Chapter 8 Resource Masters - farragutcareeracademy.org · ©Glencoe/McGraw-Hill iv Glencoe Geometry Teacher’s Guide to Using the Chapter 8 Resource Masters The Fast FileChapter

Chapter 8Resource 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 8 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: 0-07-860185-1 GeometryChapter 8 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 8-1Study Guide and Intervention . . . . . . . . 417–418Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 419Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 420Reading to Learn Mathematics . . . . . . . . . . 421Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 422







Chapter 8 AssessmentChapter 8 Test, Form 1 . . . . . . . . . . . . 459–460Chapter 8 Test, Form 2A . . . . . . . . . . . 461–462Chapter 8 Test, Form 2B . . . . . . . . . . . 463–464Chapter 8 Test, Form 2C . . . . . . . . . . . 465–466Chapter 8 Test, Form 2D . . . . . . . . . . . 467–468Chapter 8 Test, Form 3 . . . . . . . . . . . . 469–470Chapter 8 Open-Ended Assessment . . . . . . 471Chapter 8 Vocabulary Test/Review . . . . . . . 472Chapter 8 Quizzes 1 & 2 . . . . . . . . . . . . . . . 473Chapter 8 Quizzes 3 & 4 . . . . . . . . . . . . . . . 474Chapter 8 Mid-Chapter Test . . . . . . . . . . . . 475Chapter 8 Cumulative Review . . . . . . . . . . . 476Chapter 8 Standardized Test Practice . 477–478

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

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32

© Glencoe/McGraw-Hill iv Glencoe Geometry

Teacher’s Guide to Using theChapter 8 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 8 Resource Masters includes the core materials neededfor Chapter 8. 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 8-1.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 8-1.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 8Resources 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 458–459. 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 _____

88

© 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 8.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

diagonals

isosceles trapezoid

kite

median

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

parallelogram

rectangle

rhombus

square

trapezoid

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

88

Learning to Read MathematicsProof Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

88

© Glencoe/McGraw-Hill ix Glencoe Geometry

Proo

f Bu

ilderThis is a list of key theorems and postulates you will learn in Chapter 8. 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 8.1Interior Angle Sum Theorem

Theorem 8.2Exterior Angle Sum Theorem

Theorem 8.3

Theorem 8.4

Theorem 8.5

Theorem 8.7

Theorem 8.8


© Glencoe/McGraw-Hill x Glencoe Geometry

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 8.12

Theorem 8.13

Theorem 8.15

Theorem 8.17

Theorem 8.18

Theorem 8.19

Theorem 8.20

Learning to Read MathematicsProof Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

88

Study Guide and InterventionAngles of Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

© Glencoe/McGraw-Hill 417 Glencoe Geometry

Less

on

8-1

Sum of Measures of Interior Angles The segments that connect thenonconsecutive sides of a polygon are called diagonals. Drawing all of the diagonals fromone vertex of an n-gon separates the polygon into n � 2 triangles. The sum of the measuresof the interior angles of the polygon can be found by adding the measures of the interiorangles of those n � 2 triangles.

Interior Angle If a convex polygon has n sides, and S is the sum of the measures of its interior angles, Sum Theorem then S � 180(n � 2).

A convex polygon has 13 sides. Find the sum of the measuresof the interior angles.S � 180(n � 2)

� 180(13 � 2)� 180(11)� 1980

The measure of aninterior angle of a regular polygon is120. Find the number of sides.The number of sides is n, so the sum of themeasures of the interior angles is 120n.

S � 180(n � 2)120n � 180(n � 2)120n � 180n � 360

�60n � �360n � 6

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find the sum of the measures of the interior angles of each convex polygon.

1. 10-gon 2. 16-gon 3. 30-gon

4. 8-gon 5. 12-gon 6. 3x-gon

The measure of an interior angle of a regular polygon is given. Find the numberof sides in each polygon.

7. 150 8. 160 9. 175

10. 165 11. 168.75 12. 135

13. Find x.

7x �

(4x � 10)�

(4x � 5)�

(6x � 10)�

(5x � 5)�

A B

C

D

E


Sum of Measures of Exterior Angles There is a simple relationship among theexterior angles of a convex polygon.

Exterior Angle If a polygon is convex, then the sum of the measures of the exterior angles, Sum Theorem one at each vertex, is 360.

Find the sum of the measures of the exterior angles, one at eachvertex, of a convex 27-gon.For any convex polygon, the sum of the measures of its exterior angles, one at each vertex,is 360.

Find the measure of each exterior angle of regular hexagon ABCDEF.The sum of the measures of the exterior angles is 360 and a hexagon has 6 angles. If n is the measure of each exterior angle, then

6n � 360n � 60

Find the sum of the measures of the exterior angles of each convex polygon.

1. 10-gon 2. 16-gon 3. 36-gon

Find the measure of an exterior angle for each convex regular polygon.

4. 12-gon 5. 36-gon 6. 2x-gon

Find the measure of an exterior angle given the number of sides of a regularpolygon.

7. 40 8. 18 9. 12

10. 24 11. 180 12. 8

A B

C

DE

F

Study Guide and Intervention (continued)

Angles of Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticeAngles of Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1


Less

on

8-1


1. nonagon 2. heptagon 3. decagon


4. 108 5. 120 6. 150

Find the measure of each interior angle using the given information.

7. 8.

9. quadrilateral STUW with �S � �T, 10. hexagon DEFGHI with �U � �W, m�S � 2x � 16, �D � �E � �G � �H, �F � �I,m�U � x � 14 m�D � 7x, m�F � 4x

Find the measures of an interior angle and an exterior angle for each regularpolygon.

11. quadrilateral 12. pentagon 13. dodecagon

Find the measures of an interior angle and an exterior angle given the number ofsides of each regular polygon. Round to the nearest tenth if necessary.

14. 8 15. 9 16. 13

H G

FI

D ES T

UW

2x �

(2x � 20)� (3x � 10)�

(2x � 10)�

L M

NP

x �

x �

(2x � 15)�

(2x � 15)�

A B

CD



1. 11-gon 2. 14-gon 3. 17-gon


4. 144 5. 156 6. 160

Find the measure of each interior angle using the given information.

7. 8. quadrilateral RSTU with m�R � 6x � 4, m�S � 2x � 8

Find the measures of an interior angle and an exterior angle for each regularpolygon. Round to the nearest tenth if necessary.

9. 16-gon 10. 24-gon 11. 30-gon

Find the measures of an interior angle and an exterior angle given the number ofsides of each regular polygon. Round to the nearest tenth if necessary.

12. 14 13. 22 14. 40

15. CRYSTALLOGRAPHY Crystals are classified according to seven crystal systems. Thebasis of the classification is the shapes of the faces of the crystal. Turquoise belongs tothe triclinic system. Each of the six faces of turquoise is in the shape of a parallelogram.Find the sum of the measures of the interior angles of one such face.

R S

TU

x �

(2x � 15)� (3x � 20)�

(x � 15)�

J K

MN

Practice Angles of Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

Reading to Learn MathematicsAngles of Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1


Less

on

8-1

Pre-Activity How does a scallop shell illustrate the angles of polygons?

Read the introduction to Lesson 8-1 at the top of page 404 in your textbook.

• How many diagonals of the scallop shell shown in your textbook can bedrawn from vertex A?

• How many diagonals can be drawn from one vertex of an n-gon? Explainyour reasoning.

Reading the Lesson1. Write an expression that describes each of the following quantities for a regular n-gon. If

the expression applies to regular polygons only, write regular. If it applies to all convexpolygons, write all.

a. the sum of the measures of the interior angles

b. the measure of each interior angle

c. the sum of the measures of the exterior angles (one at each vertex)

d. the measure of each exterior angle

2. Give the measure of an interior angle and the measure of an exterior angle of each polygon.a. equilateral triangle c. squareb. regular hexagon d. regular octagon

3. Underline the correct word or phrase to form a true statement about regular polygons.a. As the number of sides increases, the sum of the measures of the interior angles

(increases/decreases/stays the same).b. As the number of sides increases, the measure of each interior angle

(increases/decreases/stays the same).c. As the number of sides increases, the sum of the measures of the exterior angles

(increases/decreases/stays the same).d. As the number of sides increases, the measure of each exterior angle

(increases/decreases/stays the same).e. If a regular polygon has more than four sides, each interior angle will be a(n)

(acute/right/obtuse) angle, and each exterior angle will be a(n) (acute/right/obtuse) angle.

Helping You Remember4. A good way to remember a new mathematical idea or formula is to relate it to something

you already know. How can you use your knowledge of the Angle Sum Theorem (for atriangle) to help you remember the Interior Angle Sum Theorem?


TangramsThe tangram puzzle is composed of seven pieces that form a square, as shown at theright. This puzzle has been a popularamusement for Chinese students for hundreds and perhaps thousands of years.

Make a careful tracing of the figure above. Cut out the pieces and rearrange them to form each figure below. Record each answer by drawing lines within each figure.

1. 2.

3. 4.

5. 6.

7. Create a different figure using the seven tangram pieces. Trace the outline.Then challenge another student to solve the puzzle.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

Study Guide and InterventionParallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2


Less

on

8-2

Sides and Angles of Parallelograms A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Here are fourimportant properties of parallelograms.

If PQRS is a parallelogram, then

The opposite sides of a P�Q� � S�R� and P�S� �Q�R�parallelogram are congruent.

The opposite angles of a �P � �R and �S � �Qparallelogram are congruent.

The consecutive angles of a �P and �S are supplementary; �S and �R are supplementary; parallelogram are supplementary. �R and �Q are supplementary; �Q and �P are supplementary.

If a parallelogram has one right If m�P � 90, then m�Q � 90, m�R � 90, and m�S � 90.angle, then it has four right angles.

If ABCD is a parallelogram, find a and b.A�B� and C�D� are opposite sides, so A�B� � C�D�.2a � 34a � 17

�A and �C are opposite angles, so �A � �C.8b � 112b � 14

Find x and y in each parallelogram.

1. 2.

3. 4.

5. 6.

72x

30x 150

2y60�

55� 5x�

2y�

6x� 12x�

3y�3y

12

6x

8y

88

6x�

4y�

3x�

BA

D C34

2a8b�

112�

P Q

RS

ExampleExample

ExercisesExercises


Diagonals of Parallelograms Two important properties of parallelograms deal with their diagonals.

If ABCD is a parallelogram, then:

The diagonals of a parallelogram bisect each other. AP � PC and DP � PB

Each diagonal separates a parallelogram �ACD � �CAB and �ADB � �CBDinto two congruent triangles.

Find x and y in parallelogram ABCD.The diagonals bisect each other, so AE � CE and DE � BE.6x � 24 4y � 18x � 4 y � 4.5

Find x and y in each parallelogram.

1. 2. 3.

4. 5. 6.

Complete each statement about �ABCD.Justify your answer.

7. �BAC �

8. D�E� �

9. �ADC �

10. A�D� ||

A B

CDE

x

17

4y3x� 2y

12

3x

30�10

y

2x�

60� 4y�

4x

2y28

3x4y

812

A BE

CD

6x

1824

4y

A B

CD

P


Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2

ExampleExample

ExercisesExercises

Skills PracticeParallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2


Less

on

8-2

Complete each statement about �DEFG. Justify your answer.

1. D�G� ||

2. D�E� �

3. G�H� �

4. �DEF �

5. �EFG is supplementary to .

6. �DGE �

ALGEBRA Use �WXYZ to find each measure or value.

7. m�XYZ � 8. m�WZY �

9. m�WXY � 10. a �

COORDINATE GEOMETRY Find the coordinates of the intersection of thediagonals of parallelogram HJKL given each set of vertices.

11. H(1, 1), J(2, 3), K(6, 3), L(5, 1) 12. H(�1, 4), J(3, 3), K(3, �2), L(�1, �1)

13. PROOF Write a paragraph proof of the theorem Consecutive angles in a parallelogramare supplementary.

D C

BA

YZ

XWA

50�30

2a

70�

?

?

?

?

?

? FG

ED H


Complete each statement about �LMNP. Justify your answer.

1. L�Q� �

2. �LMN �

3. �LMP �

4. �NPL is supplementary to .

5. L�M� �

ALGEBRA Use �RSTU to find each measure or value.

6. m�RST � 7. m�STU �

8. m�TUR � 9. b �

COORDINATE GEOMETRY Find the coordinates of the intersection of thediagonals of parallelogram PRYZ given each set of vertices.

10. P(2, 5), R(3, 3), Y(�2, �3), Z(�3, �1) 11. P(2, 3), R(1, �2), Y(�5, �7), Z(�4, �2)

12. PROOF Write a paragraph proof of the following.Given: �PRST and �PQVUProve: �V � �S

13. CONSTRUCTION Mr. Rodriquez used the parallelogram at the right to design a herringbone pattern for a paving stone. He will use the pavingstone for a sidewalk. If m�1 is 130, find m�2, m�3, and m�4.

1 2

34

P R

S

U V

Q

T

TU

SRB30� 23

4b � 1

25�

?

?

?

?

?P N

MQ

L

Practice Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2

Reading to Learn MathematicsParallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2


Less

on

8-2

Pre-Activity How are parallelograms used to represent data?


• What is the name of the shape of the top surface of each wedge of cheese?

• Are the three polygons shown in the drawing similar polygons? Explain yourreasoning.

Reading the Lesson

1. Underline words or phrases that can complete the following sentences to make statementsthat are always true. (There may be more than one correct choice for some of the sentences.)

a. Opposite sides of a parallelogram are (congruent/perpendicular/parallel).

b. Consecutive angles of a parallelogram are (complementary/supplementary/congruent).

c. A diagonal of a parallelogram divides the parallelogram into two(acute/right/obtuse/congruent) triangles.

d. Opposite angles of a parallelogram are (complementary/supplementary/congruent).

e. The diagonals of a parallelogram (bisect each other/are perpendicular/are congruent).

f. If a parallelogram has one right angle, then all of its other angles are(acute/right/obtuse) angles.

2. Let ABCD be a parallelogram with AB � BC and with no right angles.

a. Sketch a parallelogram that matches the descriptionabove and draw diagonal B�D�.

In parts b–f, complete each sentence.

b. A�B� || and A�D� || .

c. A�B� � and B�C� � .

d. �A � and �ABC � .

e. �ADB � because these two angles are

angles formed by the two parallel lines and and the

transversal .

f. �ABD � .

Helping You Remember

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

DA

CB


TessellationsA tessellation is a tiling pattern made of polygons. The pattern can be extended so that the polygonal tiles cover the plane completely withno gaps. A checkerboard and a honeycomb pattern are examples oftessellations. Sometimes the same polygon can make more than onetessellation pattern. Both patterns below can be formed from anisosceles triangle.

Draw a tessellation using each polygon.

1. 2. 3.

Draw two different tessellations using each polygon.

4. 5.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2

Study Guide and InterventionTests for Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-38-3


Less

on

8-3

Conditions for a Parallelogram There are many ways to establish that a quadrilateral is a parallelogram.

If: If:

both pairs of opposite sides are parallel, A�B� || D�C� and A�D� || B�C�,

both pairs of opposite sides are congruent, A�B� � D�C� and A�D� � B�C�,

both pairs of opposite angles are congruent, �ABC � �ADC and �DAB � �BCD,

the diagonals bisect each other, A�E� � C�E� and D�E� � B�E�,

one pair of opposite sides is congruent and parallel, A�B� || C�D� and A�B� � C�D�, or A�D� || B�C� and A�D� �B�C�,

then: the figure is a parallelogram. then: ABCD is a parallelogram.

Find x and y so that FGHJ is a parallelogram.FGHJ is a parallelogram if the lengths of the opposite sides are equal.

6x � 3 � 15 4x � 2y � 26x � 12 4(2) � 2y � 2x � 2 8 � 2y � 2

�2y � �6y � 3

Find x and y so that each quadrilateral is a parallelogram.

1. 2.

3. 4.

5. 6. 6y�

3x�

(x � y)�2x�

30�

24�

9x�6y

45�185x�5y�

25�

11x�

5y� 55�2x � 2

2y 8

12

J H

GF 6x � 3

154x � 2y 2

A B

CD

E

ExercisesExercises

ExampleExample


Parallelograms on the Coordinate Plane On the coordinate plane, the DistanceFormula and the Slope Formula can be used to test if a quadrilateral is a parallelogram.

Determine whether ABCD is a parallelogram.The vertices are A(�2, 3), B(3, 2), C(2, �1), and D(�3, 0).

Method 1: Use the Slope Formula, m � �yx

2

2

�

�

yx

1

1�.

slope of A�D� � ��2

3�

�

(0�3)

� � �31� � 3 slope of B�C� � �

23�

�

(�21)

� � �31� � 3

slope of A�B� � �3

2�

�

(�32)

� � ��15� slope of C�D� � �

2�

�

1(�

�

03)

� � ��15�

Opposite sides have the same slope, so A�B� || C�D� and A�D� || B�C�. Both pairs of opposite sidesare parallel, so ABCD is a parallelogram.

Method 2: Use the Distance Formula, d � �(x2 ��x1)2 �� ( y2 �� y1)2�.

AB � �(�2 �� 3)2 �� (3 ��2)2� � �25 ��1� or �26�

CD � �(2 � (��3))2�� (�1� � 0)2� � �25 ��1� or �26�

AD � �(�2 �� (�3))�2 � (3� � 0)2� � �1 � 9� or �10�

BC � �(3 � 2�)2 � (�2 � (��1))2� � �1 � 9� or �10�Both pairs of opposite sides have the same length, so ABCD is a parallelogram.

Determine whether a figure with the given vertices is a parallelogram. Use themethod indicated.

1. A(0, 0), B(1, 3), C(5, 3), D(4, 0); 2. D(�1, 1), E(2, 4), F(6, 4), G(3, 1);Slope Formula Slope Formula

3. R(�1, 0), S(3, 0), T(2, �3), U(�3, �2); 4. A(�3, 2), B(�1, 4), C(2, 1), D(0, �1);Distance Formula Distance and Slope Formulas

5. S(�2, 4), T(�1, �1), U(3, �4), V(2, 1); 6. F(3, 3), G(1, 2), H(�3, 1), I(�1, 4);Distance and Slope Formulas Midpoint Formula

7. A parallelogram has vertices R(�2, �1), S(2, 1), and T(0, �3). Find all possiblecoordinates for the fourth vertex.

x

y

O

C

A

D

B


Tests for Parallelograms

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ExampleExample

ExercisesExercises

Skills PracticeTests for Parallelograms

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Determine whether each quadrilateral is a parallelogram. Justify your answer.

1. 2.

3. 4.

COORDINATE GEOMETRY Determine whether a figure with the given vertices is aparallelogram. Use the method indicated.

5. P(0, 0), Q(3, 4), S(7, 4), Y(4, 0); Slope Formula

6. S(�2, 1), R(1, 3), T(2, 0), Z(�1, �2); Distance and Slope Formula

7. W(2, 5), R(3, 3), Y(�2, �3), N(�3, 1); Midpoint Formula

ALGEBRA Find x and y so that each quadrilateral is a parallelogram.

8. 9.

10. 11.

3x � 14

x � 20

3y � 2y � 20

(4x � 35)�

(3x � 10)�(2y � 5)�

(y � 15)�

2y � 11

y � 3 4x � 3

3x

x � 16

y � 192y

2x � 8


Determine whether each quadrilateral is a parallelogram. Justify your answer.

1. 2.

3. 4.

COORDINATE GEOMETRY Determine whether a figure with the given vertices is aparallelogram. Use the method indicated.

5. P(�5, 1), S(�2, 2), F(�1, �3), T(2, �2); Slope Formula

6. R(�2, 5), O(1, 3), M(�3, �4), Y(�6, �2); Distance and Slope Formula

ALGEBRA Find x and y so that each quadrilateral is a parallelogram.

7. 8.

9. 10.

11. TILE DESIGN The pattern shown in the figure is to consist of congruent parallelograms. How can the designer be certain that the shapes areparallelograms?

�4y � 2

x � 12

�2x � 6

y � 23�4x � 6

�6x

7y � 312y � 7

3y � 5�3x � 4

�4x � 22y � 8(5x � 29)�

(7x � 11)�(3y � 15)�

(5y � 9)�

118� 62�

118�62�

Practice Tests for Parallelograms

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Reading to Learn MathematicsTests for Parallelograms

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Pre-Activity How are parallelograms used in architecture?


Make two observations about the angles in the roof of the covered bridge.

Reading the Lesson1. Which of the following conditions guarantee that a quadrilateral is a parallelogram?

A. Two sides are parallel.B. Both pairs of opposite sides are congruent.C. The diagonals are perpendicular.D. A pair of opposite sides is both parallel and congruent.E. There are two right angles.F. The sum of the measures of the interior angles is 360.G. All four sides are congruent.H. Both pairs of opposite angles are congruent.I. Two angles are acute and the other two angles are obtuse.J. The diagonals bisect each other.K. The diagonals are congruent.L. All four angles are right angles.

2. Determine whether there is enough given information to know that each figure is aparallelogram. If so, state the definition or theorem that justifies your conclusion.

a. b.

c. d.

Helping You Remember3. A good way to remember a large number of mathematical ideas is to think of them in

groups. How can you state the conditions as one group about the sides of quadrilateralsthat guarantee that the quadrilateral is a parallelogram?


Tests for ParallelogramsBy definition, a quadrilateral is a parallelogram if and only if both pairs of opposite sidesare parallel. What conditions other than both pairs of opposite sides parallel will guaranteethat a quadrilateral is a parallelogram? In this activity, several possibilities will beinvestigated by drawing quadrilaterals to satisfy certain conditions. Remember that any test that seems to work is not guaranteed to work unless it can be formally proven.

Complete.

1. Draw a quadrilateral with one pair of opposite sides congruent.Must it be a parallelogram?

2. Draw a quadrilateral with both pairs of opposite sides congruent.Must it be a parallelogram?

3. Draw a quadrilateral with one pair of opposite sides parallel and the other pair of opposite sides congruent. Must it be a parallelogram?

4. Draw a quadrilateral with one pair of opposite sides both parallel and congruent. Must it be a parallelogram?

5. Draw a quadrilateral with one pair of opposite angles congruent.Must it be a parallelogram?

6. Draw a quadrilateral with both pairs of opposite angles congruent.Must it be a parallelogram?

7. Draw a quadrilateral with one pair of opposite sides parallel and one pair of opposite angles congruent. Must it be a parallelogram?

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Properties of Rectangles A rectangle is a quadrilateral with four right angles. Here are the properties of rectangles.

A rectangle has all the properties of a parallelogram.

• Opposite sides are parallel.• Opposite angles are congruent.• Opposite sides are congruent.• Consecutive angles are supplementary.• The diagonals bisect each other.

Also:

• All four angles are right angles. �UTS, �TSR, �SRU, and �RUT are right angles.• The diagonals are congruent. T�R� � U�S�

T S

RU

Q

In rectangle RSTUabove, US � 6x � 3 and RT � 7x � 2.Find x.The diagonals of a rectangle bisect eachother, so US � RT.

6x � 3 � 7x � 23 � x � 25 � x

In rectangle RSTUabove, m�STR � 8x � 3 and m�UTR �16x � 9. Find m�STR.�UTS is a right angle, so m�STR � m�UTR � 90.

8x � 3 � 16x � 9 � 9024x � 6 � 90

24x � 96x � 4

m�STR � 8x � 3 � 8(4) � 3 or 35

Example 1Example 1 Example 2Example 2

ExercisesExercises

ABCD is a rectangle.

1. If AE � 36 and CE � 2x � 4, find x.

2. If BE � 6y � 2 and CE � 4y � 6, find y.

3. If BC � 24 and AD � 5y � 1, find y.

4. If m�BEA � 62, find m�BAC.

5. If m�AED � 12x and m�BEC � 10x � 20, find m�AED.

6. If BD � 8y � 4 and AC � 7y � 3, find BD.

7. If m�DBC � 10x and m�ACB � 4x2� 6, find m� ACB.

8. If AB � 6y and BC � 8y, find BD in terms of y.

9. In rectangle MNOP, m�1 � 40. Find the measure of each numbered angle.

M N

OP

K

1 23

45

6

B C

DA

E


Prove that Parallelograms Are Rectangles The diagonals of a rectangle arecongruent, and the converse is also true.

If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

In the coordinate plane you can use the Distance Formula, the Slope Formula, andproperties of diagonals to show that a figure is a rectangle.

Determine whether A(�3, 0), B(�2, 3), C(4, 1),and D(3, �2) are the vertices of a rectangle.

Method 1: Use the Slope Formula.

slope of A�B� � ��2

3�

�

(0�3)

� � �31� or 3 slope of A�D� � �

3�

�

2(�

�

03)

� � ��62� or ��

13�

slope of C�D� � ��32��

41

� � ��

31� or 3 slope of B�C� � �

41�

�

(�32)

� � ��62� or ��

13�

Opposite sides are parallel, so the figure is a parallelogram. Consecutive sides areperpendicular, so ABCD is a rectangle.

Method 2: Use the Midpoint and Distance Formulas.

The midpoint of A�C� is ��32� 4�, �

0 �2

1��

12�, �

12�� and the midpoint of B�D� is

��22� 3�, �

3 �2

2��

12�, �

12��. The diagonals have the same midpoint so they bisect each other.

Thus, ABCD is a parallelogram.

