Connect these points to form�XYZ. Since each segment of �XYZ is a midsegment of �PQR, itslength will be half the corresponding side, and therefore the perimeterwill be half the perimeter of �PQR.

b.

c. Each of the triangles are similar by SSS (since their sides are half thelength of the preceding sides), so the figure is strictly self-similar andthus a fractal.

d. 6

P R

Q

X

Y Z

P R

Q

X

Y Z

FD

CA

B

1

1E

4–3

3–4


Chapter 6 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 344 Page 345 Page 346

An

swer

s

Quiz 2Page 345

Quiz 4Page 346

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

ratio

scale factor

midsegment

extremes

means

fractal

similar polygons

iteration

proportion

self-similar

Sample answer: Given

a proportion �ba

� � �dc

�,

ad � bc.

Sample answer: Eachpart of the figure, no

matter where it islocated or what size isselected, contains the

same figure as thewhole.

1.

2.

3.

4.

5.

1:30

33 in.

�BAC � �QPR or�ABC � �QPR by

SAS.

x � 4;scale factor 5:4

B

1.

2.

3.

4.

5.

yes; AA

No; the sides arenot proportional.

�ABD � �CDE or�ADB � �CDF;

8

22�12

� ft

6

1.

2.

3.

4.

5.

22�12

�

13�13

�

20

9

14

1.

2.

3.

4.

81

7, 27, 107

6, 30, 870

Stage 3


Chapter 6 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 347 Page 348

Part I

Part II

6.

7.

8.

9.

10.

No; SAS doesnot apply.yes; SAS

28; �83

�

120

3.6 in.

1.

2.

3.

4.

5.

B

D

C

D

B

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

(�5.5, �8)

11

true

6

perpendicular

y � 4x � 47

yes; TU � WX � �40�,UV � XY � �53�, and

VT � YW � �41�

�NMQ � �NPR;

L�M� � L�P�

x � 0.5, y � 2, z � 1

m�BEG � m�CEG

�H � �H, J�L� || K�M�,and �HJL � �HKM,thus �HJL � �HKM

by AA Similarity.

9

a � 1; b � 2


An

swer

s

Chapter 6 Assessment Answer KeyStandardized Test Practice

Page 349 Page 350

1.

2.

3.

4.

5.

6.

7.

8. E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D9. 10.

11. 12.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

13.

14.

15.

16.

�3

85.5

11 � x � 53

64.5 cm

1 1 9 2 5

1 2 8 1 6

Documents

Chapter 6 Resource Masters - Math Problem Solving - Homejaeproblemsolving.weebly.com/uploads/5/1/9/6/... · ©Glencoe/McGraw-Hill iv Glencoe Geometry Teacher’s Guide to Using the