Chapter 3 Resource Masters - Math Problem Solving · ©Glencoe/McGraw-Hill iv Glencoe Geometry Teacher’s Guide to Using the Chapter 3 Resource Masters The Fast FileChapter Resource

Chapter 3Resource 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 3 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: 0-07-860180-0 GeometryChapter 3 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 3-1Study Guide and Intervention . . . . . . . . 125–126Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 127Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Reading to Learn Mathematics . . . . . . . . . . 129Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 130






Chapter 3 AssessmentChapter 3 Test, Form 1 . . . . . . . . . . . . 161–162Chapter 3 Test, Form 2A . . . . . . . . . . . 163–164Chapter 3 Test, Form 2B . . . . . . . . . . . 165–166Chapter 3 Test, Form 2C . . . . . . . . . . . 167–168Chapter 3 Test, Form 2D . . . . . . . . . . . 169–170Chapter 3 Test, Form 3 . . . . . . . . . . . . 171–172Chapter 3 Open-Ended Assessment . . . . . . 173Chapter 3 Vocabulary Test/Review . . . . . . . 174Chapter 3 Quizzes 1 & 2 . . . . . . . . . . . . . . . 175Chapter 3 Quizzes 3 & 4 . . . . . . . . . . . . . . . 176Chapter 3 Mid-Chapter Test . . . . . . . . . . . . 177Chapter 3 Cumulative Review . . . . . . . . . . . 178Chapter 3 Standardized Test Practice . 179–180Unit 1 Test/Review (Ch. 1–3) . . . . . . . . 181–182

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

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A30

© Glencoe/McGraw-Hill iv Glencoe Geometry

Teacher’s Guide to Using theChapter 3 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 3 Resource Masters includes the core materials neededfor Chapter 3. 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 3-1.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 3-1.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 3Resources 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 172–173. 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 _____

33

© 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 3.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

alternate exterior angles

alternate interior angles

consecutive interior angles

corresponding angles

equidistant

ee·kwuh·DIS·tuhnt

non-Euclidean geometry

yoo·KLID·ee·yuhn

parallel lines

parallel planes

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

plane Euclidean geometry

point-slope form

rate of change

skew lines

slope

slope-intercept form

spherical geometry

SFIR·uh·kuhl

transversal

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

33

Learning to Read MathematicsProof Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

33

© Glencoe/McGraw-Hill ix Glencoe Geometry

Proo

f Bu

ilderThis is a list of key theorems and postulates you will learn in Chapter 3. 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 3.1Alternate Interior Angles Theorem

Theorem 3.2Consecutive Interior Angles Theorem

Theorem 3.3Alternate Exterior Angles Theorem

Theorem 3.4Perpendicular Transversal Theorem

Theorem 3.5

Theorem 3.6

Theorem 3.7


© Glencoe/McGraw-Hill x Glencoe Geometry

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 3.8

Theorem 3.9

Postulate 3.1Corresponding Angles Postulate

Postulate 3.2

Postulate 3.3

Postulate 3.4

Postulate 3.5Parallel Postulate

Learning to Read MathematicsProof Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

33

Study Guide and InterventionParallel Lines and Transversals

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

© Glencoe/McGraw-Hill 125 Glencoe Geometry

Less

on

3-1

Relationships Between Lines and Planes When two lines lie in the same plane and do not intersect, they are parallel.Lines that do not intersect and are not coplanar are skew lines.In the figure, � is parallel to m, or � || m. You can also write PQ�� || RS��. Similarly, if two planes do not intersect, they are parallel planes.

a. Name all planes that are parallel to plane ABD.plane EFH

b. Name all segments that are parallel to C�G�.B�F�, D�H�, and A�E�

c. Name all segments that are skew to E�H�.B�F�, C�G�, B�D�, C�D�, and A�B�

For Exercises 1–3, refer to the figure at the right.