AC � �(�3 �� 4)2 �� (0 ��1)2� � �49 ��1� or �50�BD � �(�2 �� 3)2 �� (3 ��(�2))2� � �25 ��25� or �50�The diagonals are congruent. ABCD is a parallelogram with diagonals that bisect eachother, so it is a rectangle.

Determine whether ABCD is a rectangle given each set of vertices. Justify youranswer.

1. A(�3, 1), B(�3, 3), C(3, 3), D(3, 1) 2. A(�3, 0), B(�2, 3), C(4, 5), D(3, 2)

3. A(�3, 0), B(�2, 2), C(3, 0), D(2, �2) 4. A(�1, 0), B(0, 2), C(4, 0), D(3, �2)

5. A(�1, �5), B(�3, 0), C(2, 2), D(4, �3) 6. A(�1, �1), B(0, 2), C(4, 3), D(3, 0)

7. A parallelogram has vertices R(�3, �1), S(�1, 2), and T(5, �2). Find the coordinates ofU so that RSTU is a rectangle.

x

y

O

D

B

A

E C


Rectangles

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ALGEBRA ABCD is a rectangle.

1. If AC � 2x � 13 and DB � 4x � 1, find x.

2. If AC � x � 3 and DB � 3x � 19, find AC.

3. If AE � 3x � 3 and EC � 5x � 15, find AC.

4. If DE � 6x � 7 and AE � 4x � 9, find DB.

5. If m�DAC � 2x � 4 and m�BAC � 3x � 1, find x.

6. If m�BDC � 7x � 1 and m�ADB � 9x � 7, find m�BDC.

7. If m�ABD � x2 � 7 and m�CDB � 4x � 5, find x.

8. If m�BAC � x2 � 3 and m�CAD � x � 15, find m�BAC.

PRST is a rectangle. Find each measure if m�1 � 50.

9. m�2 10. m�3

11. m�4 12. m�5

13. m�6 14. m�7

15. m�8 16. m�9

COORDINATE GEOMETRY Determine whether TUXY is a rectangle given each setof vertices. Justify your answer.

17. T(�3, �2), U(�4, 2), X(2, 4), Y(3, 0)

18. T(�6, 3), U(0, 6), X(2, 2), Y(�4, �1)

19. T(4, 1), U(3, �1), X(�3, 2), Y(�2, 4)

T

1 2 3 4

567 89

S

RP

D C

BAE


ALGEBRA RSTU is a rectangle.

1. If UZ � x � 21 and ZS � 3x � 15, find US.

2. If RZ � 3x � 8 and ZS � 6x � 28, find UZ.

3. If RT � 5x � 8 and RZ � 4x � 1, find ZT.

4. If m�SUT � 3x � 6 and m�RUS � 5x � 4, find m�SUT.

5. If m�SRT � x2 � 9 and m�UTR � 2x � 44, find x.

6. If m�RSU � x2 � 1 and m�TUS � 3x � 9, find m�RSU.

GHJK is a rectangle. Find each measure if m�1 � 37.

7. m�2 8. m�3

9. m�4 10. m�5

11. m�6 12. m�7

COORDINATE GEOMETRY Determine whether BGHL is a rectangle given each setof vertices. Justify your answer.

13. B(�4, 3), G(�2, 4), H(1, �2), L(�1, �3)

14. B(�4, 5), G(6, 0), H(3, �6), L(�7, �1)

15. B(0, 5), G(4, 7), H(5, 4), L(1, 2)

16. LANDSCAPING Huntington Park officials approved a rectangular plot of land for aJapanese Zen garden. Is it sufficient to know that opposite sides of the garden plot arecongruent and parallel to determine that the garden plot is rectangular? Explain.

K

2 1 5

436

7

J

HG

U T

SRZ

Practice Rectangles

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Pre-Activity How are rectangles used in tennis?


Are the singles court and doubles court similar rectangles? Explain youranswer.

Reading the Lesson1. Determine whether each sentence is always, sometimes, or never true.

a. If a quadrilateral has four congruent angles, it is a rectangle.

b. If consecutive angles of a quadrilateral are supplementary, then the quadrilateral is a rectangle.

c. The diagonals of a rectangle bisect each other.

d. If the diagonals of a quadrilateral bisect each other, the quadrilateral is a rectangle.

e. Consecutive angles of a rectangle are complementary.

f. Consecutive angles of a rectangle are congruent.

g. If the diagonals of a quadrilateral are congruent, the quadrilateral is a rectangle.

h. A diagonal of a rectangle bisects two of its angles.

i. A diagonal of a rectangle divides the rectangle into two congruent right triangles.

j. If the diagonals of a quadrilateral bisect each other and are congruent, thequadrilateral is a rectangle.

k. If a parallelogram has one right angle, it is a rectangle.

l. If a parallelogram has four congruent sides, it is a rectangle.

2. ABCD is a rectangle with AD � AB.Name each of the following in this figure.

a. all segments that are congruent to B�E�

b. all angles congruent to �1

c. all angles congruent to �7

d. two pairs of congruent triangles

Helping You Remember3. It is easier to remember a large number of geometric relationships and theorems if you

are able to combine some of them. How can you combine the two theorems aboutdiagonals that you studied in this lesson?

A1

2 3 4 5

678D

CBE


Counting Squares and RectanglesEach puzzle below contains many squares and/or rectangles.Count them carefully. You may want to label each region so youcan list all possibilities.

How many rectangles are in the figure at the right?

Label each small region with a letter. Then list each rectangle by writing the letters of regions it contains.

A, B, C, D, AB, CD, AC, BD, ABCD

There are 9 rectangles.

How many squares are in each figure?

1. 2. 3.

How many rectangles are in each figure?

1. 2. 3.

A B

C D

Enrichment

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Study Guide and InterventionRhombi and Squares

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Properties of Rhombi A rhombus is a quadrilateral with four congruent sides. Opposite sides are congruent, so a rhombus is also aparallelogram and has all of the properties of a parallelogram. Rhombi also have the following properties.

The diagonals are perpendicular. M�H� ⊥ R�O�

Each diagonal bisects a pair of opposite angles. M�H� bisects �RMO and �RHO.R�O� bisects �MRH and �MOH.

If the diagonals of a parallelogram are If RHOM is a parallelogram and R�O� ⊥ M�H�,perpendicular, then the figure is a rhombus. then RHOM is a rhombus.

In rhombus ABCD, m�BAC � 32. Find the measure of each numbered angle.ABCD is a rhombus, so the diagonals are perpendicular and �ABEis a right triangle. Thus m�4 � 90 and m�1 � 90 � 32 or 58. Thediagonals in a rhombus bisect the vertex angles, so m�1 � m�2.Thus, m�2 � 58.

A rhombus is a parallelogram, so the opposite sides are parallel. �BAC and �3 arealternate interior angles for parallel lines, so m�3 � 32.

ABCD is a rhombus.

1. If m�ABD � 60, find m�BDC. 2. If AE � 8, find AC.

3. If AB � 26 and BD � 20, find AE. 4. Find m�CEB.

5. If m�CBD � 58, find m�ACB. 6. If AE � 3x � 1 and AC � 16, find x.

7. If m�CDB � 6y and m�ACB � 2y � 10, find y.

8. If AD � 2x � 4 and CD � 4x � 4, find x.

9. a. What is the midpoint of F�H�?

b. What is the midpoint of G�J�?

c. What kind of figure is FGHJ? Explain.

d. What is the slope of F�H�?

e. What is the slope of G�J�?

f. Based on parts c, d, and e, what kind of figure is FGHJ ? Explain.

x

y

O

J

GF

H

C

A D

EB

CA

D

E

B

32�

1 2

34

M O

HRB

ExampleExample

ExercisesExercises


Properties of Squares A square has all the properties of a rhombus and all theproperties of a rectangle.

Find the measure of each numbered angle of square ABCD.

Using properties of rhombi and rectangles, the diagonals are perpendicular and congruent. �ABE is a right triangle, so m�1 � m�2 � 90.

Each vertex angle is a right angle and the diagonals bisect the vertex angles,so m�3 � m�4 � m�5 � 45.

Determine whether the given vertices represent a parallelogram, rectangle,rhombus, or square. Explain your reasoning.

1. A(0, 2), B(2, 4), C(4, 2), D(2, 0)

2. D(�2, 1), E(�1, 3), F(3, 1), G(2, �1)

3. A(�2, �1), B(0, 2), C(2, �1), D(0, �4)

4. A(�3, 0), B(�1, 3), C(5, �1), D(3, �4)

5. S(�1, 4), T(3, 2), U(1, �2), V(�3, 0)

6. F(�1, 0), G(1, 3), H(4, 1), I(2, �2)

7. Square RSTU has vertices R(�3, �1), S(�1, 2), and T(2, 0). Find the coordinates ofvertex U.

C

A D

1 2345

EB


Rhombi and Squares

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Use rhombus DKLM with AM � 4x, AK � 5x � 3, and DL � 10.

1. Find x. 2. Find AL.

3. Find m�KAL. 4. Find DM.

Use rhombus RSTV with RS � 5y � 2, ST � 3y � 6, and NV � 6.

5. Find y. 6. Find TV.

7. Find m�NTV. 8. Find m�SVT.

9. Find m�RST. 10. Find m�SRV.

COORDINATE GEOMETRY Given each set of vertices, determine whether �QRSTis a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning.

11. Q(3, 5), R(3, 1), S(�1, 1), T(�1, 5)

12. Q(�5, 12), R(5, 12), S(�1, 4), T(�11, 4)

13. Q(�6, �1), R(4, �6), S(2, 5), T(�8, 10)

14. Q(2, �4), R(�6, �8), S(�10, 2), T(�2, 6)

R V

NS T

M L

AD K


Use rhombus PRYZ with RK � 4y � 1, ZK � 7y � 14,PK � 3x � 1, and YK � 2x � 6.

1. Find PY. 2. Find RZ.

3. Find RY. 4. Find m�YKZ.

Use rhombus MNPQ with PQ � 3�2�, PA � 4x � 1, and AM � 9x � 6.

5. Find AQ. 6. Find m�APQ.

7. Find m�MNP. 8. Find PM.

COORDINATE GEOMETRY Given each set of vertices, determine whether �BEFGis a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning.

9. B(�9, 1), E(2, 3), F(12, �2), G(1, �4)

10. B(1, 3), E(7, �3), F(1, �9), G(�5, �3)

11. B(�4, �5), E(1, �5), F(�7, �1), G(�2, �1)

12. TESSELATIONS The figure is an example of a tessellation. Use a ruler or protractor to measure the shapes and then name the quadrilaterals used to form the figure.

AN

M Q

P

K

P

YR

Z

Practice Rhombi and Squares

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Reading to Learn MathematicsRhombi and Squares

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Pre-Activity How can you ride a bicycle with square wheels?


If you draw a diagonal on the surface of one of the square wheels shown inthe picture in your textbook, how can you describe the two triangles that areformed?

Reading the Lesson

1. Sketch each of the following.

a. a quadrilateral with perpendicular b. a quadrilateral with congruent diagonals that is not a rhombus diagonals that is not a rectangle

c. a quadrilateral whose diagonals are perpendicular and bisect each other,but are not congruent

2. List all of the following special quadrilaterals that have each listed property:parallelogram, rectangle, rhombus, square.

a. The diagonals are congruent.

b. Opposite sides are congruent.

c. The diagonals are perpendicular.

d. Consecutive angles are supplementary.

e. The quadrilateral is equilateral.

f. The quadrilateral is equiangular.

g. The diagonals are perpendicular and congruent.

h. A pair of opposite sides is both parallel and congruent.

3. What is the common name for a regular quadrilateral? Explain your answer.


4. A good way to remember something is to explain it to someone else. Suppose that yourclassmate Luis is having trouble remembering which of the properties he has learned inthis chapter apply to squares. How can you help him?


Creating Pythagorean PuzzlesBy drawing two squares and cutting them in a certain way, youcan make a puzzle that demonstrates the Pythagorean Theorem.A sample puzzle is shown. You can create your own puzzle byfollowing the instructions below.

1. Carefully construct a square and label the length of a side as a. Then construct a smaller square to the right of it and label the length of a side as b,as shown in the figure above. The bases should be adjacent and collinear.

2. Mark a point X that is b units from the left edge of the larger square. Then draw the segments from the upper left corner of the larger square to point X, and from point X to the upper right corner of the smaller square.

3. Cut out and rearrange your five pieces to form a larger square. Draw a diagram to show your answer.

4. Verify that the length of each side is equal to

�a2 � b�2�.

a

a

b X

b

b

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Properties of Trapezoids A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called basesand the nonparallel sides are called legs. If the legs are congruent,the trapezoid is an isosceles trapezoid. In an isosceles trapezoid both pairs of base angles are congruent. STUR is an isosceles trapezoid.

S�R� � T�U�; �R � �U, �S ��T

The vertices of ABCD are A(�3, �1), B(�1, 3),C(2, 3), and D(4, �1). Verify that ABCD is a trapezoid.

slope of A�B� � ��

31�

�

(�(�

13))

� � �42� � 2

slope of A�D� � ��

41�

�

(�(�

31))

� � �07� � 0

slope of B�C� � �2

3�

�

(�31)

� � �03� � 0

slope of C�D� � ��41��

23

� � ��24� � �2

Exactly two sides are parallel, A�D� and B�C�, so ABCD is a trapezoid. AB � CD, so ABCD isan isosceles trapezoid.

In Exercises 1–3, determine whether ABCD is a trapezoid. If so, determinewhether it is an isosceles trapezoid. Explain.

1. A(�1, 1), B(2, 1), C(3, �2), and D(2, �2)

2. A(3, �3), B(�3, �3), C(�2, 3), and D(2, 3)

3. A(1, �4), B(�3, �3), C(�2, 3), and D(2, 2)

4. The vertices of an isosceles trapezoid are R(�2, 2), S(2, 2), T(4, �1), and U(�4, �1).Verify that the diagonals are congruent.

x

y

O

B

DA

C

T

R U

base

base

Sleg

AB � �(�3 �� (�1))�2 � (��1 � 3�)2�� 4 � 1�6� � �20� � 2�5�

CD � �(2 � 4�)2 � (�3 � (��1))2�� 4 � 1�6� � �20� � 2�5�

ExercisesExercises

ExampleExample


Medians of Trapezoids The median of a trapezoid is the segment that joins the midpoints of the legs. It is parallel to the bases, and its length is one-half the sum of the lengths of the bases.

In trapezoid HJKL, MN � �12�(HJ � LK ).

M�N� is the median of trapezoid RSTU. Find x.

MN � �12�(RS � UT)

30 � �12�(3x � 5 � 9x � 5)

30 � �12�(12x)

30 � 6x5 � x

M�N� is the median of trapezoid HJKL. Find each indicated value.

1. Find MN if HJ � 32 and LK � 60.

2. Find LK if HJ � 18 and MN � 28.

3. Find MN if HJ � LK � 42.

4. Find m�LMN if m�LHJ � 116.

5. Find m�JKL if HJKL is isosceles and m�HLK � 62.

6. Find HJ if MN � 5x � 6, HJ � 3x � 6, and LK � 8x.

7. Find the length of the median of a trapezoid with vertices A(�2, 2), B(3, 3), C(7, 0), and D(�3, �2).

L K

NM

H J

U T

NM

R 3x � 5

30

9x � 5

S

L K

NM

H J


Trapezoids

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ExampleExample

ExercisesExercises

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COORDINATE GEOMETRY ABCD is a quadrilateral with vertices A(�4, �3),B(3, �3), C(6, 4), D(�7, 4).

1. Verify that ABCD is a trapezoid.

2. Determine whether ABCD is an isosceles trapezoid. Explain.

COORDINATE GEOMETRY EFGH is a quadrilateral with vertices E(1, 3), F(5, 0),G(8, �5), H(�4, 4).

3. Verify that EFGH is a trapezoid.

4. Determine whether EFGH is an isosceles trapezoid. Explain.

COORDINATE GEOMETRY LMNP is a quadrilateral with vertices L(�1, 3), M(�4, 1),N(�6, 3), P(0, 7).

5. Verify that LMNP is a trapezoid.

6. Determine whether LMNP is an isosceles trapezoid. Explain.

ALGEBRA Find the missing measure(s) for the given trapezoid.

7. For trapezoid HJKL, S and T are 8. For trapezoid WXYZ, P and Q aremidpoints of the legs. Find HJ. midpoints of the legs. Find WX.

9. For trapezoid DEFG, T and U are 10. For isosceles trapezoid QRST, find themidpoints of the legs. Find TU, m�E, length of the median, m�Q, and m�S.and m�G.

ST

RQ125�

60

25

D E

FG

T U85�

35�42

14

W X

YZ

P Q12

19

H J

KL

S T72

86


COORDINATE GEOMETRY RSTU is a quadrilateral with vertices R(�3, �3), S(5, 1),T(10, �2), U(�4, �9).

1. Verify that RSTU is a trapezoid.

2. Determine whether RSTU is an isosceles trapezoid. Explain.

COORDINATE GEOMETRY BGHJ is a quadrilateral with vertices B(�9, 1), G(2, 3),H(12, �2), J(�10, �6).

3. Verify that BGHJ is a trapezoid.

4. Determine whether BGHJ is an isosceles trapezoid. Explain.

ALGEBRA Find the missing measure(s) for the given trapezoid.

5. For trapezoid CDEF, V and Y are 6. For trapezoid WRLP, B and C aremidpoints of the legs. Find CD. midpoints of the legs. Find LP.

7. For trapezoid FGHI, K and M are 8. For isosceles trapezoid TVZY, find the midpoints of the legs. Find FI, m�F, length of the median, m�T, and m�Z.and m�I.

9. CONSTRUCTION A set of stairs leading to the entrance of a building is designed in theshape of an isosceles trapezoid with the longer base at the bottom of the stairs and theshorter base at the top. If the bottom of the stairs is 21 feet wide and the top is 14, findthe width of the stairs halfway to the top.

10. DESK TOPS A carpenter needs to replace several trapezoid-shaped desktops in aclassroom. The carpenter knows the lengths of both bases of the desktop. What othermeasurements, if any, does the carpenter need?

V

Y Z

T 60�34

5

H

K M

G

IF

140� 125�

21

36

W R

LP

B C42

66F E

DC

V Y28

18

Practice Trapezoids

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Pre-Activity How are trapezoids used in architecture?


How might trapezoids be used in the interior design of a home?

Reading the Lesson

1. In the figure at the right, EFGH is a trapezoid, I is the midpoint of F�E�, and J is the midpoint of G�H�. Identify each of the following segments or angles in the figure.

a. the bases of trapezoid EFGH

b. the two pairs of base angles of trapezoid EFGH

c. the legs of trapezoid EFGH

d. the median of trapezoid EFGH

2. Determine whether each statement is true or false. If the statement is false, explain why.

a. A trapezoid is a special kind of parallelogram.

b. The diagonals of a trapezoid are congruent.

c. The median of a trapezoid is parallel to the legs.

d. The length of the median of a trapezoid is the average of the length of the bases.

e. A trapezoid has three medians.

f. The bases of an isosceles trapezoid are congruent.

g. An isosceles trapezoid has two pairs of congruent angles.

h. The median of an isosceles trapezoid divides the trapezoid into two smaller isoscelestrapezoids.


3. A good way to remember a new geometric theorem is to relate it to one you alreadyknow. Name and state in words a theorem about triangles that is similar to the theoremin this lesson about the median of a trapezoid.

HE

JIGF


Quadrilaterals in Construction

Quadrilaterals are often used in construction work.

1. The diagram at the right represents a roof frame and shows many quadrilaterals. Find the following shapes in the diagram and shade in their edges.

a. isosceles triangle

b. scalene triangle

c. rectangle

d. rhombus

e. trapezoid (not isosceles)

f. isosceles trapezoid

2. The figure at the right represents a window. The wooden part between the panes of glass is 3 inches wide. The frame around the outer edge is 9 inches wide. The outside measurements of the frame are 60 inches by 81 inches. The height of the top and bottom panes is the same. The top three panes are the same size.

a. How wide is the bottom pane of glass?

b. How wide is each top pane of glass?

c. How high is each pane of glass?

3. Each edge of this box has been reinforced with a piece of tape. The box is 10 inches high, 20 inches wide, and 12 inches deep.What is the length of the tape that has been used?

10 in.

12 in.

20 in.

60 in.

81 in.

9 in.

3 in.

3 in.

3 in.

9 in.

9 in.

jack rafters

hip rafter

common rafters

Roof Frame

ridge boardvalley rafter

plate

Enrichment

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Study Guide and InterventionCoordinate Proof with Quadrilaterals

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Position Figures Coordinate proofs use properties of lines and segments to provegeometric properties. The first step in writing a coordinate proof is to place the figure on thecoordinate plane in a convenient way. Use the following guidelines for placing a figure onthe coordinate plane.

1. Use the origin as a vertex, so one set of coordinates is (0, 0), or use the origin as the center of the figure.

2. Place at least one side of the quadrilateral on an axis so you will have some zero coordinates.

3. Try to keep the quadrilateral in the first quadrant so you will have positive coordinates.

4. Use coordinates that make the computations as easy as possible. For example, use even numbers if you are going to be finding midpoints.

Position and label a rectangle with sides aand b units long on the coordinate plane.

• Place one vertex at the origin for R, so one vertex is R(0, 0).• Place side R�U� along the x-axis and side R�S� along the y-axis,

with the rectangle in the first quadrant.• The sides are a and b units, so label two vertices S(0, a) and

U(b, 0).• Vertex T is b units right and a units up, so the fourth vertex is T(b, a).

Name the missing coordinates for each quadrilateral.

1. 2. 3.

Position and label each quadrilateral on the coordinate plane.

4. square STUV with 5. parallelogram PQRS 6. rectangle ABCD withside s units with congruent diagonals length twice the width

x

y

D(2a, 0)

C(2a, a)B(0, a)

A(0, 0)x

y

S(a, 0)

R(a, b)Q(0, b)

P(0, 0)x

y

V(s, 0)

U(s, s)T(0, s)

S(0, 0)

x

y

D(a, 0)

C(?, c)B(b, ?)

A(0, 0)x

y

Q(a, 0)

P(a � b, c)N(?, ?)

M(0, 0)x

y

B(c, 0)

C(?, ?)D(0, c)

A(0, 0)

x

y

U(b, 0)

T(b, a)S(0, a)

R(0, 0)

ExercisesExercises

ExampleExample


Prove Theorems After a figure has been placed on the coordinate plane and labeled, acoordinate proof can be used to prove a theorem or verify a property. The Distance Formula,the Slope Formula, and the Midpoint Theorem are often used in a coordinate proof.

Write a coordinate proof to show that the diagonals of a square are perpendicular.The first step is to position and label a square on the coordinate plane. Place it in the first quadrant, with one side on each axis.Label the vertices and draw the diagonals.

Given: square RSTUProve: S�U� ⊥ R�T�

Proof: The slope of S�U� is �0a��

a0� � �1, and the slope of R�T� is �aa

��

00� � 1.

The product of the two slopes is �1, so S�U� ⊥ R�T�.

Write a coordinate proof to show that the length of the median of a trapezoid ishalf the sum of the lengths of the bases.

D(2a, 0)

B(2b, 2c)

M N

C(2d, 2c)

A(0, 0)

x

y

U(a, 0)

T(a, a)S(0, a)

R(0, 0)


Coordinate Proof With Quadrilaterals

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ExerciseExercise

Skills PracticeCoordinate Proof with Quadrilaterals

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1. rectangle with length 2a units and 2. isosceles trapezoid with height a units,height a units bases c � b units and b � c units


3. rectangle 4. rectangle

5. parallelogram 6. isosceles trapezoid

Position and label the figure on the coordinate plane. Then write a coordinateproof for the following.

7. The segments joining the midpoints of the sides of a rhombus form a rectangle.

B(0, 2b)

D(0, �2b)P

NM

Q

C(2a, 0)

A(�2a, 0)

x

y

T(b, 0)

W(?, c)Y(?, ?)

S(a, 0)Ox

y

S(a, ?)

R(?, ?)P(b, c)

T(0, 0)O

x

y

E(5b, 0)

F(5b, 2b)G(?, ?)

D(0, 0)Ox

y

U(c, 0)

C(?, ?)D(0, a)

A(0, 0)O

B(b � c, 0)

B(b, a) C(c, a)

A(0, 0)B(2a, 0)

D(0, a) C(2a, a)

A(0, 0)



1. parallelogram with side length b units 2. isosceles trapezoid with height b units,and height a units bases 2c � a units and 2c � a units


3. parallelogram 4. isosceles trapezoid

Position and label the figure on the coordinate plane. Then write a coordinateproof for the following.

5. The opposite sides of a parallelogram are congruent.

6. THEATER A stage is in the shape of a trapezoid. Write a coordinate proof to prove that T�R� and S�F� are parallel.

x

y

T(10, 0)

S(30, 25)F(0, 25)

R(20, 0)O

B(a, 0)

D(b, c) C(a � b, c)

A(0, 0)

x

y

W(a, 0)

Z(b, c)Y(?, ?)

X(?, ?) Ox

y

J(2b, 0)

K(?, a)L(c, ?)

H(0, 0)O

B(a � 2c, 0)

D(a, b) C(2c, b)

A(0, 0)B(b, 0)

D(c, a) C(b � c, a)

A(0, 0)

Practice Coordinate Proof with Quadrilaterals

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8-7

Pre-Activity How can you use a coordinate plane to prove theorems aboutquadrilaterals?


What special kinds of quadrilaterals can be placed on a coordinate system sothat two sides of the quadrilateral lie along the axes?

Reading the Lesson1. Find the missing coordinates in each figure. Then write the coordinates of the four

vertices of the quadrilateral.

a. isosceles trapezoid b. parallelogram

2. Refer to quadrilateral EFGH.

a. Find the slope of each side.

b. Find the length of each side.

c. Find the slope of each diagonal.

d. Find the length of each diagonal.

e. What can you conclude about the sides of EFGH?

f. What can you conclude about the diagonals of EFGH?

g. Classify EFGH as a parallelogram, a rhombus, or a square. Choose the most specificterm. Explain how your results from parts a-f support your conclusion.

Helping You Remember3. What is an easy way to remember how best to draw a diagram that will help you devise

a coordinate proof?

x

y

O

E

FG

H

x

y

Q(b, ?)

P(?, c � a)

N(?, a)

M(?, ?)Ox

y

U(a, ?)

T(b, ?)S(?, c)

R(?, ?) O


Coordinate ProofsAn important part of planning a coordinate proof is correctly placing and labeling thegeometric figure on the coordinate plane.

Draw a diagram you would use in a coordinate proof of thefollowing theorem.

The median of a trapezoid is parallel to the bases of the trapezoid.

In the diagram, note that one vertex, A, is placed at the origin. Also,the coordinates of B, C, and D use 2a, 2b, and 2c in order to avoid the use of fractions when finding the coordinates of the midpoints,M and N.

When doing coordinate proofs, the following strategies may be helpful.

1. If you are asked to prove that segments are parallel or perpendicular, use slopes.

2. If you are asked to prove that segments are congruent or have related measures,use the distance formula.

3. If you are asked to prove that a segment is bisected, use the midpoint formula.

For each of the following theorems, a diagram has been provided to be used in aformal proof. Name the missing coordinates in the diagram. Then, using the Givenand the Prove statements, prove the theorem.

1. The median of a trapezoid is parallel to the bases of the trapezoid. (Use the diagramgiven in the example above.)

Coordinates of M and N:

Given: Trapezoid ABCD has median M�N�.Prove: B�C� || M�N� and A�D� || M�N�

Proof:

2. The medians to the legs of an isosceles triangle are congruent.

Coordinates of T and K:

Given: �ABC is isosceles with medians T�B� and K�A�.Prove: T�B� � K�A�

Proof:x

y

O

T K

A(0, 0) B(4a, 0)

C(2a, 2b)

x

y

O

M N

A(0, 0) D(2a, 0)

C(2d, 2c)B(2b, 2c)

Enrichment

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ExampleExample

Chapter 8 Test, Form 188


Ass

essm

ents

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

1. Find the sum of the measures of the interior angles of a convex 30-gon.A. 5400 B. 5040 C. 360 D. 168

2. Find the sum of the measures of the exterior angles of a convex 21-gon.A. 21 B. 180 C. 360 D. 3420

3. If the measure of each interior angle of a regular polygon is 108, find themeasure of each exterior angle.A. 18 B. 72 C. 90 D. 108

4. For parallelogram ABCD, find x.A. 4 B. 10.25C. 16 D. 21.5

5. Which of the following is a property of a parallelogram?A. The diagonals are congruent. B. The diagonals bisect the angles.C. The diagonals are perpendicular. D. The diagonals bisect each other.

6. Find x and y so that ABCD will be a parallelogram.A. x � 6, y � 42B. x � 6, y � 22C. x � 20, y � 42D. x � 20, y � 22

7. Find x so that this quadrilateral is a parallelogram.A. 44 B. 46C. 90 D. 134

8. ABCD is a parallelogram with A(0, 0), B(2, 4), and C(10, 4). Find the possiblecoordinates of D.A. (8, 0) B. (10, 0) C. (0, 4) D. (10, 8)

9. Which of the following is a property of all rectangles?A. four congruent sides B. diagonals bisect the anglesC. diagonals are perpendicular D. four right angles

10. ABCD is a rectangle with diagonals A�C� and B�D�. If AC � 2x � 10 and BD � 56, find x.A. 23 B. 33 C. 78 D. 122

11. ABCD is a rectangle with B(�5, 0), C(7, 0) and D(7, 3). Find the coordinates of A.A. (�5, 7) B. (3, 5)C. (�5, 3) D. (7, �3)

x� 134�

134� 46�

4x�(y � 10)�32�

24�

5x � 12

3x � 20

A D

CB

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 8 Test, Form 1 (continued)88

12.

13.

14.

15.

16.

17.

18.

19.

20.

12. Given rhombus ABCD, find m�1.A. 45 B. 60C. 90 D. 120

13. Find m�PRS in square PQRS.A. 30 B. 45C. 60 D. 90

14. Choose a pair of base angles of trapezoid ABCD.A. �A, �C B. �B, �DC. �A, �D D. �D, �C

15. In trapezoid DEFG, find m�D.A. 44 B. 72C. 108 D. 136

16. The bases of a trapezoid are 12 and 26. Find the length of the median.A. 12 B. 19 C. 26 D. 38

17. If the length of one base of a trapezoid is 44, the median is 36, and the otherbase is 2x � 10, find x.A. 9 B. 17 C. 21 D. 40

18. Given trapezoid ABCD with median E�F�, which of the following is true?

A. EF � �12�AD B. AE � FD

C. AB � EF D. EF � �BC �

2AD

�

19. ABCD is a rectangle with A(0, 0), B(b, 0), and D(0, a). Find the coordinates of C.A. C(a, b) B. C(b, a) C. C(�b, a) D. C(a � b, a)

20. To prove that the diagonals of a square bisect each other, you would position andlabel a square in the coordinate plane and then find which of the following?A. measures of the angles B. midpoints of the diagonalsC. lengths of the diagonals D. slopes of the diagonals

Bonus Find x and m�WYZ in rhombus XYZW.

X Y

ZW

(3x � 20)�

(x2)�

DA

B CFE

D G

E F136�

72�

D C

A B

P S

Q R

1

A D

CB

B:

NAME DATE PERIOD

Chapter 8 Test, Form 2A88


Ass

essm

ents


1. Find the sum of the measures of the interior angles of a convex 45-gon.A. 8100 B. 7740 C. 360 D. 172

2. Find x.A. 30 B. 66C. 102 D. 138

3. Find the sum of the measures of the exterior angles of a convex 39-gon.A. 39 B. 90 C. 180 D. 360

4. Which of the following is a property of a parallelogram?A. Each pair of opposite sides is congruent.B. Only one pair of opposite angles is congruent.C. Each pair of opposite angles is supplementary.D. There are four right angles.

5. Find m�1 in parallelogram ABCD.A. 60 B. 54C. 36 D. 18

6. ABCD is a parallelogram with diagonals intersecting at E. If AE � 3x � 12and EC � 27, find x.A. 5 B. 17 C. 27 D. 47

7. Find x and y so that this quadrilateral is a parallelogram.A. x � 13, y � 24 B. x � 13, y � 6C. x � 7, y � 24 D. x � 7, y � 6

8. Find x so that this quadrilateral is a parallelogram.A. 12 B. 24C. 36 D. 132

9. Given A(8, 2), B(6, �4), C(�5, �4), find the coordinates of D so that ABCD isa parallelogram.A. D(�5, 2) B. D(�3, 2) C. D(�2, 2) D. D(�4, 8)

10. ABCD is a rectangle. If AC � 5x � 2 and BD � x � 22, find x.A. 5 B. 6 C. 11 D. 26

11. Which of the following is true for all rectangles?A. The diagonals are perpendicular.B. The diagonals bisect the angles.C. The consecutive sides are congruent.D. The consecutive sides are perpendicular.

(2x � 60)�

(4x � 12)�

(2x � 30)�

(5x � 9)�

(y � 12)�1–2y �

120�

36�

1

A D

CB

(2x � 10)�(x � 40)�

(x � 20)�x�

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 8 Test, Form 2A (continued)88

12.

13.

14.

15.

16.

17.

18.

19.

20.

12. ABCD is a rectangle with B(�4, 6), C(�4, 2), and D(10, 2). Find thecoordinates of A.A. A(6, 4) B. A(10, 4) C. A(2, 6) D. A(10, 6)

13. Find m�1 in rhombus GHJK.A. 22 B. 44C. 68 D. 90

14. The diagonals of square ABCD intersect at E. If AE � 2x � 6 and BD � 6x � 10, find AC.A. 11 B. 28 C. 56 D. 90

15. ABCD is an isosceles trapezoid with A(10, �1), B(8, 3), and C(�1, 3). Find thecoordinates of D.A. (�3, �1) B. (�10, �11) C. (�1, 8) D. (�3, 3)

16. For isosceles trapezoid MNOP, find m�MNP.A. 44 B. 64C. 80 D. 116

17. The length of one base of a trapezoid is 19 inches and the length of themedian is 16 inches. Find the length of the other base.A. 35 in. B. 19 in. C. 17.5 in. D. 13 in.

18. E�F� is the median of isosceles trapezoid ABCD.Find x.A. 2x � 46 B. 32C. 46 D. 68

19. What type of quadrilateral has vertices at (0, 0), (a, b), (c, b), and (c � a, 0)?A. parallelogram B. rectangleC. rhombus D. trapezoid

20. To prove that the diagonals of a rhombus are perpendicular to each other, youwould position and label a rhombus on a coordinate plane and then findwhich of the following?A. measures of the angles B. slopes of the diagonalsC. lengths of the diagonals D. midpoints of the diagonals

Bonus Find the possible value(s) of x in rectangle JKLM.

J K

LM

x2 � 8

3x � 36

A D

CB

E F3x � 72

x � 20

P

ON

M 36�44�

22�

1G H

JK

B:

NAME DATE PERIOD

Chapter 8 Test, Form 2B88


Ass

essm

ents


1. Find the sum of the measures of the interior angles of a convex 50-gon.A. 9000 B. 8640 C. 360 D. 172.8

2. Find x.A. 16 B. 34C. 50 D. 70

3. Find the sum of the measures of the exterior angles of a convex 65-gon.A. 5.54 B. 90 C. 180 D. 360

4. Which of the following is a property of all parallelograms?A. Each pair of opposite angles is congruent.B. Only one pair of opposite sides is congruent.C. Each pair of opposite angles is supplementary.D. There are four right angles.

5. Find m�1 in parallelogram ABCD.A. 19 B. 38C. 52 D. 56

6. ABCD is a parallelogram with diagonals intersecting at E. If AE � 4x � 8and EC � 36, find x.A. 7 B. 11 C. 15.5 D. 38

7. Find x and y so that this quadrilateral is a parallelogram.A. x � 27, y � 90 B. x � 27, y � 40C. x � 13, y � 90 D. x � 13, y � 40

8. Find x so that this quadrilateral is a parallelogram.

A. 7�13� B. 8

C. 12 D. 66

9. ABCD is a parallelogram with A(5, 4), B(�1, �2), C(8, �2). Find thecoordinates of D.A. D(�5, 4) B. D(8, 2) C. D(14, 4) D. D(4, 1)

10. ABCD is a rectangle. If AB � 7x � 6 and CD � 5x � 30, find x.

A. 5�13� B. 12 C. 13 D. 18

11. Which of the following is true for all rectangles?A. The diagonals are perpendicular.B. The consecutive angles are supplementary.C. The opposite sides are supplementary.D. The opposite angles are complementary.

6x � 6 3x � 30

(4x � 4)�

(3x � 23)�

(y � 60)�

1–3y �

38�

124�1

A D

B C

(2x � 10)�

(x � 30)�

(x � 70)� x �

2x �

2x �

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 8 Test, Form 2B (continued)88

12.

13.

14.

15.

16.

17.

18.

19.

20.

12. ABCD is a rectangle with B(�7, 3), C(5, 3), and D(5, �8). Find the coordinatesof A.A. (�8, �7) B. (�7, �8) C. (�5, �3) D. (�8, �5)

13. Find m�1 in rhombus GHJK.A. 90 B. 64C. 52 D. 38

14. The diagonals of square ABCD intersect at E. If AE � 3x � 4 and BD � 10x � 48, find AC.A. 90 B. 52 C. 26 D. 10

15. ABCD is an isosceles trapezoid with A(0, �1), B(�2, 3), and D(6, �1). Findthe coordinates of C.A. C(6, 1) B. C(9, 4) C. C(2, 3) D. C(8, 3)

16. Find m�MNP in isosceles trapezoid MNOP.A. 42 B. 70C. 82 D. 98

17. The length of one base of a trapezoid is 19 meters and the length of themedian is 23 meters. Find the length of the other base.A. 15 m B. 21 m C. 27 m D. 42 m

18. E�F� is the median of isosceles trapezoid ABCD.Find x.A. 22 B. 18.5C. 42.5 D. 82

19. What type of quadrilateral has vertices at (0, 0), (a, b), (a � c, b), and (c, 0)?A. parallelogram B. rectangleC. rhombus D. trapezoid

20. To prove that the diagonals of a rectangle are congruent, you would positionand label a rectangle on a coordinate plane and then find which of thefollowing?A. measures of the angles B. slopes of the diagonalsC. lengths of the diagonals D. midpoints of the diagonals

Bonus The sum of the measures of the interior angles of a convex polygon is ten times the sum of the measures of its exterior angles. Find the number of sides of the polygon.

A D

CB

E F3x � 52

x � 15

28�42�

P

ON

M

128�

1

G K

JH

B:

NAME DATE PERIOD

Chapter 8 Test, Form 2C88


Ass

essm

ents

1. Find the sum of the measures of the interior angles of aconvex 60-gon.

2. A convex pentagon has interior angles with measures (5x � 12)°,(2x � 100)°, (4x � 16)°, (6x � 15)°, and (3x � 41)°. Find x.

3. If the measure of each interior angle of a regular polygon is171, find the number of sides of the polygon.

4. In parallelogram ABCD,m�1 � x � 12, and m�2 � 6x � 18.Find m�1.

5. Find the measure of each exterior angle of a regular 45-gon.

6. In parallelogram ABCD, m�A � 58. Find m�B.

7. Find the coordinates of the intersection of the diagonals ofparallelogram XYZW with vertices X(2, 2), Y(3, 6), Z(10, 6),and W(9, 2).

8. Determine whether ABCD is a parallelogram. Justify your answer.

9. Use the Slope Formula to determine whether A(5, 7), B(1, �2),C(�6, �3), and D(2, 5) are the coordinates of the vertices ofparallelogram ABCD.

10. If the slope of A�B� is �14�, the slope of B�C� is ��

23�, and the slope of

C�D� is �14�, find the slope of D�A� so that ABCD will be a

parallelogram.

11. Given rectangle ABCD, find x.

12. ABCD is a parallelogram and A�C� � B�D�. Determine whetherABCD is a rectangle. Justify your answer.

13. ABCD is a rhombus with diagonals intersecting at E. Ifm�ABC � 3m�BAD, find m�EBC.

A B

CD

(2x � 10)�

(3x � 30)�

A D

CB

12 C

D

A

B

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

NAME DATE PERIOD

SCORE


Chapter 8 Test, Form 2C (continued)88

14. TUVW is a square with U(10, 2), V(8, 8), and W(2, 6). Find thecoordinates of T.

15. Find m�MNQ in isosceles trapezoid MNOP.

16. ABCD is a quadrilateral with A(8, 3), B(6, 7), C(�1, 5), andD(�6, �1). Determine whether ABCD is a trapezoid. Justifyyour answer.

17. The length of the median of trapezoid EFGH is 13 feet. If thebases have lengths 2x � 4 and 10x � 50, find x.

18. Name the missing coordinates for rhombus ABCD. Thendetermine the relationshipbetween A�C� and B�D�.

For Questions 19–25, write true or false.

19. A rectangle is always a parallelogram.

20. The diagonals of a rhombus are always perpendicular.

21. The diagonals of a square always bisect each other.

22. A trapezoid always has two congruent sides.

23. The median of a trapezoid is always parallel to the bases.

24. A quadrilateral with vertices (a, 0), (b, c), (�b, c), and (�a, 0)is an isosceles trapezoid.

25. If the diagonals of a parallelogram are perpendicular, then theparallelogram is a rectangle.

Bonus In parallelogram ABCD, AB � 2x � 7, BC � x � 3y,CD � x � y, and AD � 2x � y � 1. Find x and y.

x

y

A(?, ?)

B(0, b)

C(�a, 0)

D(0, �b)

O

N

Q

O

PM 59�

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

NAME DATE PERIOD

Chapter 8 Test, Form 2D88


Ass

essm

ents


2. A convex octagon has interior angles with measures (x � 55)°,(3x � 20)°, 4x°, (4x � 10)°, (6x � 55)°, (3x � 52)°, 3x°, and (2x � 30)°. Find x.

3. If the measure of each interior angle of a regular polygon is 176find the number of sides on the polygon.

4. In parallelogram ABCD,m�1 � x � 25, and m�2 � 2x.Find m�2.

5. Find the measure of each exterior angle of a regular 100-gon.

6. If m�A � 63 in parallelogram ABCD, find m�B.

7. Find the coordinates of the intersection of the diagonals ofparallelogram XYZW with vertices X(3, 0), Y(3, 8), Z(�2, 6),and W(�2, �2).

8. Determine whether this quadrilateral is a parallelogram. Justify your answer.

9. Use the Slope Formula to determine whether A(5, 7), B(1, �1),C(�6, �3), and D(�2, 5) are the coordinates of the vertices ofa parallelogram.

10. If the slope of A�B� is �12�, the slope of B�C� is �4, and the slope of

C�D� is �12�, find the slope of D�A� so that ABCD is a parallelogram.

11. Given rectangle ABCD, find x.

12. ABCD is a parallelogram and m�A � 90. Determine whetherABCD is a rectangle. Justify your answer.

B C

DA

(6x � 20)�

(3x � 2)�

1

2A D

CB

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

NAME DATE PERIOD

SCORE


Chapter 8 Test, Form 2D (continued)88

13. ABCD is a rhombus with diagonals intersecting at E. Ifm�ABC � 4m�BAD, find m�EBC.

14. PQRS is a square with Q(�2, 8), R(5, 7), and S(4, 0). Find thecoordinates of P.

15. Find m�MNQ in isosceles trapezoid MNOP.

16. ABCD is a quadrilateral with A(8, �1), B(10, �7), C(�6, �1),and D(0, 2). Determine whether ABCD is a trapezoid. Justifyyour answer.

17. The length of the median of trapezoid EFGH is 17 centimeters.If the bases have lengths 2x � 4 and 8x � 50, find x.

18. Name the missing coordinates for square ABCD. Then determine thecoordinates of the midpoints of thediagonals.


19. A parallelogram always has four right angles.

20. The diagonals of a rhombus always bisect the angles.

21. A rhombus is always a square.

22. A rectangle is always a square.

23. The diagonals of an isosceles trapezoid are always congruent.

24. The median of a trapezoid always bisects the angles.

25. A quadrilateral with vertices (0, 0), (a, 0), (a � c, b), and (b, c)is a parallelogram.

Bonus The measure of each interior angle of a regular polygon is24 more than 38 times the measure of each exterior angle.Find the number of sides of the polygon.

x

y

A(a, 0)

C(0, a)

D(0, 0)

B(?, ?)

O

M P

ON

Q74�

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

NAME DATE PERIOD

Chapter 8 Test, Form 388


Ass

essm

ents


2. A convex hexagon has interior angles with measures x°,(5x � 103)°, (2x � 60)°, (7x � 31)°, (6x � 6)°, and (9x � 100)°.Find x and the measure of each angle.

3. Find the measure of each exterior angle of a regular 2x-gon.

4. Find m�1 in parallelogram ABCD.

5. ABCD is a parallelogram with diagonals that intersect eachother at E. If AE � x2 and EC � 6x � 8, find all possiblevalues of AC.


7. Given the slope of A�B� is �23� and the slope of B�C� is �2, find the

slopes of C�D� and D�A� so that ABCD will be a parallelogram.

8. In rectangle ABCD, find m�1.

9. The diagonals of rhombus ABCD intersect at E. If m�BAE �

�23�(m�ABE), find m�BCD.

10. The diagonals of square ABCD intersect at E. If AE � 2, findthe perimeter of ABCD.

11. Find AE in isosceles trapezoid ABCD.

12. Points G and H are midpoints of A�F� and D�E� inregular hexagon ABCDEF. If AB � 6 find GH.

B C

F

A DHG

E

A D

CB

E F60�

8

B C

DA(3x � 16)�

1(2x � 10)�

96� 31�1

D C

BA

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

NAME DATE PERIOD

SCORE


Chapter 8 Test, Form 3 (continued)88

13. If the vertices of trapezoid ABCD are A(10, �1), B(6, 6),C(�8, �1), and D(�2, 6), find the length of the median.

14. A parallelogram has three vertices (0, 0), (b, c), and (d, 0).What are the possible coordinates of the fourth vertex?

15. Determine whether ABCD is a rectangle given A(6, 2), B(2, 10),C(�6, 6), and D(�2, �2). Justify your answer.

16. Use the Distance Formula to determine whether ABCD is aparallelogram given A(1, 6), B(7, 6), C(2, �3), and D(�4, �3).

17. Name the missing coordinates for ABCD. Then determine therelationship between AC and thesegment formed by connecting the midpoints of B�C� and A�B�.

For Questions 18 and 19, complete the two-column proof bysupplying the missing information for each correspondinglocation.Given: ABCD is a parallelogram.

B�Q� � D�S�, P�A� � R�C�Prove: PQRS is a parallelogram.

Statements Reasons

1. ABCD is a �. 1. Given2. A�D� � C�B� 2.3. P�A� � R�C� 3. Given4. P�D� � R�B� 4. Seg. Sub. Prop.5. A�B� � C�D� 5. Opp. sides of a � are �.6. B�Q� � D�S� 6. Given7. A�Q� � C�S� 7. Seg. Sub. Prop.8. �B � �D, �A � �C 8. Opp. � of a � are �.9. �QBR � �SDP, �PAQ � �RCS 9. SAS

10. Q�P� � R�S�, Q�R� � P�S� 10. CPCTC11. PQRS is a parallelogram. 11.

20. In isosceles trapezoid ABCD, AE � 2x � 5,EC � 3x � 12, and BD � 4x � 20. Find x.

Bonus If three of the interior angles of a convex hexagon eachmeasure 140, a fourth angle measures 84, and themeasure of the fifth angle is 3 times the measure of thesixth angle, find the measure of the sixth angle.

B C

EDA

(Question 19)

(Question 18)

A P

R

D

C

SQ

B

x

y

A(2a, 0)

B(0, 2b)

C(?, ?)

D(0, �2b)

O

13.

14.

15.

16.

17.

18.

19.

20.

B:

NAME DATE PERIOD

Chapter 8 Open-Ended Assessment88


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. a. Draw a regular convex polygon and a convex polygon that is notregular, each with the same number of sides.

b. Label the measures of each exterior angle on your figures.

c. Find the sum of the exterior angles for each figure. What conjecture canbe made?

2. Draw a rectangle. Connect the midpoints of the consecutive sides. Whattype of quadrilateral is formed? How do you know?

3. Draw an example to show why one pair of opposite sides congruent and theother pair of opposite sides parallel is not sufficient to form aparallelogram.

4. a. Name a property that is true for a square and not always true for arectangle.

b. Name a property that is true for a square and not always true for arhombus.

c. Name a property that is true for a rectangle and not always true for aparallelogram.

5. John drew this figure to use in a coordinate proof.Explain whether John’s diagram could actually be used to prove that the diagonals of a rhombus are perpendicular.

x

y

D(a, 0)

B(b, c)

A(0, 0)

C(a � b, c)

O

NAME DATE PERIOD

SCORE


Chapter 8 Vocabulary Test/Review88

Choose from the terms above to complete each sentence.

1. A quadrilateral with only one pair of opposite sides paralleland the other pair of opposite sides congruent is a(n) .

2. A quadrilateral with two pairs of opposite sides parallel is a(n).

3. A quadrilateral with only one pair of opposite sides parallel isa(n) .

4. A quadrilateral that is both a rectangle and a rhombus is a(n).

5. A quadrilateral with four congruent sides is a(n) .

6. A quadrilateral with four right angles is a(n) .

7. A quadrilateral with two pairs of congruent consecutive sidesis a(n) .

8. Segments that join opposite vertices in a quadrilateral arecalled .

9. The segment joining the midpoints of the nonparallel sides ofa trapezoid is called the .

10. The parallel sides of a trapezoid are called the .

Define each term.

11. base angles of an isosceles trapezoid

12. legs of a trapezoid

?

?

?

?

?

?

?

?

?

?1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

basesdiagonalsisosceles trapezoid

kitemedianparallelogram

rectanglerhombus

squaretrapezoid

NAME DATE PERIOD

SCORE

Chapter 8 Quiz (Lessons 8–1 and 8–2)

88


Ass

essm

ents

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.


2. If the measure of each interior angle of a regular polygon is172, find the number of sides of the polygon.

3. If the measure of each exterior angle of a regular polygon is18, find the number of sides of the polygon.

4. Given parallelogram ABCD with C(5, 4), find the coordinatesof A if the diagonals intersect at (2, 7).

5. STANDARDIZED TEST PRACTICE Find m�1 in parallelogram ABCD.A. 64 B. 58C. 46 D. 36

110�

46� 1

D C

BA


88

1.

2.

3.

4.

5.