1. Name all planes that intersect plane OPT.

2. Name all segments that are parallel to N�U�.

3. Name all segments that intersect M�P�.


4. Name all segments parallel to Q�X�.

5. Name all planes that intersect plane MHE.

6. Name all segments parallel to Q�R�.

7. Name all segments skew to A�G�.

T

SG

A

X

EN

MH

O

Q

R

M

N O

P

R

U T

S

A

B C

D

E

F G

H

P Q

R S

n

m

�

ExampleExample

ExercisesExercises


Angle Relationships A line that intersects two or more other lines in a plane is calleda transversal. In the figure below, t is a transversal. Two lines and a transversal form eightangles. Some pairs of the angles have special names. The following chart lists the pairs ofangles and their names.

Angle Pairs Name

�3, �4, �5, and �6 interior angles

�3 and �5; �4 and �6 alternate interior angles

�3 and �6; �4 and �5 consecutive interior angles

�1, �2, �7, and �8 exterior angles

�1 and �7; �2 and �8 alternate exterior angles

�1 and �5; �2 and �6; �3 and �7; �4 and �8

corresponding angles

Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutiveinterior angles.

a. �10 and �16 b. �4 and �12alternate exterior angles corresponding angles

c. �12 and �13 d. �3 and �9consecutive interior angles alternate interior angles

Use the figure in the Example for Exercises 1–12.

Name the transversal that forms each pair of angles.

1. �9 and �13 2. �5 and �14 3. �4 and �6

Identify each pair of angles as alternate interior, alternate exterior,corresponding, or consecutive interior angles.

4. �1 and �5 5. �6 and �14 6. �2 and �8

7. �3 and �11 8. �12 and �3 9. �4 and �6

10. �6 and �16 11. �11 and �14 12. �10 and �16

�

p q

n1 2 9 10111234

141365151678

t

n

m1 2

34

5 678

Study Guide and Intervention (continued)

Parallel Lines and Transversals

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

ExampleExample

ExercisesExercises

Skills PracticeParallel Lines and Transversals

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1


Less

on

3-1


1. Name all planes that are parallel to plane DEH.

2. Name all segments that are parallel to A�B�.

3. Name all segments that intersect G�H�.

4. Name all segments that are skew to C�D�.

Identify the sets of lines to which each given line is a transversal.

5. r

6. s

7. w

Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles.

8. �2 and �8 9. �3 and �6

10. �1 and �9 11. �3 and �9

12. �6 and �12 13. �7 and �11

Name the transversal that forms each pair of angles. Then identify the special name for the angle pair.

14. �4 and �10 15. �2 and �12

16. �7 and �3 17. �13 and �10

18. �8 and �14 19. �6 and �14

h

g

j

k

12

910

1112

13 1514

1634

65 7

8

a

b

cd

1 2

9 101112

3465

78

rs

t

w

z

A

E F

B

D

H G

C



1. Name all planes that intersect plane STX.

2. Name all segments that intersect Q�U�.

3. Name all segments that are parallel to X�Y�.

4. Name all segments that are skew to V�W�.

Identify the sets of lines to which each given line is a transversal.

5. e

6. h

Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles.

7. �9 and �13 8. �6 and �16

9. �3 and �10 10. �8 and �14

Name the transversal that forms each pair of angles. Then identify the special name for the angle pair.

11. �2 and �12 12. �6 and �18

13. �13 and �19 14. �11 and �7

FURNITURE For Exercises 15–16, refer to the drawing of the end table.

15. Find an example of parallel planes.

16. Find an example of parallel lines.

cd

b

a1

9 101112

13 141516

1719

1820

56

78

234

m

�

pn

1

9 1011

1213

1415

16

5 678

234

g

he

i

f

Q R

ST

U

V W

X

Y

Z

Practice Parallel Lines and Transversals

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

Reading to Learn MathematicsParallel Lines and Transversals

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1


Less

on

3-1

Pre-Activity How are parallel lines and planes used in architecture?

Read the introduction to Lesson 3-1 at the top of page 126 in your textbook.

• Give an example of parallel lines that can be found in your classroom.

• Give an example of parallel planes that can be found in your classroom.