2. A quadrilateral with two pairs of parallel sides is always arectangle.

3. The diagonals of a parallelogram are always perpendicular.

4. The slope of A�B� and C�D� is �35� and the slope of B�C� and A�D� is

��53�. ABCD is a parallelogram.

5. Find m�1, m�2, and m�3 in the rectangle at the right.

34�

1

2

3

NAME DATE PERIOD

SCORE



88

1.

2.

3.

4.

5.

1. Given M(4, 0), N(0, 6), P(�4, 0) and Q(0, �6), determine whetherMNPQ is a trapezoid, a parallelogram, a square, a rhombus, or aquadrilateral. Choose the most specific term. Explain.

For Questions 2 and 3, use trapezoid MNPQ.

2. Find m�N.

3. Find m�Q.

4. True or false. A quadrilateral that is a rectangle and arhombus is a square.

5. Find x if the bases of a trapezoid have lengths 2x � 4 and 8x � 10 and the length of the median is 3x � 21.

M Q

PN 130�

72�

NAME DATE PERIOD

SCORE

Chapter 8 Quiz (Lesson 8–7)

88

1.

2.

3.

4.

5.

1. ABCD is a rhombus with A(a, 0), B(0, b), and D(0, �b) Findthe possible coordinates of C.

2. ABCD is a square with A(a, 0), B(0, a), and C(�a, 0). Find the possible coordinates of D.

3. Given rectangle ABCD, find the coordinates of A.

4. Name the coordinates of the endpoints of the median of trapezoid ABCD.

5. Name the missing coordinates for parallelogram ABCD. Thendetermine the lengths of each of the four sides.

x

y

D(a, 0)

B(b, c)

A(0, 0)

C(?, ?)

O

x

y

A(2a, 0)

C(�2d, 2c)

Y X

D(�2e, 0)

B(2b, 2c)

O

x

y

A(?, ?)

C(a, �b)D(�d, �b)

B(a, c)

O

NAME DATE PERIOD

SCORE

Chapter 8 Mid-Chapter Test (Lessons 8–1 through 8–4)

88


Ass

essm

ents

1. Find the measure of each exterior angle of a regular 56-gon. Round to thenearest tenth.A. 3.2 B. 6.4 C. 173.6 D. 9720

2. Given BE � 2x � 6 and ED � 5x � 12 in parallelogram ABCD, find BD.A. 6 B. 12C. 18 D. 36

3. If the slope of P�Q� is �23� and the slope of R�S� is �

23�, find the slope of Q�R� and S�P�

so that PQRS is a rectangle.

A. ��32� B. �

32� C. ��

23� D. �

23�

4. Find m�1 in rectangle RSTW.A. 163 B. 146C. 34 D. 17

5. Find the sum of the measures of the interior angles of a convex 48-gon.A. 172.5 B. 360 C. 8280 D. 8640

R S

TW

17�

1

D

ECB

A

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

Part II

6. Find x.

7. ABCD is a parallelogram with m�A � 138. Find m�B.

8. Determine whether ABCD is a parallelogram if AB � 6,BC � 12, CD � 6, and DA � 12.

9. In rectangle XYZW, find m�1, m�2,m�3, and m�4.

10. In quadrilateral ABCD, B�C� � A�D� and B�D� and E�F� bisect each other at G.By what theorem is ABCD aparallelogram? D

E

F

G

CB

A

Y Z

WX12� 1 2

34

(3x � 2)�(2x � 12)�

(x � 10)�(x � 30)�

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


Chapter 8 Cumulative Review(Chapters 1–8)

88

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

1. Point B lies on RT�� between points R and T. Name twoopposite rays. (Lesson 1-4)

2. Write an equation in slope-intercept form for the line that contains (�12, 9) and is perpendicular to the line y � �

23�x � 5.

(Lesson 3-4)

3. If �ABC � �WXY, AB � 72, BC � 65, CA � 13, XY � 7x � 12,and WX � 19y � 34, find x and y. (Lesson 4-3)

4. Freda bought two bells for just over $90 before tax. State theassumption you would make to write an indirect proof to showthat at least one of the bells costs more than $45. (Lesson 5-3)

5. A plot of land has dimensions 82 feet, 132 feet, 75 feet, and120 feet. A draftsman applies a scale of 2 inches � 8 feet tomake a scale drawing of the land. Find the dimensions of thedrawing. (Lesson 6-2)

6. Triangle QRT has vertices Q(1, 2), R(5, �1), and T(�2, �4),and A�B� is a midsegment parallel to R�T�. Find the coordinatesof A and B. (Lesson 6-4)

7. Name the angles of elevation and depression. (Lesson 7-5)

8. Use the Law of Sines to find c to the nearest whole number. (Lesson 7-6)

For Questions 10 and 11, use rectangle ABDC and parallelogram CDGF.

9. If m�CDF � 21, m�FDG � 34,CF � 3x � 8, and DG � 7, find m�GFC,m�FCD, and x. (Lesson 8-2)

10. Find AE if AD � 12y � 7 and BC � 21 � 2y. (Lesson 8-4)

11. Quadrilateral LMNP has vertices L(3, 0), M(7, �3), N(�7, �9),and P(�4, �3). Verify that LMNP is a trapezoid. Explain.(Lesson 8-6)

A B

DC

F G

E

A

B C

bc

69�

96�

46.89

W X

ZY

NAME DATE PERIOD

SCORE

Standardized Test Practice (Chapters 1–8)


1. Find the coordinates of X if V(0.5, 5) is the midpoint of U�X� withU(15, �10). (Lesson 1-3)

A. (�14, 20) B. (7.75, �2.5)C. (0, 0) D. (15.5, �5)

2. Which of the following are possible measures for vertical angles Gand H? (Lesson 2-8)

E. m�G � 125 and m�H � 55F. m�G � 125 and m�H � 125G. m�G � 55 and m�H � 45H. m�G � 55 and m�H � 152.5

3. Determine which lines are parallel.(Lesson 3-5)

A. NS�� || PT�� B. NP�� || ST��

C. QR�� || ST�� D. NP�� || QR��

4. Find the coordinates of B, the midpoint of A�C�, if A(2a, b) and C(0, �b). (Lesson 4-7)

E. (2a, �b) F. (a, b) G. (a, 0) H. (0, b)

5. If R�V� is an angle bisector, find m�UVT. (Lesson 5-1)

A. 10 B. 34C. 68 D. 136

6. Which is not a valid postulate or theorem for proving twotriangles similar? (Lesson 6-3)

E. SSA Theorem F. SAS TheoremG. SSS Theorem H. AA Postulate

7. Find m�K to the nearest tenth.(Lesson 7-7)

A. 22.3 B. 32.6C. 54.8 D. 125.2

8. Which statement ensures that quadrilateral QRST is a parallelogram? (Lesson 8-3)

E. �Q � �S F. Q�R� � T�S� and Q�R� || T�S�G. Q�T� || R�S� H. m�Q � m�S � 180

Q R

ST

41

2719H

J K

(2y � 14)�

(4y � 6)�T VS

R

U

67�

113�

65�

N P

ST

RQ

NAME DATE PERIOD

SCORE 88

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


Standardized Test Practice (continued)

9. AB�� || JK��. Find the slope of AB�� if J(14, 8) andK(�7, 5). (Lesson 3-3)

10. If �UVW is an isosceles triangle, U�V� � W�U�,UV � 16b � 40, VW � 6b, and WU � 10b � 2,find b. (Lesson 4-1)

11. If �RTU � �GEF and S�U� and F�H� arealtitudes of �RTU and �GEF respectively, findFH. (Lesson 6-5)

12. Find the geometric mean between 25 and 4.(Lesson 7-1)

13. Find the sum of the measures of the interiorangles for a convex heptagon. (Lesson 8-1)

35 50 16

ST

E

FGH

R

U

NAME DATE PERIOD

88

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. 11.

11. 12.

13.

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

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

14. A polygon has six congruent sides. Lines containing two of itssides contain points in its interior. Name the polygon by itsnumber of sides, and then classify it as convex or concave andregular or irregular. (Lesson 1-6)

15. If R�T� � Q�M� and RT � 88.9 centimeters, find QM. (Lesson 2-7)

16. Which segment is the shortest segment from D to JM��? (Lesson 5-4)

D

J K L H M

14.

15.

16.

1 / 7 7

1 1 . 2

9 0 0

1 0

Standardized Test PracticeStudent Record Sheet (Use with pages 458–459 of the Student Edition.)

88

© 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

3 6 DCBADCBA

DCBADCBA

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 Question 11, also enter your answer by writing each number or symbol in abox. Then fill in the corresponding oval for that number or symbol.

8 11

9

10

11 (grid in)

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

Record your answers for Questions 12–13 on the back of this paper.


Stu

dy G

uid

e a

nd I

nte

rven

tion

An

gle

s o

f P

oly

go

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1

©G

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w-H

ill41

7G

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oe G

eom

etry

Lesson 8-1

Sum

of

Mea

sure

s o

f In

teri

or

An

gle

sT

he

segm

ents

th

at c

onn

ect

the

non

con

secu

tive

sid

es o

f a

poly

gon

are

cal

led

dia

gon

als.

Dra

win

g al

l of

th

e di

agon

als

from

one

vert

ex o

f an

n-g

onse

para

tes

the

poly

gon

in

to n

�2

tria

ngl

es.T

he

sum

of

the

mea

sure

sof

th

e in

teri

or a

ngl

es o

f th

e po

lygo

n c

an b

e fo

un

d by

add

ing

the

mea

sure

s of

th

e in

teri

oran

gles

of

thos

e n

�2

tria

ngl

es.

Inte

rio

r A

ng

leIf

a co

nvex

pol

ygon

has

nsi

des,

and

Sis

the

sum

of

the

mea

sure

s of

its

inte

rior

angl

es,

Su

m T

heo

rem

then

S�

180(

n�

2).

A c

onve

x p

olyg

on h

as

13 s

ides

.Fin

d t

he

sum

of

the

mea

sure

sof

th

e in

teri

or a

ngl

es.

S�

180(

n�

2)�

180(

13 �

2)�

180(

11)

�19

80

Th

e m

easu

re o

f an

inte

rior

an

gle

of a

reg

ula

r p

olyg

on i

s12

0.F

ind

th

e n

um

ber

of

sid

es.

Th

e n

um

ber

of s

ides

is

n,s

o th

e su

m o

f th

em

easu

res

of t

he

inte

rior

an

gles

is

120n

.S

�18

0(n

�2)

120n

�18

0(n

�2)

120n

�18

0n�

360

�60

n�

�36

0n

�6

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d t

he

sum

of

the

mea

sure

s of

th

e in

teri

or a

ngl

es o

f ea

ch c

onve

x p

olyg

on.

1.10

-gon

2.16

-gon

3.30

-gon

1440

2520

5040

4.8-

gon

5.12

-gon

6.3x

-gon

1080

1800

180(

3x�

2)

Th

e m

easu

re o

f an

in

teri

or a

ngl

e of

a r

egu

lar

pol

ygon

is

give

n.F

ind

th

e n

um

ber

of s

ides

in

eac

h p

olyg

on.

7.15

08.

160

9.17

5

1218

72

10.1

6511

.168

.75

12.1

35

2432

8

13.F

ind

x.

20

7x�

( 4x

� 1

0)�

( 4x

� 5

) � ( 6x

� 1

0)�

( 5x

� 5

) �

AB

C

D

E

©G

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8G

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eom

etry

Sum

of

Mea

sure

s o

f Ex

teri

or

An

gle

sT

her

e is

a s

impl

e re

lati

onsh

ip a

mon

g th

eex

teri

or a

ngl

es o

f a

con

vex

poly

gon

.

Ext

erio

r A

ng

leIf

a po

lygo

n is

con

vex,

the

n th

e su

m o

f th

e m

easu

res

of t

he e

xter

ior

angl

es,

Su

m T

heo

rem

one

at e

ach

vert

ex,

is 3

60.

Fin

d t

he

sum

of

the

mea

sure

s of

th

e ex

teri

or a

ngl

es,o

ne

at e

ach

vert

ex,o

f a

con

vex

27-g

on.

For

an

yco

nve

x po

lygo

n,t

he

sum

of

the

mea

sure

s of

its

ext

erio

r an

gles

,on

e at

eac

h v

erte

x,is

360

.

Fin

d t

he

mea

sure

of

each

ext

erio

r an

gle

of

regu

lar

hex

agon

AB

CD

EF

.T

he

sum

of

the

mea

sure

s of

th

e ex

teri

or a

ngl

es i

s 36

0 an

d a

hex

agon

h

as 6

an

gles

.If

nis

th

e m

easu

re o

f ea

ch e

xter

ior

angl

e,th

en

6n�

360

n�

60

Fin

d t

he

sum

of

the

mea

sure

s of

th

e ex

teri

or a

ngl

es o

f ea

ch c

onve

x p

olyg

on.

1.10

-gon

2.16

-gon

3.36

-gon

360

360

360

Fin

d t

he

mea

sure

of

an e

xter

ior

angl

e fo

r ea

ch c

onve

x re

gula

r p

olyg

on.

4.12

-gon

5.36

-gon

6.2x

-gon

3010

�18 x0 �

Fin

d t

he

mea

sure

of

an e

xter

ior

angl

e gi

ven

th

e n

um

ber

of

sid

es o

f a

regu

lar

pol

ygon

.

7.40

8.18

9.12

920

30

10.2

411

.180

12.8

152

45

AB

C

DE

F

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

An

gle

s o

f P

oly

go

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Answers (Lesson 8-1)


An

swer

s

Skil

ls P

ract

ice

An

gle

s o

f P

oly

go

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1

©G

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9G

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eom

etry

Lesson 8-1

Fin

d t

he

sum

of

the

mea

sure

s of

th

e in

teri

or a

ngl

es o

f ea

ch c

onve

x p

olyg

on.

1.n

onag

on2.

hep

tago

n3.

deca

gon

1260

900

1440

Th

e m

easu

re o

f an

in

teri

or a

ngl

e of

a r

egu

lar

pol

ygon

is

give

n.F

ind

th

e n

um

ber

of s

ides

in

eac

h p

olyg

on.

4.10

85.

120

6.15

0

56

12

Fin

d t

he

mea

sure

of

each

in

teri

or a

ngl

e u

sin

g th

e gi

ven

in

form

atio

n.

7.8.

m�

A�

115,

m�

B�

65,

m�

L�

100,

m�

M�

110,

m�

C�

115,

m�

D�

65m

�N

�70

,m�

P�

80

9.qu

adri

late

ral

ST

UW

wit

h �

S�

�T

,10

.hex

agon

DE

FG

HI

wit

h

�U

��

W,m

�S

�2x

�16

,�

D�

�E

� �

G�

�H

,�F

��

I,m

�U

�x

�14

m�

D�

7x,m

�F

�4x

m�

S�

116,

m�

T�

116,

m�

D�

140,

m�

E�

140,

m�

U�

64,m

�W

�64

m�

F�

80,m

�G

�14

0,m

�H

�14

0,m

�I�

80

Fin

d t

he

mea

sure

s of

an

in

teri

or a

ngl

e an

d a

n e

xter

ior

angl

e fo

r ea

ch r

egu

lar

pol

ygon

.

11.q

uad

rila

tera

l12

.pen

tago

n13

.dod

ecag

on

90,9

010

8,72

150,

30

Fin

d t

he

mea

sure

s of

an

in

teri

or a

ngl

e an

d a

n e

xter

ior

angl

e gi

ven

th

e n

um

ber

of

sid

es o

f ea

ch r

egu

lar

pol

ygon

.Rou

nd

to

the

nea

rest

ten

th i

f n

eces

sary

.

14.8

15.9

16.1

3

135,

4514

0,40

152.

3,27

.7

HG

FI

DE

ST

UW

2x�( 2x

� 2

0)�

( 3x

� 1

0)�

( 2x

� 1

0)�

LM

NP

x�

x�

( 2x

� 1

5)�

( 2x

� 1

5)�

AB

CD

©G

lenc

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w-H

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0G

lenc

oe G

eom

etry

Fin

d t

he

sum

of

the

mea

sure

s of

th

e in

teri

or a

ngl

es o

f ea

ch c

onve

x p

olyg

on.

1.11

-gon

2.14

-gon

3.17

-gon

1620

2160

2700

Th

e m

easu

re o

f an

in

teri

or a

ngl

e of

a r

egu

lar

pol

ygon

is

give

n.F

ind

th

e n

um

ber

of s

ides

in

eac

h p

olyg

on.

4.14

45.

156

6.16

0

1015

18

Fin

d t

he

mea

sure

of

each

in

teri

or a

ngl

e u

sin

g th

e gi

ven

in

form

atio

n.

7.8.

quad

rila

tera

l R

ST

Uw

ith

m

�R

�6x

�4,

m�

S�

2x�

8

m�

J�

115,

m�

K�

130,

m�

M�

50,m

�N

�65

m�

R�

128,

m�

S�

52,

m�

T�

128,

m�

U�

52

Fin

d t

he

mea

sure

s of

an

in

teri

or a

ngl

e an

d a

n e

xter

ior

angl

e fo

r ea

ch r

egu

lar

pol

ygon

.Rou

nd

to

the

nea

rest

ten

th i

f n

eces

sary

.

9.16

-gon

10.2

4-go

n11

.30-

gon

157.

5,22

.516

5,15

168,

12

Fin

d t

he

mea

sure

s of

an

in

teri

or a

ngl

e an

d a

n e

xter

ior

angl

e gi

ven

th

e n

um

ber

of

sid

es o

f ea

ch r

egu

lar

pol

ygon

.Rou

nd

to

the

nea

rest

ten

th i

f n

eces

sary

.

12.1

413

.22

14.4

0

154.

3,25

.716

3.6,

16.4

171,

9

15.C

RYST

ALL

OG

RA

PHY

Cry

stal

s ar

e cl

assi

fied

acc

ordi

ng

to s

even

cry

stal

sys

tem

s.T

he

basi

s of

th

e cl

assi

fica

tion

is

the

shap

es o

f th

e fa

ces

of t

he

crys

tal.

Tu

rqu

oise

bel

ongs

to

the

tric

lin

ic s

yste

m.E

ach

of

the

six

face

s of

tu

rqu

oise

is

in t

he

shap

e of

a p

aral

lelo

gram

.F

ind

the

sum

of

the

mea

sure

s of

th

e in

teri

or a

ngl

es o

f on

e su

ch f

ace.

360

RS

TU

x�

( 2x

� 1

5)�

( 3x

� 2

0)�

( x �

15)

�

JK

MN

Pra

ctic

e (

Ave

rag

e)

An

gle

s o

f P

oly

go

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1



Readin

g t

o L

earn

Math

em

ati

csA

ng

les

of

Po

lyg

on

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1

©G

lenc

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ill42

1G

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eom

etry

Lesson 8-1

Pre-

Act

ivit

yH

ow d

oes

a sc

allo

p s

hel

l il

lust

rate

th

e an

gles

of

pol

ygon

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-1

at

the

top

of p

age

404

in y

our

text

book

.

•H

ow m

any

diag

onal

s of

th

e sc

allo

p sh

ell

show

n i

n y

our

text

book

can

be

draw

n f

rom

ver

tex

A?

9•

How

man

y di

agon

als

can

be

draw

n f

rom

on

e ve

rtex

of

an n

-gon

? E

xpla

inyo

ur r

easo

ning

.n

�3;

Sam

ple

an

swer

:An

n-g

on

has

nve

rtic

es,

but

dia

go

nal

s ca

nn

ot

be

dra

wn

fro

m a

ver

tex

to it

self

or

toei

ther

of

the

two

ad

jace

nt

vert

ices

.Th

eref

ore

,th

e n

um

ber

of

dia

go

nal

s fr

om

on

e ve

rtex

is 3

less

th

an t

he

nu

mb

er o

fve

rtic

es.

Rea

din

g t

he

Less

on

1.W

rite

an

exp

ress

ion

th

at d

escr

ibes

eac

h o

f th

e fo

llow

ing

quan

titi

es f

or a

reg

ula

r n

-gon

.If

the

expr

essi

on a

ppli

es t

o re

gula

r po

lygo

ns

only

,wri

te r

egu

lar.

If i

t ap

plie

s to

all

con

vex

poly

gon

s,w

rite

all

.

a.th

e su

m o

f th

e m

easu

res

of t

he

inte

rior

an

gles

180(

n�

2);

all

b.

the

mea

sure

of

each

in

teri

or a

ngl

e�18

0(n n

�2)

�;

reg

ula

r

c.th

e su

m o

f th

e m

easu

res

of t

he

exte

rior

an

gles

(on

e at

eac

h v

erte

x)36

0;al

l

d.

the

mea

sure

of

each

ext

erio

r an

gle

�3 n60 �;

reg

ula

r

2.G

ive

the

mea

sure

of

an in

teri

or a

ngle

and

the

mea

sure

of

an e

xter

ior

angl

e of

eac

h po

lygo

n.a.

equ

ilat

eral

tri

angl

e60

;12

0c.

squ

are

90;

90b

.re

gula

r h

exag

on12

0;60

d.r

egu

lar

octa

gon

135;

45

3.U

nde

rlin

e th

e co

rrec

t w

ord

or p

hra

se t

o fo

rm a

tru

e st

atem

ent

abou

t re

gula

r po

lygo

ns.

a.A

s th

e n

um

ber

of s

ides

in

crea

ses,

the

sum

of

the

mea

sure

s of

th

e in

teri

or a

ngl

es(i

ncr

ease

s/de

crea

ses/

stay

s th

e sa

me)

.b

.A

s th

e n

um

ber

of s

ides

in

crea

ses,

the

mea

sure

of

each

in

teri

or a

ngl

e(i

ncr

ease

s/de

crea

ses/

stay

s th

e sa

me)

.c.

As

the

nu

mbe

r of

sid

es i

ncr

ease

s,th

e su

m o

f th

e m

easu

res

of t

he

exte

rior

an

gles

(in

crea

ses/

decr

ease

s/st

ays

the

sam

e).

d.

As

the

nu

mbe

r of

sid

es i

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8-2

8-2



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Pre-

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Rea

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to

Les

son

8-2

at

the

top

of p

age

411

in y

our

text

book

.

•W

hat

is

the

nam

e of

th

e sh

ape

of t

he

top

surf

ace

of e

ach

wed

ge o

f ch

eese

?p

aral

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m•

Are

the

thr

ee p

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ons

show

n in

the

dra

win

g si

mila

r po

lygo

ns?

Exp

lain

you

rre

ason

ing.

No

;sam

ple

an

swer

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eir

sid

es a

re n

ot

pro

po

rtio

nal

.

Rea

din

g t

he

Less

on

1.U

nder

line

wor

ds o

r ph

rase

s th

at c

an c

ompl

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the

foll

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g se

nten

ces

to m

ake

stat

emen

tsth

at a

re a

lway

s tr

ue.(

The

re m

ay b

e m

ore

than

one

cor

rect

cho

ice

for

som

e of

the

sen

tenc

es.)

a.O

ppos

ite

side

s of

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are

(co

ngr

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t/pe

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para

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).

b.

Con

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of

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rall

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are

(co

mpl

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tary

/su

pple

men

tary

/con

gru

ent)

.

c.A

dia

gon

al o

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para

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ivid

es t

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para

llel

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nto

tw

o(a

cute

/rig

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obtu

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ongr

uen

t) t

rian

gles

.

d.

Opp

osit

e an

gles

of

a pa

rall

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ram

are

(co

mpl

emen

tary

/su

pple

men

tary

/con

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ent)

.

e.T

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diag

onal

s of

a p

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lelo

gram

(bi

sect

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h o

ther

/are

per

pen

dicu

lar/

are

con

gru

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.

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lelo

gram

has

on

e ri

ght

angl

e,th

en a

ll o

f it

s ot

her

an

gles

are

(acu

te/r

igh

t/ob

tuse

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2.L

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BC

Dbe

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es.

a.S

ketc

h a

par

alle

logr

am t

hat

mat

ches

th

e de

scri

ptio

nS

amp

le a

nsw

er:

abov

e an

d dr

aw d

iago

nal

B�D�

.

In p

arts

b–f

,com

plet

e ea

ch s

ente

nce

.

b.

A�B�

|| an

d A�

D�||

.

c.A�

B��

and

B �C�

�B

.

d.

�A

�an

d �

AB

C�

.

e.�

AD

B�

beca

use

th

ese

two

angl

es a

re

angl

es f

orm

ed b

y th

e tw

o pa

rall

el l

ines

an

d an

d th

e

tran

sver

sal

.

f.�

AB

D�

.

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

new

th

eore

ms

in g

eom

etry

is

to r

elat

e th

em t

o th

eore

ms

you

lear

ned

ear

lier

.Nam

e a

theo

rem

abo

ut

para

llel

lin

es t

hat

can

be

use

d to

rem

embe

r th

eth

eore

m t

hat

say

s,“I

f a

para

llel

ogra

m h

as o

ne

righ

t an

gle,

it h

as f

our

righ

t an

gles

.”P

erp

end

icu

lar T

ran

sver

sal T

heo

rem

�C

DBBD

��

BC

��

AD

��

inte

rio

ral

tern

ate

�C

BD

�C

DA

�C

A�D�

C�D�

B�C�

C�D�

DA

CB

©G

lenc

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w-H

ill42

8G

lenc

oe G

eom

etry

Tess

ella

tio

ns

A t

esse

llat

ion

is

a ti

lin

g pa

tter

n m

ade

of p

olyg

ons.

Th

e pa

tter

n c

an

be e

xten

ded

so t

hat

th

e po

lygo

nal

til

es c

over

th

e pl

ane

com

plet

ely

wit

hn

o ga

ps.A

ch

ecke

rboa

rd a

nd

a h

oney

com

b pa

tter

n a

re e

xam

ples

of

tess

ella

tion

s.S

omet

imes

th

e sa

me

poly

gon

can

mak

e m

ore

than

on

ete

ssel

lati

on p

atte

rn.B

oth

pat

tern

s be

low

can

be

form

ed f

rom

an

isos

cele

s tr

ian

gle.

Dra

w a

tes

sell

atio

n u

sin

g ea

ch p

olyg

on.

1.2.

3.

Dra

w t

wo

dif

fere

nt

tess

ella

tion

s u

sin

g ea

ch p

olyg

on.

4.5.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-2

8-2



Stu

dy G

uid

e a

nd I

nte

rven

tion

Test

s fo

r P

aral

lelo

gra

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-3

8-3

©G

lenc

oe/M

cGra

w-H

ill42

9G

lenc

oe G

eom

etry

Lesson 8-3

Co

nd

itio

ns

for

a Pa

ralle

log

ram

Th

ere

are

man

y w

ays

to

esta

blis

h t

hat

a q

uad

rila

tera

l is

a p

aral

lelo

gram

.

If:

If:

both

pai

rs o

f op

posi

te s

ides

are

par

alle

l,A�

B�|| D�

C�an

d A�

D�|| B�

C�,

both

pai

rs o

f op

posi

te s

ides

are

con

grue

nt,

A�B�

� D�

C�an

d A�

D��

B�C�

,

both

pai

rs o

f op

posi

te a

ngle

s ar

e co

ngru

ent,

�A

BC

��

AD

Can

d �

DA

B�

�B

CD

,

the

diag

onal

s bi

sect

eac

h ot

her,

A�E�

� C�

E�an

d D�

E��

B�E�

,

one

pair

of o

ppos

ite s

ides

is c

ongr

uent

and

par

alle

l,A�

B�|| C�

D�an

d A�

B��

C�D�

, or

A�D�

|| B�C�

and

A�D�

�B�

C�,

then

:th

e fig

ure

is a

par

alle

logr

am.

then

:A

BC

Dis

a p

aral

lelo

gram

.

Fin

d x

and

yso

th

at F

GH

Jis

a

par

alle

logr

am.

FG

HJ

is a

par

alle

logr

am i

f th

e le

ngt

hs

of t

he

oppo

site

si

des

are

equ

al.

6x�

3 �

154x

�2y

�2

6x�

124(

2) �

2y�

2x

�2

8 �

2y�

2�

2y�

�6

y�

3

Fin

d x

and

yso

th

at e

ach

qu

adri

late

ral

is a

par

alle

logr

am.

1.x

�7;

y�

42.

x�

5;y

�25

3.x

�31

;y

�5

4.x

�5;

y�

3

5.x

�15

;y

�9

6.x

�30

;y

�15

6y�

3 x�

( x �

y) �

2x�

30�

24�

9x�

6y45

�18

5x�

5y�

25�

11x�

5y�

55�

2x �

2

2 y8

12

JHG

F6x

� 3

154x

� 2

y2

AB

CD

E

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

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cGra

w-H

ill43

0G

lenc

oe G

eom

etry

Para

llelo

gra

ms

on

th

e C

oo

rdin

ate

Plan

eO

n t

he

coor

din

ate

plan

e,th

e D

ista

nce

For

mu

la a

nd

the

Slo

pe F

orm

ula

can

be

use

d to

tes

t if

a q

uad

rila

tera

l is

a p

aral

lelo

gram

.

Det

erm

ine

wh

eth

er A

BC

Dis

a p

aral

lelo

gram

.T

he

vert

ices

are

A(�

2,3)

,B(3

,2),

C(2

,�1)

,an

d D

(�3,

0).

Met

hod

1:U

se t

he

Slo

pe F

orm

ula

,m�

�y x2 2

� �

y x1 1�

.

slop

e of

A �D�

��

23 ��

(0 �3)

��

�3 1��

3sl

ope

of B�

C��

�23�

�(�21)

��

�3 1��

3

slop

e of

A �B�

�� 3

2 �

� (�3 2)

��

��1 5�

slop

e of

C�D�

�� 2�

�1(� �

0 3)�

��

�1 5�

Opp

osit

e si

des

hav

e th

e sa

me

slop

e,so

A�B�

|| C�D�

and

A�D�

|| B�C�

.Bot

h p

airs

of

oppo

site

sid

esar

e pa

rall

el,s

o A

BC

Dis

a p

aral

lelo

gram

.

Met

hod

2:U

se t

he

Dis

tan

ce F

orm

ula

,d�

�(x

2�

�x 1

)2�

�(y

2�

�y 1

)2�

.

AB

��

(�2

��

3)2

��

(3 �

�2)

2�

��

25 �

�1�

or �

26�

CD

��

(2 �

(�

�3)

)2�

�(�

1�

�0)

2�

��

25 �

�1�

or �

26�

AD

��

(�2

��

(�3)

)�

2�

(3�

�0)

2�

��

1 �

9�

or �

10�

BC

��

(3 �

2�

)2�

(�

2 �

(��

1))2

��

�1

�9

�or

�10�

Bot

h p

airs

of

oppo

site

sid

es h

ave

the

sam

e le

ngt

h,s

o A

BC

Dis

a p

aral

lelo

gram

.

Det

erm

ine

wh

eth

er a

fig

ure

wit

h t

he

give

n v

erti

ces

is a

par

alle

logr

am.U

se t

he

met

hod

in

dic

ated

.

1.A

(0,0

),B

(1,3

),C

(5,3

),D

(4,0

);2.

D(�

1,1)

,E(2

,4),

F(6

,4),

G(3

,1);

Slo

pe F

orm

ula

Slo

pe F

orm

ula

yes

yes

3.R

(�1,

0),S

(3,0

),T

(2,�

3),U

(�3,

�2)

;4.

A(�

3,2)

,B(�

1,4)

,C(2

,1),

D(0

,�1)

;D

ista

nce

For

mu

laD

ista

nce

an

d S

lope

For

mu

las

no

yes

5.S

(�2,

4),T

(�1,

�1)

,U(3

,�4)

,V(2

,1);

6.F

(3,3

),G

(1,2

),H

(�3,

1),I

(�1,

4);

Dis

tan

ce a

nd

Slo

pe F

orm

ula

sM

idpo

int

For

mu

la

yes

no

7.A

par

alle

logr

am h

as v

erti

ces

R(�

2,�

1),S

(2,1

),an

d T

(0,�

3).F

ind

all

poss

ible

coor

din

ates

for

th

e fo

urt

h v

erte

x.

(4,�

1),(

0,3)

,or

(�4,

�5)

x

y

O

C

A

D

B

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Test

s fo

r P

aral

lelo

gra

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-3

8-3

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Test

s fo

r P

aral

lelo

gra

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-3

8-3

©G

lenc

oe/M

cGra

w-H

ill43

1G

lenc

oe G

eom

etry

Lesson 8-3

Det

erm

ine

wh

eth

er e

ach

qu

adri

late

ral

is a

par

alle

logr

am.J

ust

ify

you

r an

swer

.

1.2.

Yes;

a p

air

of

op

po

site

sid

esYe

s;b

oth

pai

rs o

f o

pp

osi

teis

par

alle

l an

d c

on

gru

ent.

ang

les

are

con

gru

ent.

3.4.

No

;n

on

e o

f th

e te

sts

for

Yes;

bo

th p

airs

of

op

po

site

sid

esp

aral

lelo

gra

ms

is f

ulf

illed

.ar

e co

ng

ruen

t.

CO

OR

DIN

ATE

GEO

MET

RYD

eter

min

e w

het

her

a f

igu

re w

ith

th

e gi

ven

ver

tice

s is

ap

aral

lelo

gram

.Use

th

e m

eth

od i

nd

icat

ed.

5.P

(0,0

),Q

(3,4

),S

(7,4

),Y

(4,0

);S

lope

For

mu

la

yes

6.S

(�2,

1),R

(1,3

),T

(2,0

),Z

(�1,

�2)

;Dis

tan

ce a

nd

Slo

pe F

orm

ula

yes

7.W

(2,5

),R

(3,3

),Y

(�2,

�3)

,N(�

3,1)

;Mid

poin

t F

orm

ula

no

ALG

EBR

AF

ind

xan

d y

so t

hat

eac

h q

uad

rila

tera

l is

a p

aral

lelo

gram

.

8.9.

x�

24,y

�19

x�

3,y

�14

10.

11.

x�

45,y

�20

x�

17,y

�9

3x �

14

x �

20

3y �

2y

� 2

0( 4

x �

35)

�

( 3x

� 1

0)�

( 2y

� 5

) �

( y �

15)

�

2y �

11

y � 3

4x � 3

3x

x �

16

y �

19

2y

2x �

8

©G

lenc

oe/M

cGra

w-H

ill43

2G

lenc

oe G

eom

etry

Det

erm

ine

wh

eth

er e

ach

qu

adri

late

ral

is a

par

alle

logr

am.J

ust

ify

you

r an

swer

.

1.2.

Yes;

the

dia

go

nal

s b

isec

tN

o;

no

ne

of

the

test

s fo

rea

ch o

ther

.p

aral

lelo

gra

ms

is f

ulf

illed

.

3.4.

Yes;

bo

th p

airs

of

op

po

site

No

;n

on

e o

f th

e te

sts

for

ang

les

are

con

gru

ent.

par

alle

log

ram

s is

fu

lfill

ed.

CO

OR

DIN

ATE

GEO

MET

RYD

eter

min

e w

het

her

a f

igu

re w

ith

th

e gi

ven

ver

tice

s is

ap

aral

lelo

gram

.Use

th

e m

eth

od i

nd

icat

ed.

5.P

(�5,

1),S

(�2,

2),F

(�1,

�3)

,T(2

,�2)

;Slo

pe F

orm

ula

no

6.R

(�2,

5),O

(1,3

),M

(�3,

�4)

,Y(�

6,�

2);D

ista

nce

an

d S

lope

For

mu

la

yes

ALG

EBR

AF

ind

xan

d y

so t

hat

eac

h q

uad

rila

tera

l is

a p

aral

lelo

gram

.

7.8.

x�

20,y

�12

x�

�6,

y�

13

9.10

.

x�

�3,

y�

2x

��

2,y

��

5

11.T

ILE

DES

IGN

Th

e pa

tter

n s

how

n i

n t

he

figu

re i

s to

con

sist

of

con

gru

ent

para

llel

ogra

ms.

How

can

th

e de

sign

er b

e ce

rtai

n t

hat

th

e sh

apes

are

para

llel

ogra

ms?

Sam

ple

an

swer

:C

on

firm

th

at b

oth

pai

rs o

f o

pp

osi

te �

s ar

e �

.

�4y

� 2

x � 12

�2x

� 6

y �

23

�4x

� 6

�6x

7y �

312

y �

7

3y �

5�

3x

� 4

�4x

� 2

2y �

8( 5

x �

29)

�

( 7x

� 1

1)�

( 3y

� 1

5)�

( 5y

� 9

) �

118�

62�

118�

62�

Pra

ctic

e (

Ave

rag

e)

Test

s fo

r P

aral

lelo

gra

ms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-3

8-3



Readin

g t

o L

earn

Math

em

ati

csTe

sts

for

Par

alle

log

ram

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-3

8-3

©G

lenc

oe/M

cGra

w-H

ill43

3G

lenc

oe G

eom

etry

Lesson 8-3

Pre-

Act

ivit

yH

ow a

re p

aral

lelo

gram

s u

sed

in

arc

hit

ectu

re?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-3

at

the

top

of p

age

417

in y

our

text

book

.

Mak

e tw

o ob

serv

atio

ns

abou

t th

e an

gles

in

th

e ro

of o

f th

e co

vere

d br

idge

.S

amp

le a

nsw

er:

Op

po

site

an

gle

s ap

pea

r co

ng

ruen

t.C

on

secu

tive

an

gle

s ap

pea

r su

pp

lem

enta

ry.

Rea

din

g t

he

Less

on

1.W

hic

h o

f th

e fo

llow

ing

con

diti

ons

guar

ante

e th

at a

qu

adri

late

ral

is a

par

alle

logr

am?

A.

Tw

o si

des

are

para

llel

.B

,D,G

,H,J

,LB

.Bot

h p

airs

of

oppo

site

sid

es a

re c

ongr

uen

t.C

.T

he

diag

onal

s ar

e pe

rpen

dicu

lar.

D.A

pai

r of

opp

osit

e si

des

is b

oth

par

alle

l an

d co

ngr

uen

t.E

.T

her

e ar

e tw

o ri

ght

angl

es.

F.T

he

sum

of

the

mea

sure

s of

th

e in

teri

or a

ngl

es i

s 36

0.G

.All

fou

r si

des

are

con

gru

ent.

H.B

oth

pai

rs o

f op

posi

te a

ngl

es a

re c

ongr

uen

t.I.

Tw

o an

gles

are

acu

te a

nd

the

oth

er t

wo

angl

es a

re o

btu

se.

J.T

he

diag

onal

s bi

sect

eac

h o

ther

.K

.Th

e di

agon

als

are

con

gru

ent.

L.

All

fou

r an

gles

are

rig

ht

angl

es.

2.D

eter

min

e w

het

her

th

ere

is e

nou

gh g

iven

in

form

atio

n t

o kn

ow t

hat

eac

h f

igu

re i

s a

para

llel

ogra

m.I

f so

,sta

te t

he

defi

nit

ion

or

theo

rem

th

at ju

stif

ies

you

r co

ncl

usi

on.

a.b

.

no

Yes;

if t

he

dia

go

nal

s b

isec

t ea

cho

ther

,th

en t

he

qu

adri

late

ral i

s a

par

alle

log

ram

.c.

d.

Yes;

if b

oth

pai

rs o

f o

pp

osi

ten

o

sid

es a

re �

,th

en q

uad

rila

tera

l is

�.

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r a

larg

e n

um

ber

of m

ath

emat

ical

ide

as i

s to

th

ink

of t

hem

in

grou

ps.H

ow c

an y

ou s

tate

th

e co

ndi

tion

s as

on

e gr

oup

abou

t th

e si

des

of q

uad

rila

tera

lsth

at g

uar

ante

e th

at t

he

quad

rila

tera

l is

a p

aral

lelo

gram

?S

amp

le a

nsw

er:

bo

thp

airs

of

op

po

site

sid

es p

aral

lel,

bo

th p

airs

of

op

po

site

sid

es c

on

gru

ent,

or

on

e p

air

of

op

po

site

sid

es b

oth

par

alle

l an

d c

on

gru

ent

©G

lenc

oe/M

cGra

w-H

ill43

4G

lenc

oe G

eom

etry

Test

s fo

r P

aral

lelo

gra

ms

By

defi

nit

ion

,a q

uad

rila

tera

l is

a p

aral

lelo

gram

if

and

on

ly i

fbo

th p

airs

of

oppo

site

sid

esar

e pa

rall

el.W

hat

con

diti

ons

oth

er t

han

bot

h p

airs

of

oppo

site

sid

es p

aral

lel

wil

l gu

aran

tee

that

a q

uad

rila

tera

l is

a p

aral

lelo

gram

? In

th

is a

ctiv

ity,

seve

ral

poss

ibil

itie

s w

ill

bein

vest

igat

ed b

y dr

awin

g qu

adri

late

rals

to

sati

sfy

cert

ain

con

diti

ons.

Rem

embe

r th

at a

ny

test

th

at s

eem

s to

wor

k is

not

gu

aran

teed

to

wor

k u

nle

ss i

t ca

n b

e fo

rmal

ly p

rove

n.

Com

ple

te.

1.D

raw

a q

uad

rila

tera

l w

ith

on

e pa

ir o

f op

posi

te s

ides

con

gru

ent.

Mu

st i

t be

a p

aral

lelo

gram

?n

o

2.D

raw

a q

uad

rila

tera

l w

ith

bot

h p

airs

of

oppo

site

sid

es c

ongr

uen

t.M

ust

it

be a

par

alle

logr

am?

yes

3.D

raw

a q

uad

rila

tera

l w

ith

on

e pa

ir o

f op

posi

te s

ides

par

alle

l an

d th

e ot

her

pai

r of

opp

osit

e si

des

con

gru

ent.

Mu

st i

t be

a

para

llel

ogra

m?

no

4.D

raw

a q

uad

rila

tera

l w

ith

on

e pa

ir o

f op

posi

te s

ides

bot

h p

aral

lel

and

con

gru

ent.

Mu

st i

t be

a p

aral

lelo

gram

?ye

s

5.D

raw

a q

uad

rila

tera

l w

ith

on

e pa

ir o

f op

posi

te a

ngl

es c

ongr

uen

t.M

ust

it

be a

par

alle

logr

am?

no

6.D

raw

a q

uad

rila

tera

l w

ith

bot

h p

airs

of

oppo

site

an

gles

con

gru

ent.

Mu

st i

t be

a p

aral

lelo

gram

?ye

s

7.D

raw

a q

uad

rila

tera

l w

ith

on

e pa

ir o

f op

posi

te s

ides

par

alle

l an

d on

e pa

ir o

f op

posi

te a

ngl

es c

ongr

uen

t.M

ust

it

be a

par

alle

logr

am?

yes

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-3

8-3



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Rec

tan

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-4

8-4

©G

lenc

oe/M

cGra

w-H

ill43

5G

lenc

oe G

eom

etry

Lesson 8-4

Pro

per

ties

of

Rec

tan

gle

sA

rec

tan

gle

is a

qu

adri

late

ral

wit

h f

our

righ

t an

gles

.Her

e ar

e th

e pr

oper

ties

of

rect

angl

es.

A r

ecta

ngl

e h

as a

ll t

he

prop

erti

es o

f a

para

llel

ogra

m.

•O

ppos

ite

side

s ar

e pa

rall

el.

•O

ppos

ite

angl

es a

re c

ongr

uen

t.•

Opp

osit

e si

des

are

con

gru

ent.

•C

onse

cuti

ve a

ngl

es a

re s

upp

lem

enta

ry.

•T

he

diag

onal

s bi

sect

eac

h o

ther

.

Als

o:

•A

ll f

our

angl

es a

re r

igh

t an

gles

.�

UT

S,�

TS

R,�

SR

U,a

nd

�R

UT

are

righ

t an

gles

.•

Th

e di

agon

als

are

con

gru

ent.

T �R�

� U�

S�

TS R

U

Q

In r

ecta

ngl

e R

ST

Uab

ove,

US

�6x

�3

and

RT

�7x

�2.

Fin

d x

.T

he

diag

onal

s of

a r

ecta

ngl

e bi

sect

eac

hot

her

,so

US

�R

T.

6x�

3�

7x�

23

�x

�2

5�

x

In r

ecta

ngl

e R

ST

Uab

ove,

m�

ST

R�

8x�

3 an

d m

�U

TR

�16

x�

9.F

ind

m�

ST

R.

�U

TS

is a

rig

ht

angl

e,so

m

�S

TR

�m

�U

TR

�90

.

8x�

3 �

16x

�9

�90

24x

�6

�90

24x

�96

x�

4

m�

ST

R�

8x�

3 �

8(4)

�3

or 3

5

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

AB

CD

is a

rec

tan

gle.

1.If

AE

�36

an

d C

E�

2x�

4,fi

nd

x.20

2.If

BE

�6y

�2

and

CE

�4y

�6,

fin

d y.

2

3.If

BC

�24

an

d A

D�

5y�

1,fi

nd

y.5

4.If

m�

BE

A�

62,f

ind

m�

BA

C.

59

5.If

m�

AE

D�

12x

and

m�

BE

C�

10x

�20

,fin

d m

�A

ED

.12

0

6.If

BD

�8y

�4

and

AC

�7y

�3,

fin

d B

D.

52

7.If

m�

DB

C�

10x

and

m�

AC

B�

4x2 �

6,fi

nd

m�

AC

B.

30

8.If

AB

�6y

and

BC

�8y

,fin

d B

Din

ter

ms

of y

.10

y

9.In

rec

tan

gle

MN

OP

,m�

1 �

40.F

ind

the

mea

sure

of

each

n

um

bere

d an

gle.

m�

2 �

40;

m�

3 �

50;

m�

4 �

50;

m�

5 �

80;

m�

6 �

100

MN O

P

K 12

345

6

BC D

A

E

©G

lenc

oe/M

cGra

w-H

ill43

6G

lenc

oe G

eom

etry

Pro

ve t

hat

Par

alle

log

ram

s A

re R

ecta

ng

les

Th

e di

agon

als

of a

rec

tan

gle

are

con

gru

ent,

and

the

con

vers

e is

als

o tr

ue.

If th

e di

agon

als

of a

par

alle

logr

am a

re c

ongr

uent

, th

en t

he p

aral

lelo

gram

is a

rec

tang

le.

In t

he

coor

din

ate

plan

e yo

u c

an u

se t

he

Dis

tan

ce F

orm

ula

,th

e S

lope

For

mu

la,a

nd

prop

erti

es o

f di

agon

als

to s

how

th

at a

fig

ure

is

a re

ctan

gle.

Det

erm

ine

wh

eth

er A

(�3,

0),B

(�2,

3),C

(4,1

),an

d D

(3,�

2) a

re t

he

vert

ices

of

a re

ctan

gle.

Met

hod

1:U

se t

he

Slo

pe F

orm

ula

.

slop

e of

A �B�

��

23 ��

(0 �3)

��

�3 1�or

3sl

ope

of A�

D��

� 3�

�2(� �

0 3)�

�� 62 �

or �

�1 3�

slop

e of

C�D�

�� 32 ��

41�

��

3 1�or

3sl

ope

of B�

C��

� 41 �

� (�3 2)

��

�� 62 �or

��1 3�

Opp

osit

e si

des

are

para

llel

,so

the

figu

re i

s a

para

llel

ogra

m.C

onse

cuti

ve s

ides

are

perp

endi

cula

r,so

AB

CD

is a

rec

tan

gle.

Met

hod

2:U

se t

he

Mid

poin

t an

d D

ista

nce

For

mu

las.

Th

e m

idpo

int

of A�

C�is

��3 2�

4�

,�0

� 21

��

��1 2� ,�1 2� �

and

the

mid

poin

t of

B�D�

is

��2 2�

3�

,�3

� 22

��

��1 2� ,�1 2� �.

Th

e di

agon

als

hav

e th

e sa

me

mid

poin

t so

th

ey b

isec

t ea

ch o

ther

.

Th

us,

AB

CD

is a

par

alle

logr

am.

AC

��

(�3

��

4)2

��

(0 �

�1)

2�

��

49 �

�1�

or �

50�B

D�

�(�

2 �

�3)

2�

�(3

��

(�2)

)2�

��

25 �

�25�

or �

50�T

he

diag

onal

s ar

e co

ngr

uen

t.A

BC

Dis

a p

aral

lelo

gram

wit

h d

iago

nal

s th

at b

isec

t ea

chot

her

,so

it i

s a

rect

angl

e.

Det

erm

ine

wh

eth

er A

BC

Dis

a r

ecta

ngl

e gi

ven

eac

h s

et o

f ve

rtic

es.J

ust

ify

you

ran

swer

.S

amp

le ju

stif

icat

ion

s ar

e g

iven

.

1.A

(�3,

1),B

(�3,

3),C

(3,3

),D

(3,1

)2.

A(�

3,0)

,B(�

2,3)

,C(4

,5),

D(3

,2)

Yes;

con

secu

tive

sid

es a

re ⊥

.N

o;

con

secu

tive

sid

es a

re n

ot

⊥.

3.A

(�3,

0),B

(�2,

2),C

(3,0

),D

(2,�

2)4.

A(�

1,0)

,B(0

,2),

C(4

,0),

D(3

,�2)

No

;co

nse

cuti

ve s

ides

are

no

t ⊥.

Yes;

con

secu

tive

sid

es a

re ⊥

.

5.A

(�1,

�5)

,B(�

3,0)

,C(2

,2),

D(4

,�3)

6.A

(�1,

�1)

,B(0

,2),

C(4

,3),

D(3

,0)

Yes;

the

dia

go

nal

s ar

e co

ng

ruen

t N

o;

the

dia

go

nal

s ar

e n

ot

�.

and

bis

ect

each

oth

er.

7.A

par

alle

logr

am h

as v

erti

ces

R(�

3,�

1),S

(�1,

2),a

nd

T(5

,�2)

.Fin

d th

e co

ordi

nat

es o

fU

so t

hat

RS

TU

is a

rec

tan

gle.

(3,�

5)

x

y O

D

B

A

EC

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Rec

tan

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-4

8-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Rec

tan

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-4

8-4

©G

lenc

oe/M

cGra

w-H

ill43

7G

lenc

oe G

eom

etry

Lesson 8-4

ALG

EBR

AA

BC

Dis

a r

ecta

ngl

e.

1.If

AC

�2x

�13

an

d D

B�

4x�

1,fi

nd

x.7

2.If

AC

�x

�3

and

DB

�3x

�19

,fin

d A

C.

14

3.If

AE

�3x

�3

and

EC

�5x

�15

,fin

d A

C.

60

4.If

DE

�6x

�7

and

AE

�4x

�9,

fin

d D

B.