Reading the Lesson

1. Write a geometrical term that matches each definition.

a. two planes that do not intersect

b. lines that are not coplanar and do not intersect

c. two coplanar lines that do not intersect

d. a line that intersects two or more lines in a plane at different points

e. a pair of angles determined by two lines and a transversal consisting of an interiorangle and an exterior angle that have different vertices and that lie on the same sideof the transversal

2. Refer to the figure at the right. Give the special name for each angle pair.

a. �3 and �5

b. �6 and �12

c. �4 and �8

d. �2 and �3

e. �8 and �12

f. �5 and �9

g. �4 and �10

h. �6 and �7

Helping You Remember

3. A good way to remember new mathematical terms is to relate them to words that youuse in everyday life. Many words start with the prefix trans-, which is a Latin rootmeaning across. List four English words that start with trans-. How can the meaning ofthis prefix help you remember the meaning of transversal?

�m

np

12

910

1112

34

65

78


Perspective DrawingsTo draw three-dimensional objects, artists makeperspective drawingssuch as the ones shown.To indicate depth in aperspective drawing, someparallel lines are drawn as converging lines. Thedotted lines in the figuresbelow each extend to avanishing point, or spotwhere parallel lines appear to meet.

Draw lines to locate the vanishing point in each drawing of a box.

1. 2. 3.

4. The fronts of two cubes are shown below. Using point P as the vanishing point for both cubes, complete the perspective drawings of the cubes.

5. Find an example of a perspective drawing in a newspaper or magazine.Trace the drawing and locate a vanishing point.

P

Vanishing points

Railroad tracks Cube Cabinet

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

Study Guide and InterventionAngles and Parallel Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2


Less

on

3-2

Parallel Lines and Angle Pairs When two parallel lines are cut by a transversal,the following pairs of angles are congruent.

• corresponding angles• alternate interior angles• alternate exterior angles

Also, consecutive interior angles are supplementary.

In the figure, m�2 � 75. Find the measures of the remaining angles.m�1 � 105 �1 and �2 form a linear pair.m�3 � 105 �3 and �2 form a linear pair.m�4 � 75 �4 and �2 are vertical angles.m�5 � 105 �5 and �3 are alternate interior angles.m�6 � 75 �6 and �2 are corresponding angles.m�7 � 105 �7 and �3 are corresponding angles.m�8 � 75 �8 and �6 are vertical angles.

In the figure, m�3 � 102. Find the measure of each angle.

1. �5 2. �6

3. �11 4. �7

5. �15 6. �14

In the figure, m�9 � 80 and m�5 � 68. Find the measure of each angle.

7. �12 8. �1

9. �4 10. �3

11. �7 12. �16

w v

p

q

1 234

6578

9 101112

1413

1516

p q

m

n

1 234

6578

9 101112

14131516

p

m

n

1 234

6578

ExampleExample

ExercisesExercises


Algebra and Angle Measures Algebra can be used to find unknown values in angles formed by a transversal and parallel lines.

If m�1 � 3x � 15, m�2 � 4x � 5, m�3 � 5y,and m�4 � 6z � 3, find x and y.

p || q, so m�1 � m�2 because they are corresponding angles.

3x � 15 � 4x � 53x � 15 � 3x � 4x � 5 � 3x

15 � x � 515 � 5 � x � 5 � 5

20 � x

p q

r

s

1 2

34


Angles and Parallel Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2

ExampleExample

r || s, so m�2 � m�3 because they are corresponding angles.

m�2 � m�375 � 5y

�755� � �

55y�

15 � y

ExercisesExercises

Find x and y in each figure.

1. 2.

3. 4.

Find x, y, and z in each figure.

5. 6.

2x�

2y�

90� x�

z�

2y�106�

x�

(4z � 6)�

(5x � 20)�

3x�2y �

4y �

(11x � 4)�

(13y � 5)�

(5y � 5)�

5x�

(15x � 30)�

(3y � 18)� 10x�

90�

(5x � 5)�(6y � 4)�

(4x � 10)�

Skills PracticeAngles and Parallel Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2


Less

on

3-2


1. �3 2. �5

3. �8 4. �1

5. �4 6. �6


7. �9 8. �6

9. �8 10. �2

11. �5 12. �11


13. �2 14. �5

15. �7 16. �15

17. �14 18. �9


19. 20.