82

5.If

m�

DA

C�

2x�

4 an

d m

�B

AC

�3x

�1,

fin

d x.

17

6.If

m�

BD

C�

7x�

1 an

d m

�A

DB

�9x

�7,

fin

d m

�B

DC

.43

7.If

m�

AB

D�

x2�

7 an

d m

�C

DB

�4x

�5,

fin

d x.

6

8.If

m�

BA

C�

x2�

3 an

d m

�C

AD

�x

�15

,fin

d m

�B

AC

.67

or

84

PR

ST

is a

rec

tan

gle.

Fin

d e

ach

mea

sure

if

m�

1 �

50.

9.m

�2

4010

.m�

340

11.m

�4

5012

.m�

540

13.m

�6

4014

.m�

750

15.m

�8

100

16.m

�9

80

CO

OR

DIN

ATE

GEO

MET

RYD

eter

min

e w

het

her

TU

XY

is a

rec

tan

gle

give

n e

ach

set

of v

erti

ces.

Ju

stif

y yo

ur

answ

er.

17.T

(�3,

�2)

,U(�

4,2)

,X(2

,4),

Y(3

,0)

No

;sa

mp

le a

nsw

er:

An

gle

s ar

e n

ot

rig

ht

ang

les.

18.T

(�6,

3),U

(0,6

),X

(2,2

),Y

(�4,

�1)

Yes;

sam

ple

an

swer

:O

pp

osi

te s

ides

are

co

ng

ruen

t an

d d

iag

on

als

are

con

gru

ent.

19.T

(4,1

),U

(3,�

1),X

(�3,

2),Y

(�2,

4)

Yes;

sam

ple

an

swer

:O

pp

osi

te s

ides

are

par

alle

l an

d c

on

secu

tive

sid

esar

e p

erp

end

icu

lar.

T

12

34

56

78

9

SRPD

CBA

E

©G

lenc

oe/M

cGra

w-H

ill43

8G

lenc

oe G

eom

etry

ALG

EBR

AR

ST

Uis

a r

ecta

ngl

e.

1.If

UZ

�x

�21

an

d Z

S�

3x�

15,f

ind

US

.78

2.If

RZ

�3x

�8

and

ZS

�6x

�28

,fin

d U

Z.

44

3.If

RT

�5x

�8

and

RZ

�4x

�1,

fin

d Z

T.

9

4.If

m�

SU

T�

3x�

6 an

d m

�R

US

�5x

�4,

fin

d m

�S

UT

.39

5.If

m�

SR

T�

x2�

9 an

d m

�U

TR

�2x

�44

,fin

d x.

�5

or

7

6.If

m�

RS

U�

x2�

1 an

d m

�T

US

�3x

�9,

fin

d m

�R

SU

.24

or

3

GH

JK

is a

rec

tan

gle.

Fin

d e

ach

mea

sure

if

m�

1 �

37.

7.m

�2

538.

m�

337

9.m

�4

3710

.m�

553

11.m

�6

106

12.m

�7

74

CO

OR

DIN

ATE

GEO

MET

RYD

eter

min

e w

het

her

BG

HL

is a

rec

tan

gle

give

n e

ach

set

of v

erti

ces.

Ju

stif

y yo

ur

answ

er.

13.B

(�4,

3),G

(�2,

4),H

(1,�

2),L

(�1,

�3)

Yes;

sam

ple

an

swer

:O

pp

osi

te s

ides

are

par

alle

l an

d c

on

secu

tive

sid

esar

e p

erp

end

icu

lar.

14.B

(�4,

5),G

(6,0

),H

(3,�

6),L

(�7,

�1)

Yes;

sam

ple

an

swer

:O

pp

osi

te s

ides

are

co

ng

ruen

t an

d d

iag

on

als

are

con

gru

ent.

15.B

(0,5

),G

(4,7

),H

(5,4

),L

(1,2

)

No

;sa

mp

le a

nsw

er:

Dia

go

nal

s ar

e n

ot

con

gru

ent.

16.L

AN

DSC

API

NG

Hu

nti

ngt

on P

ark

offi

cial

s ap

prov

ed a

rec

tan

gula

r pl

ot o

f la

nd

for

aJa

pan

ese

Zen

gar

den

.Is

it s

uff

icie

nt

to k

now

th

at o

ppos

ite

side

s of

th

e ga

rden

plo

t ar

eco

ngr

uen

t an

d pa

rall

el t

o de

term

ine

that

th

e ga

rden

plo

t is

rec

tan

gula

r? E

xpla

in.

No

;if

yo

u o

nly

kn

ow

th

at o

pp

osi

te s

ides

are

co

ng

ruen

t an

d p

aral

lel,

the

mo

st y

ou

can

co

ncl

ud

e is

th

at t

he

plo

t is

a p

aral

lelo

gra

m.

K

21

5

43

67

JHGU

TSR

Z

Pra

ctic

e (

Ave

rag

e)

Rec

tan

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-4

8-4



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csR

ecta

ng

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-4

8-4

©G

lenc

oe/M

cGra

w-H

ill43

9G

lenc

oe G

eom

etry

Lesson 8-4

Pre-

Act

ivit

yH

ow a

re r

ecta

ngl

es u

sed

in

ten

nis

?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-4

at

the

top

of p

age

424

in y

our

text

book

.

Are

th

e si

ngl

es c

ourt

an

d do

ubl

es c

ourt

sim

ilar

rec

tan

gles

? E

xpla

in y

our

answ

er.

No

;sa

mp

le a

nsw

er:

All

of

thei

r an

gle

s ar

e co

ng

ruen

t,bu

t th

eir

sid

es a

re n

ot

pro

po

rtio

nal

.Th

e d

ou

ble

s co

urt

is w

ider

in r

elat

ion

to

its

len

gth

.

Rea

din

g t

he

Less

on

1.D

eter

min

e w

het

her

eac

h s

ente

nce

is

alw

ays,

som

etim

es,o

r n

ever

tru

e.

a.If

a q

uad

rila

tera

l h

as f

our

con

gru

ent

angl

es,i

t is

a r

ecta

ngl

e.al

way

sb

.If

con

secu

tive

an

gles

of

a qu

adri

late

ral

are

supp

lem

enta

ry,t

hen

th

e qu

adri

late

ral

is a

rec

tan

gle.

som

etim

esc.

Th

e di

agon

als

of a

rec

tan

gle

bise

ct e

ach

oth

er.

alw

ays

d.

If t

he

diag

onal

s of

a q

uad

rila

tera

l bi

sect

eac

h o

ther

,th

e qu

adri

late

ral

is a

re

ctan

gle.

som

etim

ese.

Con

secu

tive

an

gles

of

a re

ctan

gle

are

com

plem

enta

ry.

nev

erf.

Con

secu

tive

an

gles

of

a re

ctan

gle

are

con

gru

ent.

alw

ays

g.If

th

e di

agon

als

of a

qu

adri

late

ral

are

con

gru

ent,

the

quad

rila

tera

l is

a

rect

angl

e.so

met

imes

h.

A d

iago

nal

of

a re

ctan

gle

bise

cts

two

of i

ts a

ngl

es.

som

etim

esi.

A d

iago

nal

of

a re

ctan

gle

divi

des

the

rect

angl

e in

to t

wo

con

gru

ent

righ

t tr

ian

gles

.al

way

sj.

If t

he

diag

onal

s of

a q

uad

rila

tera

l bi

sect

eac

h o

ther

an

d ar

e co

ngr

uen

t,th

equ

adri

late

ral

is a

rec

tan

gle.

alw

ays

k.

If a

par

alle

logr

am h

as o

ne

righ

t an

gle,

it i

s a

rect

angl

e.al

way

sl.

If a

par

alle

logr

am h

as f

our

con

gru

ent

side

s,it

is

a re

ctan

gle.

som

etim

es

2.A

BC

Dis

a r

ecta

ngl

e w

ith

AD

�A

B.

Nam

e ea

ch o

f th

e fo

llow

ing

in t

his

fig

ure

.

a.al

l se

gmen

ts t

hat

are

con

gru

ent

to B�

E�A�

E�,C�

E�,D�

E�b

.al

l an

gles

con

gru

ent

to �

1�

2,�

5,�

6c.

all

angl

es c

ongr

uen

t to

�7

�3,

�4,

�8

d.

two

pair

s of

con

gru

ent

tria

ngl

es�

AE

D�

�B

EC

,�A

EB

��

DE

C,

�B

CD

��

DA

B,�

AB

C�

�C

DA

Hel

pin

g Y

ou

Rem

emb

er3.

It i

s ea

sier

to

rem

embe

r a

larg

e n

um

ber

of g

eom

etri

c re

lati

onsh

ips

and

theo

rem

s if

you

are

able

to

com

bin

e so

me

of t

hem

.How

can

you

com

bin

e th

e tw

o th

eore

ms

abou

tdi

agon

als

that

you

stu

died

in

th

is l

esso

n?

Sam

ple

an

swer

:A

par

alle

log

ram

is a

rect

ang

le if

an

d o

nly

if it

s d

iag

on

als

are

con

gru

ent.

A12

34

5 67

8DC

BE

©G

lenc

oe/M

cGra

w-H

ill44

0G

lenc

oe G

eom

etry

Co

un

tin

g S

qu

ares

an

d R

ecta

ng

les

Eac

h p

uzz

le b

elow

con

tain

s m

any

squ

ares

an

d/or

rec

tan

gles

.C

oun

t th

em c

aref

ull

y.Yo

u m

ay w

ant

to l

abel

eac

h r

egio

n s

o yo

uca

n l

ist

all

poss

ibil

itie

s.

How

man

y re

ctan

gles

are

in

th

e fi

gure

at

th

e ri

ght?

Lab

el e

ach

sm

all

regi

on w

ith

a l

ette

r.T

hen

lis

t ea

ch r

ecta

ngl

e by

wri

tin

g th

e le

tter

s of

reg

ion

s it

con

tain

s.

A,B

,C,D

,AB

,CD

,AC

,BD

,AB

CD

Th

ere

are

9 re

ctan

gles

.

How

man

y sq

uar

es a

re i

n e

ach

fig

ure

?

1.16

2.10

3.8

How

man

y re

ctan

gles

are

in

eac

h f

igu

re?

1.19

2.25

3.21

AB

CD

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-4

8-4

Exam

ple

Exam

ple



Stu

dy G

uid

e a

nd I

nte

rven

tion

Rh

om

bi a

nd

Sq

uar

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5

©G

lenc

oe/M

cGra

w-H

ill44

1G

lenc

oe G

eom

etry

Lesson 8-5

Pro

per

ties

of

Rh

om

bi

A r

hom

bu

sis

a q

uad

rila

tera

l w

ith

fou

r co

ngr

uen

t si

des.

Opp

osit

e si

des

are

con

gru

ent,

so a

rh

ombu

s is

als

o a

para

llel

ogra

m a

nd

has

all

of

the

prop

erti

es o

f a

para

llel

ogra

m.R

hom

bi

also

hav

e th

e fo

llow

ing

prop

erti

es.

The

dia

gona

ls a

re p

erpe

ndic

ular

.M�

H�⊥

R�O�

Eac

h di

agon

al b

isec

ts a

pai

r of

opp

osite

ang

les.

M�H�

bise

cts

�R

MO

and

�R

HO

.R�

O�bi

sect

s �

MR

Han

d �

MO

H.

If th

e di

agon

als

of a

par

alle

logr

am a

re

If R

HO

Mis

a p

aral

lelo

gram

and

R�O�

⊥M�

H�,

perp

endi

cula

r, th

en t

he f

igur

e is

a r

hom

bus.

then

RH

OM

is a

rho

mbu

s.

In r

hom

bu

s A

BC

D,m

�B

AC

�32

.Fin

d t

he

mea

sure

of

each

nu

mb

ered

an

gle.

AB

CD

is a

rh

ombu

s,so

th

e di

agon

als

are

perp

endi

cula

r an

d �

AB

Eis

a r

igh

t tr

ian

gle.

Th

us

m�

4 �

90 a

nd

m�

1 �

90 �

32 o

r 58

.Th

edi

agon

als

in a

rh

ombu

s bi

sect

th

e ve

rtex

an

gles

,so

m�

1 �

m�

2.T

hu

s,m

�2

�58

.

A r

hom

bus

is a

par

alle

logr

am,s

o th

e op

posi

te s

ides

are

par

alle

l.�

BA

Can

d �

3 ar

eal

tern

ate

inte

rior

an

gles

for

par

alle

l li

nes

,so

m�

3 �

32.

AB

CD

is a

rh

omb

us.

1.If

m�

AB

D�

60,f

ind

m�

BD

C.

602.

If A

E�

8,fi

nd

AC

.16

3.If

AB

�26

an

d B

D�

20,f

ind

AE

.24

4.F

ind

m�

CE

B.

90

5.If

m�

CB

D�

58,f

ind

m�

AC

B.

326.

If A

E�

3x�

1 an

d A

C�

16,f

ind

x.3

7.If

m�

CD

B�

6yan

d m

�A

CB

�2y

�10

,fin

d y.

10

8.If

AD

�2x

�4

and

CD

�4x

�4,

fin

d x.

4

9.a.

Wh

at i

s th

e m

idpo

int

of F�

H�?

(5,5

)

b.

Wh

at i

s th

e m

idpo

int

of G�

J�?

(5,5

)

c.W

hat

kin

d of

fig

ure

is

FG

HJ

? E

xpla

in.

FG

HJ

is a

par

alle

log

ram

bec

ause

th

e d

iag

on

als

bis

ect

each

oth

er.

d.

Wh

at i

s th

e sl

ope

of F�

H�?

�1 2�

e.W

hat

is

the

slop

e of

G�J�

?�

2

f.B

ased

on

par

ts c

,d,a

nd

e,w

hat

kin

d of

fig

ure

is

FG

HJ

? E

xpla

in.

FG

HJ

is a

par

alle

log

ram

wit

h p

erp

end

icu

lar

dia

go

nal

s,so

it is

a r

ho

mbu

s.

x

y

O

J

GF

HC

AD

EB

CA

DEB

32�

12

34

MO

HR

B

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

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w-H

ill44

2G

lenc

oe G

eom

etry

Pro

per

ties

of

Squ

ares

A s

quar

e h

as a

ll t

he

prop

erti

es o

f a

rhom

bus

and

all

the

prop

erti

es o

f a

rect

angl

e.

Fin

d t

he

mea

sure

of

each

nu

mb

ered

an

gle

of

squ

are

AB

CD

.

Usi

ng

prop

erti

es o

f rh

ombi

an

d re

ctan

gles

,th

e di

agon

als

are

perp

endi

cula

r an

d co

ngr

uen

t.�

AB

Eis

a r

igh

t tr

ian

gle,

so m

�1

�m

�2

�90

.

Eac

h v

erte

x an

gle

is a

rig

ht

angl

e an

d th

e di

agon

als

bise

ct t

he

vert

ex a

ngl

es,

so m

�3

�m

�4

�m

�5

�45

.

Det

erm

ine

wh

eth

er t

he

give

n v

erti

ces

rep

rese

nt

a p

ara

llel

ogra

m,r

ecta

ngl

e,rh

ombu

s,or

sq

ua

re.E

xpla

in y

our

reas

onin

g.

1.A

(0,2

),B

(2,4

),C

(4,2

),D

(2,0

)

Sq

uar

e;th

e fo

ur

sid

es a

re �

and

co

nse

cuti

ve s

ides

are

⊥.

2.D

(�2,

1),E

(�1,

3),F

(3,1

),G

(2,�

1)

Rec

tan

gle

;bo

th p

airs

of

op

po

site

sid

es a

re ||

and

co

nse

cuti

ve s

ides

are

⊥.

3.A

(�2,

�1)

,B(0

,2),

C(2

,�1)

,D(0

,�4)

Rh

om

bus;

the

fou

r si

des

are

�an

d c

on

secu

tive

sid

es a

re n

ot

⊥.

4.A

(�3,

0),B

(�1,

3),C

(5,�

1),D

(3,�

4)

Rec

tan

gle

;bo

th p

airs

of

op

po

site

sid

es a

re ||

and

co

nse

cuti

ve s

ides

are

⊥.

5.S

(�1,

4),T

(3,2

),U

(1,�

2),V

(�3,

0)

Sq

uar

e;th

e fo

ur

sid

es a

re �

and

co

nse

cuti

ve s

ides

are

⊥.

6.F

(�1,

0),G

(1,3

),H

(4,1

),I(

2,�

2)

Sq

uar

e;th

e fo

ur

sid

es a

re �

and

co

nse

cuti

ve s

ides

are

⊥.

7.S

quar

e R

ST

Uh

as v

erti

ces

R(�

3,�

1),S

(�1,

2),a

nd

T(2

,0).

Fin

d th

e co

ordi

nat

es o

fve

rtex

U. (

0,�

3)

C

AD

12

34

5

EB

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Rh

om

bi a

nd

Sq

uar

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5

Exer

cises

Exer

cises

Exam

ple

Exam

ple



An

swer

s

Skil

ls P

ract

ice

Rh

om

bi a

nd

Sq

uar

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5

©G

lenc

oe/M

cGra

w-H

ill44

3G

lenc

oe G

eom

etry

Lesson 8-5

Use

rh

omb

us

DK

LM

wit

h A

M�

4x,A

K�

5x�

3,an

d

DL

�10

.

1.F

ind

x.2.

Fin

d A

L.

35

3.F

ind

m�

KA

L.

4.F

ind

DM

.

9013

Use

rh

omb

us

RS

TV

wit

h R

S�

5y�

2,S

T�

3y�

6,an

d

NV

�6.

5.F

ind

y.6.

Fin

d T

V.

212

7.F

ind

m�

NT

V.

8.F

ind

m�

SV

T.

3060

9.F

ind

m�

RS

T.

10.F

ind

m�

SR

V.

120

60

CO

OR

DIN

ATE

GEO

MET

RYG

iven

eac

h s

et o

f ve

rtic

es,d

eter

min

e w

het

her

�Q

RS

Tis

a r

hom

bus,

a re

cta

ngl

e,or

a s

qu

are

.Lis

t al

l th

at a

pp

ly.E

xpla

in y

our

reas

onin

g.

11.Q

(3,5

),R

(3,1

),S

(�1,

1),T

(�1,

5)

Rh

om

bus,

rect

ang

le,s

qu

are;

all s

ides

are

co

ng

ruen

t an

d t

he

dia

go

nal

sar

e p

erp

end

icu

lar

and

co

ng

ruen

t.

12.Q

(�5,

12),

R(5

,12)

,S(�

1,4)

,T(�

11,4

)

Rh

om

bus;

all s

ides

are

co

ng

ruen

t an

d t

he

dia

go

nal

s ar

e p

erp

end

icu

lar,

but

no

t co

ng

ruen

t.

13.Q

(�6,

�1)

,R(4

,�6)

,S(2

,5),

T(�

8,10

)

Rh

om

bus;

all s

ides

are

co

ng

ruen

t an

d t

he

dia

go

nal

s ar

e p

erp

end

icu

lar,

but

no

t co

ng

ruen

t.

14.Q

(2,�

4),R

(�6,

�8)

,S(�

10,2

),T

(�2,

6)

No

ne;

op

po

site

sid

es a

re c

on

gru

ent,

but

the

dia

go

nal

s ar

e n

eith

erco

ng

ruen

t n

or

per

pen

dic

ula

r.

RV

NS

T

ML

AD

K

©G

lenc

oe/M

cGra

w-H

ill44

4G

lenc

oe G

eom

etry

Use

rh

omb

us

PR

YZ

wit

h R

K�

4y�

1,Z

K�

7y�

14,

PK

�3x

�1,

and

YK

�2x

�6.

1.F

ind

PY

.2.

Fin

d R

Z.

4042

3.F

ind

RY

.4.

Fin

d m

�Y

KZ

.

2990

Use

rh

omb

us

MN

PQ

wit

h P

Q�

3�2�,

PA

�4x

�1,

and

A

M�

9x �

6.

5.F

ind

AQ

.6.

Fin

d m

�A

PQ

.

345

7.F

ind

m�

MN

P.

8.F

ind

PM

.

906

CO

OR

DIN

ATE

GEO

MET

RYG

iven

eac

h s

et o

f ve

rtic

es,d

eter

min

e w

het

her

�B

EF

Gis

a r

hom

bus,

a re

cta

ngl

e,or

a s

qu

are

.Lis

t al

l th

at a

pp

ly.E

xpla

in y

our

reas

onin

g.

9.B

(�9,

1),E

(2,3

),F

(12,

�2)

,G(1

,�4)

Rh

om

bus;

all s

ides

are

co

ng

ruen

t an

d t

he

dia

go

nal

s ar

e p

erp

end

icu

lar,

but

no

t co

ng

ruen

t.

10.B

(1,3

),E

(7,�

3),F

(1,�

9),G

(�5,

�3)

Rh

om

bus,

rect

ang

le,s

qu

are;

all s

ides

are

co

ng

ruen

t an

d t

he

dia

go

nal

sar

e p

erp

end

icu

lar

and

co

ng

ruen

t.

11.B

(�4,

�5)

,E(1

,�5)

,F(�

7,�

1),G

(�2,

�1)

No

ne;

two

of

the

op

po

site

sid

es a

re n

ot

con

gru

ent.

12.T

ESSE

LATI

ON

ST

he

figu

re i

s an

exa

mpl

e of

a t

esse

llat

ion

.Use

a r

ule

r

or p

rotr

acto

r to

mea

sure

th

e sh

apes

an

d th

en n

ame

the

quad

rila

tera

ls

use

d to

for

m t

he

figu

re.

Th

e fi

gu

re c

on

sist

s o

f 6

con

gru

ent

rho

mb

i.

AN M

QP

K

P

YR

Z

Pra

ctic

e (

Ave

rag

e)

Rh

om

bi a

nd

Sq

uar

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5



Readin

g t

o L

earn

Math

em

ati

csR

ho

mb

i an

d S

qu

ares

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5

©G

lenc

oe/M

cGra

w-H

ill44

5G

lenc

oe G

eom

etry

Lesson 8-5

Pre-

Act

ivit

yH

ow c

an y

ou r

ide

a b

icyc

le w

ith

sq

uar

e w

hee

ls?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-5

at

the

top

of p

age

431

in y

our

text

book

.

If y

ou d

raw

a d

iago

nal

on

th

e su

rfac

e of

on

e of

th

e sq

uar

e w

hee

ls s

how

n i

nth

e pi

ctur

e in

you

r te

xtbo

ok,h

ow c

an y

ou d

escr

ibe

the

two

tria

ngle

s th

at a

refo

rmed

?S

amp

le a

nsw

er:t

wo

co

ng

ruen

t is

osc

eles

rig

ht

tria

ng

les

Rea

din

g t

he

Less

on

1.S

ketc

h e

ach

of

the

foll

owin

g.S

amp

le a

nsw

ers

are

giv

en.

a.a

quad

rila

tera

l w

ith

per

pen

dicu

lar

b.

a qu

adri

late

ral

wit

h c

ongr

uen

t di

agon

als

that

is

not

a r

hom

bus

diag

onal

s th

at i

s n

ot a

rec

tan

gle

c.a

quad

rila

tera

l w

hos

e di

agon

als

are

perp

endi

cula

r an

d bi

sect

eac

h o

ther

,bu

t ar

e n

ot c

ongr

uen

t

2.L

ist

all

of t

he

foll

owin

g sp

ecia

l qu

adri

late

rals

th

at h

ave

each

lis

ted

prop

erty

:pa

rall

elog

ram

,rec

tan

gle,

rhom

bus,

squ

are.

a.T

he

diag

onal

s ar

e co

ngr

uen

t.re

ctan

gle

,sq

uar

eb

.O

ppos

ite

side

s ar

e co

ngr

uen

t.p

aral

lelo

gra

m,r

ecta

ng

le,r

ho

mbu

s,sq

uar

ec.

Th

e di

agon

als

are

perp

endi

cula

r.rh

om

bus,

squ

are

d.

Con

secu

tive

an

gles

are

su

pple

men

tary

.p

aral

lelo

gra

m,r

ecta

ng

le,r

ho

mbu

s,sq

uar

ee.

Th

e qu

adri

late

ral

is e

quil

ater

al.

rho

mbu

s,sq

uar

ef.

Th

e qu

adri

late

ral

is e

quia

ngu

lar.

rect

ang

le,s

qu

are

g.T

he

diag

onal

s ar

e pe

rpen

dicu

lar

and

con

gru

ent.

squ

are

h.

A p

air

of o

ppos

ite

side

s is

bot

h p

aral

lel

and

con

gru

ent.

par

alle

log

ram

,rec

tan

gle

,rh

om

bus,

squ

are

3.W

hat

is

the

com

mon

nam

e fo

r a

regu

lar

quad

rila

tera

l? E

xpla

in y

our

answ

er.

Sq

uar

e;sa

mp

le a

nsw

er:

All

fou

r si

des

of

a sq

uar

e ar

e co

ng

ruen

t an

d a

ll fo

ur

ang

les

are

con

gru

ent.

Hel

pin

g Y

ou

Rem

emb

er

4.A

goo

d w

ay t

o re

mem

ber

som

eth

ing

is t

o ex

plai

n i

t to

som

eon

e el

se.S

upp

ose

that

you

rcl

assm

ate

Lu

is i

s h

avin

g tr

oubl

e re

mem

beri

ng

wh

ich

of

the

prop

erti

es h

e h

as l

earn

ed i

nth

is c

hap

ter

appl

y to

squ

ares

.How

can

you

hel

p h

im?

Sam

ple

an

swer

:A

sq

uar

e is

a p

aral

lelo

gra

m t

hat

is b

oth

a r

ecta

ng

le a

nd

a r

ho

mbu

s,so

all

pro

per

ties

of

par

alle

log

ram

s,re

ctan

gle

s,an

d r

ho

mb

i ap

ply

to

sq

uar

es.

©G

lenc

oe/M

cGra

w-H

ill44

6G

lenc

oe G

eom

etry

Cre

atin

g P

yth

ago

rean

Pu

zzle

sB

y dr

awin

g tw

o sq

uar

es a

nd

cutt

ing

them

in

a c

erta

in w

ay,y

ouca

n m

ake

a pu

zzle

th

at d

emon

stra

tes

the

Pyt

hag

orea

n T

heo

rem

.A

sam

ple

puzz

le i

s sh

own

.You

can

cre

ate

you

r ow

n p

uzz

le b

yfo

llow

ing

the

inst

ruct

ion

s be

low

.

See

stu

den

ts’w

ork

.A s

amp

le a

nsw

er is

sh

ow

n.

1.C

aref

ull

y co

nst

ruct

a s

quar

e an

d la

bel

the

len

gth

of

a s

ide

as a

.Th

en c

onst

ruct

a s

mal

ler

squ

are

to

the

righ

t of

it

and

labe

l th

e le

ngt

h o

f a

side

as

b,as

sh

own

in

th

e fi

gure

abo

ve.T

he

base

s sh

ould

be

adj

acen

t an

d co

llin

ear.

2.M

ark

a po

int

Xth

at i

s b

un

its

from

th

e le

ft e

dge

of t

he

larg

er s

quar

e.T

hen

dra

w t

he

segm

ents

fr

om t

he

upp

er l

eft

corn

er o

f th

e la

rger

squ

are

to

poin

t X

,an

d fr

om p

oin

t X

to t

he

upp

er r

igh

t co

rner

of

th

e sm

alle

r sq

uar

e.

3.C

ut

out

and

rear

ran

ge y

our

five

pie

ces

to f

orm

a

larg

er s

quar

e.D

raw

a d

iagr

am t

o sh

ow y

our

answ

er.

4.V

erif

y th

at t

he

len

gth

of

each

sid

e is

equ

al t

o

�a2

�b

�2 �.

a

a

bX

b

b

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Trap

ezo

ids

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6

©G

lenc

oe/M

cGra

w-H

ill44

7G

lenc

oe G

eom

etry

Lesson 8-6

Pro

per

ties

of

Trap

ezo

ids

A t

rap

ezoi

dis

a q

uad

rila

tera

l w

ith

ex

actl

y on

e pa

ir o

f pa

rall

el s

ides

.Th

e pa

rall

el s

ides

are

cal

led

bas

esan

d th

e n

onpa

rall

el s

ides

are

cal

led

legs

.If

the

legs

are

con

gru

ent,

the

trap

ezoi

d is

an

iso

scel

es t

rap

ezoi

d.I

n a

n i

sosc

eles

tra

pezo

id

both

pai

rs o

f b

ase

angl

esar

e co

ngr

uen

t.S

TU

Ris

an

isos

cele

s tr

apez

oid.

S�R�

�T�U�

; �

R�

�U

, �

S�

�T

Th

e ve

rtic

es o

f A

BC

Dar

e A

(�3,

�1)

,B(�

1,3)

,C

(2,3

),an

d D

(4,�

1).V

erif

y th

at A

BC

Dis

a t

rap

ezoi

d.

slop

e of

A �B�

�� 3 1� �

(� (�1 3) )

��

�4 2��

2

slop

e of

A �D�

�� 41 ��

(�(�31 ))

��

�0 7��

0

slop

e of

B �C�

�� 2

3 �

� (�3 1)

��

�0 3��

0

slop

e of

C �D�

�� 41 ��

23�

�� 24 �

��

2

Exa

ctly

tw

o si

des

are

para

llel

,A�D�

and

B�C�

,so

AB

CD

is a

tra

pezo

id.A

B�

CD

,so

AB

CD

isan

iso

scel

es t

rape

zoid

.

In E

xerc

ises

1–3

,det

erm

ine

wh

eth

er A

BC

Dis

a t

rap

ezoi

d.I

f so

,det

erm

ine

wh

eth

er i

t is

an

iso

scel

es t

rap

ezoi

d.E

xpla

in.

1.A

(�1,

1),B

(2,1

),C

(3,�

2),a

nd

D(2

,�2)

Slo

pe

of

A�B�

�0,

slo

pe

of

D�C�

�0,

slo

pe

of

A�D�

��

1,sl

op

e o

f B�

C��

�3.

Exa

ctly

tw

o s

ides

are

par

alle

l,so

AB

CD

is a

tra

pez

oid

.AD

�3�

2�an

d

BC

��

10�;

AD

�B

C,s

o A

BC

Dis

no

t is

osc

eles

.2.

A(3

,�3)

,B(�

3,�

3),C

(�2,

3),a

nd

D(2

,3)

Slo

pe

of

A�B�

�0,

slo

pe

of

D�C�

�0,

slo

pe

of

B�C�

�6,

slo

pe

of

A�D�

��

6.E

xact

ly 2

sid

es a

re || ,

so A

BC

Dis

a t

rap

ezo

id.B

C�

�37�

and

AD

��

37�;

BC

�A

D,s

o A

BC

Dis

iso

scel

es.

3.A

(1,�

4),B

(�3,

�3)

,C(�

2,3)

,an

d D

(2,2

)

Slo

pe

of

A�B�

��

�1 4� ,sl

op

e o

f D�

C��

��1 4� ,

slo

pe

of

B�C�

�6,

slo

pe

of

A�D�

�6.

Bo

th p

airs

of

op

po

site

sid

es a

re p

aral

lel,

so A

BC

Dis

no

t a

trap

ezo

id.

4.T

he

vert

ices

of

an i

sosc

eles

tra

pezo

id a

re R

(�2,

2),S

(2,2

),T

(4,�

1),a

nd

U(�

4,�

1).

Ver

ify

that

th

e di

agon

als

are

con

gru

ent.

RT

��

((4

��

(�2)

)2�

�(�

�1

�2

�)2

)��

�45�

SU

��

((�

4 �

�(2

))2

��

(�1

��

2)2

�)��

�45�

x

y

O

B

DA

C

T

RU

base

base

Sle

g

AB

��

(�3

��

(�1)

)�

2�

(��

1 �

3�

)2 ��

�4

�1

�6�

��

20��

2�5�

CD

��

(2 �

4�

)2�

(�

3 �

(��

1))2

��

�4

�1

�6�

��

20��

2�5�

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill44

8G

lenc

oe G

eom

etry

Med

ian

s o

f Tr

apez

oid

sT

he

med

ian

of a

tra

pezo

id i

s th

e se

gmen

t th

at jo

ins

the

mid

poin

ts o

f th

e le

gs.I

t is

par

alle

l to

th

e ba

ses,

and

its

len

gth

is

one-

hal

f th

e su

m o

f th

e le

ngt

hs

of t

he

base

s.

In t

rape

zoid

HJ

KL

,MN

��1 2� (

HJ

�L

K).

M�N�

is t

he

med

ian

of

trap

ezoi

d R

ST

U.F

ind

x.

MN

��1 2� (

RS

�U

T)

30�

�1 2� (3x

�5

�9x

�5)

30�

�1 2� (12

x)

30�

6x5

�x

M �N�

is t

he

med

ian

of

trap

ezoi

d H

JK

L.F

ind

eac

h i

nd

icat

ed v

alu

e.

1.F

ind

MN

if H

J�

32 a

nd

LK

�60

.

46

2.F

ind

LK

if H

J�

18 a

nd

MN

�28

.

38

3.F

ind

MN

if H

J�

LK

�42

.

21

4.F

ind

m�

LM

Nif

m�

LH

J�

116.

116

5.F

ind

m�

JK

Lif

HJ

KL

is i

sosc

eles

an

d m

�H

LK

�62

.

62

6.F

ind

HJ

if M

N�

5x�

6,H

J�

3x�

6,an

d L

K�

8x.

24

7.F

ind

the

len

gth

of

the

med

ian

of

a tr

apez

oid

wit

h v

erti

ces

A(�

2,2)

,B(3

,3),

C(7

,0),

and

D(�

3,�

2).

�3 2� �26�

LK

NM

HJ

UT

NM

R3x

� 5

30 9x �

5S

LK

NM

HJ

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Trap

ezo

ids

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Trap

ezo

ids

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6

©G

lenc

oe/M

cGra

w-H

ill44

9G

lenc

oe G

eom

etry

Lesson 8-6

CO

OR

DIN

ATE

GEO

MET

RYA

BC

Dis

a q

uad

rila

tera

l w

ith

ver

tice

s A

(�4,

�3)

,B

(3,�

3),C

(6,4

),D

(�7,

4).

1.V

erif

y th

at A

BC

Dis

a t

rape

zoid

.

A�B�

|| C�D�

2.D

eter

min

e w

het

her

AB

CD

is a

n i

sosc

eles

tra

pezo

id.E

xpla

in.

iso

scel

es;

AD

��

58�an

d B

C�

�58�

CO

OR

DIN

ATE

GEO

MET

RYE

FG

His

a q

uad

rila

tera

l w

ith

ver

tice

s E

(1,3

),F

(5,0

),G

(8,�

5),H

(�4,

4).

3.V

erif

y th

at E

FG

His

a t

rape

zoid

.

E�F�

|| G�H�

4.D

eter

min

e w

het

her

EF

GH

is a

n i

sosc

eles

tra

pezo

id.E

xpla

in.

no

t is

osc

eles

;E

H�

�26�

and

FG

��

34�

CO

OR

DIN

ATE

GEO

MET

RYL

MN

Pis

a q

uad

rila

tera

l w

ith

ver

tice

s L

(�1,

3),M

(�4,

1),

N(�

6,3)

,P(0

,7).

5.V

erif

y th

at L

MN

Pis

a t

rape

zoid

.L�M�

|| N�P�

6.D

eter

min

e w

het

her

LM

NP

is a

n i

sosc

eles

tra

pezo

id.E

xpla

in.

no

t is

osc

eles

;L

P�

�17�

and

MN

��

8�

ALG

EBR

AF

ind

th

e m

issi

ng

mea

sure

(s)

for

the

give

n t

rap

ezoi

d.

7.F

or t

rape

zoid

HJ

KL

,San

d T

are

8.F

or t

rape

zoid

WX

YZ

,Pan

d Q

are

mid

poin

ts o

f th

e le

gs.F

ind

HJ

.58

mid

poin

ts o

f th

e le

gs.F

ind

WX

.5

9.F

or t

rape

zoid

DE

FG

,Tan

d U

are

10.F

or i

sosc

eles

tra

pezo

id Q

RS

T,f

ind

the

mid

poin

ts o

f th

e le

gs.F

ind

TU

,m�

E,

len

gth

of

the

med

ian

,m�

Q,a

nd

m�

S.

and

m�

G.

28,9

5,14

542

.5,1

25,5

5

ST

RQ

125�

60 25

DE F

G

TU

85�

35�

42

14

WX

YZ

PQ

12 19

HJ

KLS

T72 86

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CO

OR

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ATE

GEO

MET

RYR

ST

Uis

a q

uad

rila

tera

l w

ith

ver

tice

s R

(�3,

�3)

,S(5

,1),

T(1

0,�

2),U

(�4,

�9)

.

1.V

erif

y th

at R

ST

Uis

a t

rape

zoid

.

R�S�

|| T�U�

2.D

eter

min

e w

het

her

RS

TU

is a

n i

sosc

eles

tra

pezo

id.E

xpla

in.

no

t is

osc

eles

;R

U�

�37�

and

ST

��

34�

CO

OR

DIN

ATE

GEO

MET

RYB

GH

Jis

a q

uad

rila

tera

l w

ith

ver

tice

s B

(�9,

1),G

(2,3

),H

(12,

�2)

,J(�

10,�

6).

3.V

erif

y th

at B

GH

Jis

a t

rape

zoid

.

B�G�

|| H�J�

4.D

eter

min

e w

het

her

BG

HJ

is a

n i

sosc

eles

tra

pezo

id.E

xpla

in.

no

t is

osc

eles

;B

J�

�50�

and

GH

��

125

�

ALG

EBR

AF

ind

th

e m

issi

ng

mea

sure

(s)

for

the

give

n t

rap

ezoi

d.

5.F

or t

rape

zoid

CD

EF

,Van

d Y

are

6.F

or t

rape

zoid

WR

LP

,Ban

d C

are

mid

poin

ts o

f th

e le

gs.F

ind

CD

.38

mid

poin

ts o

f th

e le

gs.F

ind

LP

.18

7.F

or t

rape

zoid

FG

HI,

Kan

d M

are

8.F

or i

sosc

eles

tra

pezo

id T

VZ

Y,f

ind

the

mid

poin

ts o

f th

e le

gs.F

ind

FI,

m�

F,

len

gth

of

the

med

ian

,m�

T,a

nd

m�

Z.

and

m�

I.51

,40,

5519

.5,6

0,12

0

9.C

ON

STR

UC

TIO

NA

set

of

stai

rs l

eadi

ng

to t

he

entr

ance

of

a bu

ildi

ng

is d

esig

ned

in

th

esh

ape

of a

n i

sosc

eles

tra

pezo

id w

ith

th

e lo

nge

r ba

se a

t th

e bo

ttom

of

the

stai

rs a

nd

the

shor

ter

base

at

the

top.

If t

he

bott

om o

f th

e st

airs

is

21 f

eet

wid

e an

d th

e to

p is

14,

fin

dth

e w

idth

of

the

stai

rs h

alfw

ay t

o th

e to

p.17

.5 f

t

10.D

ESK

TO

PSA

car

pen

ter

nee

ds t

o re

plac

e se

vera

l tr

apez

oid-

shap

ed d

eskt

ops

in a

clas

sroo

m.T

he

carp

ente

r kn

ows

the

len

gth

s of

bot

h b

ases

of

the

desk

top.

Wh

at o

ther

mea

sure

men

ts,i

f an

y,do

es t

he

carp

ente

r n

eed?

Sam

ple

an

swer

:th

e m

easu

res

of

the

bas

e an

gle

sV

YZ

T60

�34 5

H

KM

G

IF

140�

125�

21

36

WR

LP

BC

4266F

E

DC

VY

2818Pra

ctic

e (

Ave

rag

e)

Trap

ezo

ids

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csTr

apez

oid

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6

©G

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1G

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eom

etry

Lesson 8-6

Pre-

Act

ivit

yH

ow a

re t

rap

ezoi

ds

use

d i

n a

rch

itec

ture

?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-6

at

the

top

of p

age

439

in y

our

text

book

.

How

mig

ht

trap

ezoi

ds b

e u

sed

in t

he

inte

rior

des

ign

of

a h

ome?

Sam

ple

an

swer

:fl

oo

r ti

les

for

a ki

tch

en o

r b

ath

roo

m

Rea

din

g t

he

Less

on

1.In

th

e fi

gure

at

the

righ

t,E

FG

His

a t

rape

zoid

,Iis

th

e

mid

poin

t of

F �E�

,an

d J

is t

he

mid

poin

t of

G�H�

.Ide

nti

fy e

ach

of

th

e fo

llow

ing

segm

ents

or

angl

es i

n t

he

figu

re.

a.th

e ba

ses

of t

rape

zoid

EF

GH

F�G�,E�

H�b

.th

e tw

o pa

irs

of b

ase

angl

es o

f tr

apez

oid

EF

GH

�E

and

�H

;�

Fan

d �

Gc.

the

legs

of

trap

ezoi

d E

FG

HF�E�

,G�H�

d.

the

med

ian

of

trap

ezoi

d E

FG

HI�J�

2.D

eter

min

e w

het

her

eac

h s

tate

men

t is

tru

eor

fal

se.I

f th

e st

atem

ent

is f

alse

,exp

lain

wh

y.

a.A

tra

pezo

id i

s a

spec

ial

kin

d of

par

alle

logr

am.

Fals

e;sa

mp

le a

nsw

er:

Ap

aral

lelo

gra

m h

as t

wo

pai

rs o

f p

aral

lel s

ides

,bu

t a

trap

ezo

id h

as o

nly

on

e p

air

of

par

alle

l sid

es.

b.

Th

e di

agon

als

of a

tra

pezo

id a

re c

ongr

uen

t.Fa

lse;

sam

ple

an

swer

:Th

is is

on

lytr

ue

for

iso

scel

es t

rap

ezo

ids.

c.T

he

med

ian

of

a tr

apez

oid

is p

aral

lel

to t

he

legs

.Fa

lse;

sam

ple

an

swer

:Th

em

edia

n is

par

alle

l to

th

e b

ases

.d

.T

he

len

gth

of

the

med

ian

of

a tr

apez

oid

is t

he

aver

age

of t

he

len

gth

of

the

base

s.tr

ue

e.A

tra

pezo

id h

as t

hre

e m

edia

ns.

Fals

e;sa

mp

le a

nsw

er:

A t

rap

ezo

id h

as o

nly

on

e m

edia

n.

f.T

he

base

s of

an

iso

scel

es t

rape

zoid

are

con

gru

ent.

Fals

e:sa

mp

le a

nsw

er:T

he

leg

s ar

e co

ng

ruen

t.g.

An

iso

scel

es t

rape

zoid

has

tw

o pa

irs

of c

ongr

uen

t an

gles

.tr

ue

h.

Th

e m

edia

n o

f an

iso

scel

es t

rape

zoid

div

ides

th

e tr

apez

oid

into

tw

o sm

alle

r is

osce

les

trap

ezoi

ds.

tru

e

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

a n

ew g

eom

etri

c th

eore

m i

s to

rel

ate

it t

o on

e yo

u a

lrea

dykn

ow.N

ame

and

stat

e in

wor

ds a

th

eore

m a

bou

t tr

ian

gles

th

at i

s si

mil

ar t

o th

e th

eore

min

th

is l

esso

n a

bou

t th

e m

edia

n o

f a

trap

ezoi

d.Tr

ian

gle

Mid

seg

men

t Th

eore

m;

am

idse

gm

ent

of

a tr

ian

gle

is p

aral

lel t

o o

ne

sid

e o

f th

e tr

ian

gle

,an

d it

sle

ng

th is

on

e-h

alf

the

len

gth

of

that

sid

e.

HE

JI

GF

©G

lenc

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2G

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eom

etry

Qu

adri

late

rals

in C

on

stru

ctio

n

Qu

adri

late

rals

are

oft

en u

sed

in c

onst

ruct

ion

wor

k.

1.T

he

diag

ram

at

the

righ

t re

pres

ents

a r

oof

fram

e an

d sh

ows

man

y qu

adri

late

rals

.Fin

d th

e fo

llow

ing

shap

es i

n t

he

diag

ram

an

d sh

ade

in t

hei

r ed

ges.

See

stu

den

ts’w

ork

.

a.is

osce

les

tria

ngl

e

b.

scal

ene

tria

ngl

e

c.re

ctan

gle

d.

rhom

bus

e.tr

apez

oid

(not

iso

scel

es)

f.is

osce

les

trap

ezoi

d

2.T

he

figu

re a

t th

e ri

ght

repr

esen

ts a

win

dow

.Th

e w

oode

n p

art

betw

een

th

e pa

nes

of

glas

s is

3 i

nch

es

wid

e.T

he

fram

e ar

oun

d th

e ou

ter

edge

is

9 in

ches

w

ide.

Th

e ou

tsid

e m

easu

rem

ents

of

the

fram

e ar

e 60

in

ches

by

81 i

nch

es.T

he

hei

ght

of t

he

top

and

bott

om p

anes

is

the

sam

e.T

he

top

thre

e pa

nes

are

th

e sa

me

size

.

a.H

ow w

ide

is t

he

bott

om p

ane

of g

lass

?42

in.

b.

How

wid

e is

eac

h t

op p

ane

of g

lass

?12

in.

c.H

ow h

igh

is

each

pan

e of

gla

ss?

30 in

.

3.E

ach

edg

e of

th

is b

ox h

as b

een

rei

nfo

rced

w

ith

a p

iece

of

tape

.Th

e bo

x is

10

inch

es

hig

h,2

0 in

ches

wid

e,an

d 12

in

ches

dee

p.W

hat

is

the

len

gth

of

the

tape

th

at h

as

been

use

d?16

8 in

.10

in.

12 in

.

20 in

.60 in

.

81 in

.

9 in

.

3 in

.

3 in

.

3 in

.

9 in

.

9 in

.

jack

rafte

rs

hip

rafte

r

com

mon

rafte

rs

Ro

of

Fram

e

ridge

boa

rdva

lley

rafte

r

plat

e

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6



Stu

dy G

uid

e a

nd I

nte

rven

tion

Co

ord

inat

e P

roo

f w

ith

Qu

adri

late

rals

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7

©G

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w-H

ill45

3G

lenc

oe G

eom

etry

Lesson 8-7

Posi

tio

n F

igu

res

Coo

rdin

ate

proo

fs u

se p

rope

rtie

s of

lin

es a

nd

segm

ents

to

prov

ege

omet

ric

prop

erti

es.T

he

firs

t st

ep i

n w

riti

ng

a co

ordi

nat

e pr

oof

is t

o pl

ace

the

figu

re o

n t

he

coor

din

ate

plan

e in

a c

onve

nie

nt

way

.Use

th

e fo

llow

ing

guid

elin

es f

or p

laci

ng

a fi

gure

on

the

coor

din

ate

plan

e.

1.U

se t

he o

rigin

as

a ve

rtex

, so

one

set

of

coor

dina

tes

is (

0, 0

), o

r us

e th

e or

igin

as

the

cent

er o

f th

e fig

ure.

2.P

lace

at

leas

t on

e si

de o

f th

e qu

adril

ater

al o

n an

axi

s so

you

will

hav

e so

me

zero

coo

rdin

ates

.

3.Tr

y to

kee

p th

e qu

adril

ater

al in

the

firs

t qu

adra

nt s

o yo

u w

ill h

ave

posi

tive

coor

dina

tes.

4.U

se c

oord

inat

es t

hat

mak

e th

e co

mpu

tatio

ns a

s ea

sy a

s po

ssib

le.

For

exa

mpl

e, u

se e

ven

num

bers

if y

ou

are

goin

g to

be

findi

ng m

idpo

ints

.

Pos

itio

n a

nd

lab

el a

rec

tan

gle

wit

h s

ides

aan

d b

un

its

lon

g on

th

e co

ord

inat

e p

lan

e.

•P

lace

on

e ve

rtex

at

the

orig

in f

or R

,so

one

vert

ex i

s R

(0,0

).•

Pla

ce s

ide

R �U�

alon

g th

e x-

axis

an

d si

de R�

S�al

ong

the

y-ax

is,

wit

h t

he

rect

angl

e in

th

e fi

rst

quad

ran

t.•

Th

e si

des

are

aan

d b

un

its,

so l

abel

tw

o ve

rtic

es S

(0,a

) an

d U

(b,0

).•

Ver

tex

Tis

bu

nit

s ri

ght

and

au

nit

s u

p,so

th

e fo

urt

h v

erte

x is

T(b

,a).

Nam

e th

e m

issi

ng

coor

din

ates

for

eac

h q

uad

rila

tera

l.

1.2.

3.

C(c

,c)

N(b

,c)

B(b

,c);

C(a

�b

,c)

Pos

itio

n a

nd

lab

el e

ach

qu

adri

late

ral

on t

he

coor

din

ate

pla

ne.

Sam

ple

an

swer

sar

e g

iven

.4.

squ

are

ST

UV

wit

h

5.pa

rall

elog

ram

PQ

RS

6.re

ctan

gle

AB

CD

wit

hsi

de s

un

its

wit

h c

ongr

uen

t di

agon

als

len

gth

tw

ice

the

wid

th

x

y

D( 2

a, 0

)

C( 2

a, a

)B

( 0, a

)

A( 0

, 0)

x

y

S( a

, 0)

R( a

, b)

Q( 0

, b)

P( 0

, 0)

x

y

V( s

, 0)

U( s

, s)

T( 0

, s)

S( 0

, 0)

x

y

D( a

, 0)C

( ?, c

)B

( b, ?

)

A( 0

, 0)

x

y

Q( a

, 0)

P( a

� b

, c)

N( ?

, ?)

M( 0

, 0)

x

y

B( c

, 0)

C( ?

, ?)

D( 0

, c)

A( 0

, 0)

x

y

U( b

, 0)

T( b

, a)

S( 0

, a)

R( 0

, 0)

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

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w-H

ill45

4G

lenc

oe G

eom

etry

Pro

ve T

heo

rem

sA

fter

a f

igu

re h

as b

een

pla

ced

on t

he

coor

din

ate

plan

e an

d la

bele

d,a

coor

din

ate

proo

f ca

n b

e u

sed

to p

rove

a t

heo

rem

or

veri

fy a

pro

pert

y.T

he

Dis

tan

ce F

orm

ula

,th

e S

lope

For

mu

la,a

nd

the

Mid

poin

t T

heo

rem

are

oft

en u

sed

in a

coo

rdin

ate

proo

f.

Wri

te a

coo

rdin

ate

pro

of t

o sh

ow t

hat

th

e d

iago

nal

s of

a s

qu

are

are

per

pen

dic

ula

r.T

he

firs

t st

ep i

s to

pos

itio

n a

nd

labe

l a

squ

are

on t

he

coor

din

ate

plan

e.P

lace

it

in t

he

firs

t qu

adra

nt,

wit

h o

ne

side

on

eac

h a

xis.

Lab

el t

he

vert

ices

an

d dr

aw t

he

diag

onal

s.

Giv

en:s

quar

e R

ST

UP

rove

:S �U�

⊥R�

T�

Pro

of:T

he

slop

e of

S�U�

is �0 a

� �a 0

��

�1,

and

the

slop

e of

R�T�

is �a a

� �0 0

��

1.

Th

e pr

odu

ct o

f th

e tw

o sl

opes

is

�1,

so S�

U�⊥

R�T�

.

Wri

te a

coo

rdin

ate

pro

of t

o sh

ow t

hat

th

e le

ngt

h o

f th

e m

edia

n o

f a

trap

ezoi

d i

sh

alf

the

sum

of

the

len

gth

s of

th

e b

ases

.

Po

siti

on

A�D�

on

th

e x-

axis

an

d B�

C�p

aral

lel t

o A�

D�.

Lab

el t

wo

ver

tice

s A

(0,0

) an

d D

(2a,

0).L

abel

an

oth

erve

rtex

B(2

b,2

c).F

or

vert

ex C

,th

e y

valu

e is

th

e sa

me

as f

or

vert

ex B

,so

th

e la

st v

erte

x is

C(2

d,2

c).

Th

e co

ord

inat

es o

f m

idp

oin

t M

are

��2b2�

0�

,�0

� 22c �

��(b

,c);

the

coo

rdin

ates

of

mid

po

int

Nar

e

��2a� 2

2d�

,�2c

2�0

��

(a�

d,c

).B�

C�,M�

N�,a

nd

A�D�

are

ho

rizo

nta

l seg

men

ts,

so f

ind

th

eir

len

gth

s by

su

btr

acti

ng

th

e x-

coo

rdin

ates

.

AD

�2a

�0

�2a

,BC

�2d

�2b

,MN

�a

�d

�b

Th

en �1 2� (

AD

�B

C)

��1 2� (

2a�

2d�

2b)

�a

�d

�b

�M

N.

Th

eref

ore

,th

e le

ng

th o

f th

e m

edia

n o

f a

trap

ezo

id is

hal

f th

e su

m o

f th

ele

ng

ths

of

the

bas

es.

x

y

D( 2

a, 0

)

B( 2

b, 2

c)

MN

C( 2

d, 2

c)

A( 0

, 0)

x

y

U( a

, 0)

T( a

, a)

S( 0

, a)

R( 0

, 0)

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Co

ord

inat

e P

roo

f Wit

h Q

uad

rila

tera

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7

Exam

ple

Exam

ple

Exer

cise

Exer

cise



An

swer

s

Skil

ls P

ract

ice

Co

ord

inat

e P

roo

f w

ith

Qu

adri

late

rals

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7

©G

lenc

oe/M

cGra

w-H

ill45

5G

lenc

oe G

eom

etry

Lesson 8-7

Pos

itio

n a

nd

lab

el e

ach

qu

adri

late

ral

on t

he

coor

din

ate

pla

ne.

1.re

ctan

gle

wit

h l

engt

h 2

au

nit

s an

d 2.

isos

cele

s tr

apez

oid

wit

h h

eigh

t a

un

its,

hei

ght

au

nit

sba

ses

c�

bu

nit

s an

d b

�c

un

its

Nam

e th

e m

issi

ng

coor

din

ates

for

eac

h q

uad

rila

tera

l.

3.re

ctan

gle

C(c

,a)

4.re

ctan

gle

G(0

,2b

)

5.pa

rall

elog

ram

R(a

�b

,c),

S(a

,0)

6.is

osce

les

trap

ezoi

dW

(a�

b,c

),Y

(0,c

)

Pos

itio

n a

nd

lab

el t

he

figu

re o

n t

he

coor

din

ate

pla

ne.

Th

en w

rite

a c

oord

inat

ep

roof

for

th

e fo

llow

ing.

7.T

he

segm

ents

join

ing

the

mid

poin

ts o

f th

e si

des

of a

rh

ombu

s fo

rm a

rec

tan

gle.

Giv

en:

AB

CD

is a

rh

om

bus.

M,N

,P,a

nd

Qar

e th

e m

idp

oin

ts

of

A�B�

,B�C�

,C�D�

,an

d D�

A�,r

esp

ecti

vely

.P

rove

:M

NP

Qis

a r

ecta

ng

le.

Pro

of:

M�

(�a,

b),

N�

(a,b

),P

�(a

,�b

),an

d Q

�(�

a,�

b).

slo

pe

of

M�N�

�� a

b �� (�

b a)�

or

0sl

op

e o

f Q�

P��

�ba�

�(�ab

)�

or

0

slo

pe

of

M�Q�

��

� ab �

� (�b a)

�o

r u

nd

efin

edsl

op

e o

f N�

P��

�ba�

�(�ab

)�

or

un

def

ined

M�N�

and

Q�P�

hav

e th

e sa

me

slo

pe.

M�Q�

and

N�P�

also

hav

e th

e sa

me

slo

pe.

M�N�

is p

erp

end

icu

lar

to M�

Q�.T

her

efo

re,b

oth

pai

rs o

f o

pp

osi

te s

ides

are

par

alle

l an

d c

on

secu

tive

sid

es a

re p

erp

end

icu

lar.

Th

is m

ean

s th

at M

NP

Qis

a r

ecta

ng

le.

xO

y B( 0

, 2b)

D( 0

, �2b

)PN

M Q

C( 2

a, 0

)

A( �

2a, 0

)

x

y

T( b

, 0)

W( ?

, c)

Y( ?

, ?)

S( a

, 0)

Ox

y

S( a

, ?)

R( ?

, ?)

P( b

, c)

T( 0

, 0)

O

x

y

E( 5

b, 0

)

F( 5

b, 2

b)G

( ?, ?

)

D( 0

, 0)

Ox

y

U( c

, 0)

C( ?

, ?)

D( 0

, a)

A( 0

, 0)

O

xO

y

B( b

� c

, 0)

B( b

, a)

C( c

, a)

A( 0

, 0)

xO

y

B( 2

a, 0

)

D( 0

, a)

C( 2

a, a

)

A( 0

, 0)

©G

lenc

oe/M

cGra

w-H

ill45

6G

lenc

oe G

eom

etry

Pos

itio

n a

nd

lab

el e

ach

qu

adri

late

ral

on t

he

coor

din

ate

pla

ne.

1.pa

rall

elog

ram

wit

h s

ide

len

gth

bu

nit

s 2.

isos

cele

s tr

apez

oid

wit

h h

eigh

t b

un

its,

and

hei

ght

au

nit

sba

ses

2c�

au

nit

s an

d 2c

�a

un

its

Nam

e th

e m

issi

ng

coor

din

ates

for

eac

h q

uad

rila

tera

l.

3.pa

rall

elog

ram

4.is

osce

les

trap

ezoi

d

K(2

b�

c,a)

,L(c

,a)

X(�

a,0)

,Y(�

b,c

)

Pos

itio

n a

nd

lab

el t

he

figu

re o

n t

he

coor

din

ate

pla

ne.

Th

en w

rite

a c

oord

inat

ep

roof

for

th

e fo

llow

ing.

5.T

he

oppo

site

sid

es o

f a

para

llel

ogra

m a

re c

ongr

uen

t.

Giv

en:

AB

CD

is a

par

alle

log

ram

.P

rove

:A�

B��

C�D�

,A�D�

�B�

C�P

roo

f:

AB

��

(a�

0�

)2�

(�

0 �

0�

)2 ��

�a

2 �o

r a

CD

��

[(a

��

b)

�b

�]2

�(

�c

�c

�)2 �

��

a2 �

or

a

AD

��

(b�

0�

)2�

(�

c�

0�

)2 ��

�b

2�

�c

2 �A

B�

�[(

a�

�b

) �

a�

]2�

(c�

�0)

2�

��

b2

��

c2 �

AB

�C

Dan

d A

D�

BC

,so

A�B�

�C�

D�an

d A�

D��

B�C�

6.TH

EATE

RA

sta

ge i

s in

th

e sh

ape

of a

tra

pezo

id.W

rite

a

coor

din

ate

proo

f to

pro

ve t

hat

T �R�

and

S�F�

are

para

llel

.

Giv

en:

T(1

0,0)

,R(2

0,0)

,S(3

0,25

),F

(0,2

5)P

rove

:T�R�

|| S�F�

Pro

of:

Th

e sl

op

e o

f T�R�

�� 20 0

� �0 10

��

� 10 0�an

d

the

slo

pe

of

SF

��2 35 0� �

2 05�

�� 30 0�

.Sin

ce T�

R�an

d

S�F�

bo

th h

ave

a sl

op

e o

f 0,

T�R�|| S�

F�.

x

y T( 1

0, 0

)

S( 3

0, 2

5)F

( 0, 2

5)

R( 2

0, 0

)O

xO

y

B( a

, 0)

D( b

, c)

C( a

� b

, c)

A( 0

, 0)

x

y

W( a

, 0)

Z( b

, c)

Y( ?

, ?)

X( ?

, ?)

Ox

y

J(2b

, 0)

K( ?

, a)

L(c,

?)

H( 0

, 0)

O

xO

y

B( a

� 2

c, 0

)

D( a

, b)

C( 2

c, b

)

A( 0

, 0)

xO

y

B( b

, 0)

D( c

, a)

C( b

� c

, a)

A( 0

, 0)P

ract

ice (

Ave

rag

e)

Co

ord

inat

e P

roo

f w

ith

Qu

adri

late

rals

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7



Readin

g t

o L

earn

Math

em

ati

csC

oo

rdin

ate

Pro

of

wit

h Q

uad

rila

tera

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7

©G

lenc

oe/M

cGra

w-H

ill45

7G

lenc

oe G

eom

etry

Lesson 8-7

Pre-

Act

ivit

yH

ow c

an y

ou u

se a

coo

rdin

ate

pla

ne

to p

rove

th

eore

ms

abou

tq

uad

rila

tera

ls?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-7

at

the

top

of p

age

447

in y

our

text

book

.

Wha

t sp

ecia

l ki

nds

of q

uadr

ilat

eral

s ca

n be

pla

ced

on a

coo

rdin

ate

syst

em s

oth

at t

wo

side

s of

th

e qu

adri

late

ral

lie

alon

g th

e ax

es?

rect

ang

les

and

sq

uar

es

Rea

din

g t

he

Less

on

1.F

ind

the

mis

sin

g co

ordi

nat

es i

n e

ach

fig

ure

.Th

en w

rite

th

e co

ordi

nat

es o

f th

e fo

ur

vert

ices

of

the

quad

rila

tera

l.

a.is

osce

les

trap

ezoi

db

.par

alle

logr

am

R(�

a,0)

,S(�

b,c

),T

(b,c

),U

(a,0

)M

(0,0

),N

(0,a

),P

(b,c

�a)

,Q(b

,c)

2.R

efer

to

quad

rila

tera

l E

FG

H.

a.F

ind

the

slop

e of

eac

h s

ide.

slo

pe

E�F�

�3,

slo

pe

F�G��

�1 3� ,sl

op

e G�

H��

3,sl

op

e H�

E��

�1 3�

b.

Fin

d th

e le

ngt

h o

f ea

ch s

ide.

EF

��

10�,F

G�

�10�

,GH

��

10�,H

E�

�10�

c.F

ind

the

slop

e of

eac

h d

iago

nal

.sl

op

e E�

G��

1,sl

op

e F�H�

��

1d

.F

ind

the

len

gth

of

each

dia

gon

al.

EG

��

32�,F

H�

�8�

e.W

hat

can

you

con

clu

de a

bou

t th

e si

des

of E

FG

H?

All

fou

r si

des

are

co

ng

ruen

tan

d b

oth

pai

rs o

f o

pp

osi

te s

ides

are

par

alle

l.f.

Wh

at c

an y

ou c

oncl

ude

abo

ut

the

diag

onal

s of

EF

GH

?T

he

dia

go

nal

s ar

ep

erp

end

icu

lar

and

th

ey a

re n

ot

con

gru

ent.

g.C

lass

ify

EF

GH

as a

par

alle

logr

am,a

rh

ombu

s,or

a s

quar

e.C

hoo

se t

he

mos

t sp

ecif

icte

rm.E

xpla

in h

ow y

our

resu

lts

from

par

ts a

-f s

upp

ort

you

r co

ncl

usi

on.

EF

GH

is a

rh

om

bus.

All

fou

r si

des

are

co

ng

ruen

t an

d t

he

dia

go

nal

s ar

e p

erp

end

icu

lar.

Sin

ce t

he

dia

go

nal

s ar

e n

ot

con

gru

ent,

EF

GH

is n

ot

a sq

uar

e.

Hel

pin

g Y

ou

Rem

emb

er3.

Wh

at i

s an

eas

y w

ay t

o re

mem

ber

how

bes

t to

dra

w a

dia

gram

th

at w

ill

hel

p yo

u d

evis

ea

coor

din

ate

proo

f?S

amp

le a

nsw

er:

A k

ey p

oin

t in

th

e co

ord

inat

e p

lan

e is

the

ori

gin

.Th

e ev

eryd

ay m

ean

ing

of

ori

gin

is p

lace

wh

ere

som

eth

ing

beg

ins.

So

loo

k to

see

if t

her

e is

a g

oo

d w

ay t

o b

egin

by

pla

cin

g a

ver

tex

of

the

fig

ure

at

the

ori

gin

.

x

y

O

E

FG

H

x

y

Q( b

, ?)

P( ?

, c �

a)

N( ?

, a)

M( ?

, ?)

Ox

y

U( a

, ?)

T( b

, ?)

S( ?

, c)

R( ?

, ?)

O

©G

lenc

oe/M

cGra

w-H

ill45

8G

lenc

oe G

eom

etry

Co

ord

inat

e P

roo

fsA

n i

mpo

rtan

t pa

rt o

f pl

ann

ing

a co

ordi

nat

e pr

oof

is c

orre

ctly

pla

cin

g an

d la

beli

ng

the

geom

etri

c fi

gure

on

th

e co

ordi

nat

e pl

ane.

Dra

w a

dia

gram

you

wou

ld u

se i

n a

coo

rdin

ate

pro

of o

f th

efo

llow

ing

theo

rem

.

Th

e m

edia

n o

f a

trap

ezoi

d i

s pa

rall

el t

o th

e ba

ses

of t

he

trap

ezoi

d.

In t

he

diag

ram

,not

e th

at o

ne

vert

ex,A

,is

plac

ed a

t th

e or

igin

.Als

o,th

e co

ordi

nat

es o

f B

,C,a

nd

Du

se 2

a,2b

,an

d 2c

in o

rder

to

avoi

d th

e u

se o

f fr

acti

ons

wh

en f

indi

ng

the

coor

din

ates

of

the

mid

poin

ts,

Man

d N

.

Wh

en d

oin

g co

ordi

nat

e pr

oofs

,th

e fo

llow

ing

stra

tegi

es m

ay b

e h

elpf

ul.

1.If

you

are

ask

ed t

o pr

ove

that

seg

men

ts a

re p

aral

lel

or p

erpe

ndi

cula

r,u

se s

lope

s.

2.If

you

are

ask

ed t

o pr

ove

that

seg

men

ts a

re c

ongr

uen

t or

hav

e re

late

d m

easu

res,

use

th

e di

stan

ce f

orm

ula

.

3.If

you

are

ask

ed t

o pr

ove

that

a s

egm

ent

is b

isec

ted,

use

th

e m

idpo

int

form

ula

.

For

eac

h o

f th

e fo

llow

ing

theo

rem

s,a

dia

gram

has

bee

n p

rovi

ded

to

be

use

d i

n a

form

al p

roof

.Nam

e th

e m

issi

ng

coor

din

ates

in

th

e d

iagr

am.T

hen

,usi

ng

the

Giv

enan

d t

he

Pro

vest

atem

ents

,pro

ve t

he

theo

rem

.

1.T

he

med

ian

of

a tr

apez

oid

is p

aral

lel

to t

he

base

s of

th

e tr

apez

oid.

(Use

th

e di

agra

mgi

ven

in

th

e ex

ampl

e ab

ove.

)

Coo

rdin

ates

of

Man

d N

:M

(b,c

);N

(d�

a,c)

Giv

en:T

rape

zoid

AB

CD

has

med

ian

M�N�

.P

rove

:B �C�

|| M�N�

and

A�D�

|| M�N�

Pro

of:

Th

e sl

op

e o

f A�

D�is

� 20 a� �0 0

�o

r 0.

Th

e sl

op

e o

f B�

C�is

� 22 dc� �

2 2c b�

or

0.

Th

e sl

op

e o

f M�

N�is

� (a�c

� b)c �

b�

or

0.S

ince

all

thre

e sl

op

es a

re

equ

al,B�

C�|| M�

N�an

d A�

D�|| M�

N�.

2.T

he

med

ian

s to

th

e le

gs o

f an

iso

scel

es t

rian

gle

are

con

gru

ent.

Coo

rdin

ates

of

Tan

d K

:T(a

,b);

K(3

a,b

)

Giv

en:�

AB

Cis

iso

scel

es w

ith

med

ian

s T�

B�an

d K�

A�.

Pro

ve:T �

B��

K�A�

Pro

of:

TB

��

(4a

��

a)2

��

(0 �

�b

)2 �o

r �

9a2

��

b2

�K

A�

�(3

a�

�0)

2�

�(b

��

0)2

�o

r �

9a2

��

b2

�S

ince

TB

�K

A,T�

B��

K�A�

.

x

y

O

TK

A( 0

, 0)

B( 4

a, 0

)

C( 2

a, 2

b)

x

y

O

MN

A( 0

, 0)

D( 2

a, 0

)

C( 2

d, 2

c)B

( 2b,

2c)

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7

Exam

ple

Exam

ple



Chapter 8 Assessment Answer Key Form 1 Form 2APage 459 Page 460 Page 461


An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

B

C

B

C

D

A

B

A

D

A

C

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

C

B

D

A

B

A

D

B

B

x � 7,m�WYZ � 41

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

B

B

D

A

C

A

A

C

B

A

D


Chapter 8 Assessment Answer KeyForm 2A (continued) Form 2BPage 462 Page 463 Page 464

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

D

C

C

A

B

D

B

D

B

7 or �4

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

B

D

D

A

B

B

A

C

C

D

B

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

B

D

B

D

C

C

A

A

C

22


Chapter 8 Assessment Answer KeyForm 2CPage 465 Page 466

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

10,440

19

40

18

8

122

(6, 4)

Yes; A�B� and C�D� are|| and �.

No; the slopes are

�94

�, �17

�, 1, and �23

�.

Thus, ABCD doesnot have || sides.

��23

�

22

Yes; if the diagonalsof a � are �, then

the � is a rectangle.

67.5

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

(4, 0)

31

Yes; ABCD has onlyone pair of oppositesides ||, B�C� and A�D�.

6

A(a, 0), A�C� ⊥ B�D�

true

true

true

false

true

true

false

x � 9, y � 2


Chapter 8 Assessment Answer KeyForm 2DPage 467 Page 468

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

6120

38

90

50

3.6

117

��12

�, 3�

Yes; Both pairs ofopp. sides are �.

ABCD has two pairs

of || sides, A�B� || C�D�and B�D� || D�A�; it is a �.

�4

8

One rt. � means thatthe other � will be rt.�. If all 4 � are rt. �,the � is a rectangle.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

72

(�3, 1)

16

Yes; ABCD has onlyone pair of opp.

sides ||, A�D� and B�C�.

8

B(a, a); Themidpoint of bothdiagonals is at

��a2

�, �a2

��.

false

true

false

false

true

false

false

90


Chapter 8 Assessment Answer KeyForm 3Page 469 Page 470

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

3960

30; 30, 47, 120,179, 174, and 170

�18x0

�

65

8 or 32

Yes; the diagonalsbisect each other.

slope of C�D� � �23

�;

slope of D�A� � �2.

28

72

8�2�

4

9

13.

14.

15.

16.

17.

18.

19.

20.

B:

13

Sample answer:(b � d, c)

Yes; A�B� ⊥ B�C�,

B�C� ⊥ C�D�, C�D� ⊥ A�D�,so all � are rt. �.

ABCD has 2 pairs ofopp. sides �, A�B� �C�D� and B�C� � D�A�,so ABCD is a �.

C(�2a, 0); thesegment joining the

midpoints of B�C�and A�B� is || to A�C�.

Opp. sides of a �are �.

If both pairs of opp.sides of a quad.are �, then thequad. is a �.

27

54


Chapter 8 Assessment Answer KeyPage 471, Open-Ended Assessment

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts of anglesof polygons, properties of parallelograms, rectangles,rhombi, squares, and trapezoids, and coordinate proofs.

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

• Shows an understanding of the concepts of angles ofpolygons, properties of parallelograms, rectangles, rhombi,squares, and trapezoids, and coordinate proofs.

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

• Shows an understanding of most of the concepts ofangles of polygons, properties of parallelograms,rectangles, rhombi, squares, and trapezoids, andcoordinate proofs.

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

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

final computation.• Graphs and 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 conceptsof angles of polygons, properties of parallelograms,rectangles, rhombi, squares, and trapezoids, andcoordinate proofs.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Graphs and 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

Chapter 8 Assessment Answer KeyPage 471, Open-Ended Assessment

Sample Answers


1. a. Any type of convex polygon can bedrawn as long as one is regular andone is not regular and both have thesame number of sides.

b. Check to be sure that the exteriorangles have been properly drawn andaccurately measured.

c. 4(90) � 360; 120 � 70 � 60 � 110 �360; The sum of the exterior angles ofeach figure should be 360°. The sum ofthe exterior angles of both the regularconvex polygon and the irregularconvex polygon is 360°.

2. The student should draw a rectangle andjoin the midpoints of consecutive sides.The figure formed inside is a rhombus.Since all four small triangles can beproved to be congruent by SAS, the foursides of the interior quadrilateral arecongruent by CPCTC, making it arhombus.

3. The student should draw an isoscelestrapezoid with one pair of opposite sidesparallel and the other pair of oppositesides congruent, as in the figure below.

4. a. Possible properties:A square has four congruent sides anda rectangle may not.A square has perpendicular diagonalsand a rectangle may not.The diagonals of a square bisect theangles and those in a rectangle maynot.

b. Possible properties:A square has four right angles and arhombus may not.The diagonals of a square are congruentand those of a rhombus may not be.

c. Possible properties:A rectangle has four right angles anda parallelogram may not.The diagonals of a rectangle arecongruent and those of aparallelogram may not be.

5. The slope of diagonal A�C� is �a �c

b�, while

the slope of diagonal B�D� is �b �c

a�. These slopes are not necessarily oppositereciprocals of each other. Therefore John’sfigure could not be used to prove that thediagonals of a rhombus are perpendicular.The figure John drew on the coordinateplane has vertices that make it aparallelogram but not always a rhombus.

90�70�

60�

110�120�

90�

90�90�

In addition to the scoring rubric found on page A28, the following sample answers may be used as guidance in evaluating open-ended assessment items.

An

swer

s


Chapter 8 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 472 Page 473 Page 474

Quiz 2Page 473

Quiz 4Page 474

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

isoscelestrapezoid

parallelogram

trapezoid

square

rhombus

rectangle

kite

diagonals

median

bases

angles formed by thebase and one of thelegs of a trapezoid

the nonparallel sidesof a trapezoid

1.

2.

3.

4.

5.

12,240

45

20

(�1, 10)

A

1.

2.

3.

4.

5.

No; none of thetests for �s are

fulfilled.

false

false

true

m�1 � 68,m�2 � 56,m�3 � 34

1.

2.

3.

4.

5.

Rhombus; allsides are �.

108

50

true

12

1.

2.

3.

4.

5.

C(�a, 0)

D(0, �a)

A(�d, c)

X(a � b, c),Y(�e � d, c)

C(a � b, c),AB � CD �

�b2 ��c2�,BC � AD � a


Chapter 8 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 475 Page 476

An

swer

s

Part I

Part II

6.

7.

8.

9.

10.

50

42

yes

m�1 � 12, m�2 �78, m�3 � 12,

m�4 � 156

If one pair of opp.sides is || and �,then the quad.

is a �.

1.

2.

3.

4.

5.

B

D

A

B

C

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

BR�� and BT��

y � ��32

�x � 9

11, 2

Assume thatneither bell costmore than $45.

20.5 in., 33 in.,18.75 in., and 30 in.

A(3, 0.5) andB(�0.5, �1)

� of elevation:�XYZ; � of

depression: �WXY

13

55; 125; 5

8.5

LMNP is a trapezoid

since P�L� || M�N� and

P�N� ⁄|| L�M�.


Chapter 8 Assessment Answer KeyStandardized Test Practice

Page 477 Page 478

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. 11.

11. 12.

13.

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

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

14.

15.

16.

hexagon;concave;irregular

88.9 cm

D�L�

1 / 7 7

1 1 . 2

9 0 0

1 0

Documents

Chapter 8 Resource Masters - farragutcareeracademy.org · ©Glencoe/McGraw-Hill iv Glencoe Geometry Teacher’s Guide to Using the Chapter 8 Resource Masters The Fast FileChapter