Find m�1 in each figure.

21. 22.140�

32�

160�

1

20�

7x�

(8x � 10)�

(6y � 20)�

5x�

40�(3y � 1)�

x

w

z

y

1 2

43

65

8

9 1011 12

13 1415 16

7

sm

ut

1 2

4 3

658

91011

127

q

r

s

1 243

6587



1. �10 2. �8

3. �9 4. �5

5. �11 6. �13


7. 8.

Find m�1 in each figure.

9. 10.

11. PROOF Write a paragraph proof of Theorem 3.3.Given: � || m, m || nProve: �1 � �12

12. FENCING A diagonal brace strengthens the wire fence and prevents it from sagging. The brace makes a 50° angle with the wire as shown.Find y. 50�

y�

m

n

1 23 4

657 8

�

k

9 1011 12

144�

62�

1

100�

50�

1

3y� (2x � 13)�(5y � 4)�3x�

(9x � 12)�

(4y � 10)�

n

m

rs

12

43

65

8

910

1112

1314

1516

7

Practice Angles and Parallel Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2

Reading to Learn MathematicsAngles and Parallel Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2


Less

on

3-2

Pre-Activity How can angles and lines be used in art?

Read the introduction to Lesson 3-2 at the top of page 133 in your textbook.• Your textbook shows a painting that contains two parallel lines and a

transversal. What is the name for �1 and �2?• What is the relationship between these two angles?

Reading the Lesson1. Choose the correct word to complete each sentence.

a. If two parallel lines are cut by a transversal, then alternate exterior angles are

(congruent/complementary/supplementary).

b. If two parallel lines are cut by a transversal, then corresponding angles are


c. If parallel lines are cut by a transversal, then consecutive interior angles are


d. In a plane, if a line is perpendicular to one of two parallel lines, then it is

(parallel/perpendicular/skew) to the other.

Use the figure for Exercises 2 and 3.

2. a. Name four pairs of vertical angles.

b. Name all angles that form a linear pair with �7.

c. Name all angles that are congruent to �1.

d. Name all angles that are congruent to �4.

e. Name all angles that are supplementary to �3.

f. Name all angles that are supplementary to �2.

3. Which conclusion(s) could you make about lines u and v if m�4 � m�1?

A. t || u B. t ⊥ u C. v ⊥ u D. v ⊥ t E. v || t

Helping You Remember4. How can you use an everyday meaning of the adjective alternate to help you remember

the types of angle pairs for two lines and a transversal?

t

u

v

12

34

65

78


More Optical IllusionsIn drawings, diagonal lines may create the illusion of depth.For example, the figure at the right can be thought of as picturing a flat figure or a cube. The optical illusions on this page involve depth perception.

Answer each question.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2

1. How many cubes do you see in thedrawing?

3. Does the drawing show a view from the top or the bottom of the stairs?

2. Can this figure show an actual object?

4. Which line segment is longer, A�B� orC�D�? Measure to check your answer.

A

B C

D

5. Which person in the drawing at the right appears to be tallest? Measure to check your answer.

6. Draw two more objects the same size on the figure at the right. Does one appear larger than the other?

Study Guide and InterventionSlopes of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3


Less

on

3-3

Slope of a Line The slope m of a line containing two points with coordinates (x1, y1)

and (x2, y2) is given by the formula m � �yx

2

2

�

�

yx

1

1�, where x1 � x2.

Find the slope of each line.For line p, let (x1, y1) be (1, 2) and (x2, y2) be (�2, �2).

m � �yx

2

2

�

�

yx

1

1�

� ��

22

��

21� or �

43�

For line q, let (x1, y1) be (2, 0) and (x2, y2) be (�3, 2).

m � �yx

2

2

�

�

yx

1

1�

� ��23��

02� or ��

25�

Determine the slope of the line that contains the given points.

1. J(0, 0), K(�2, 8) 2. R(�2, �3), S(3, �5)

3. L(1, �2), N(�6, 3) 4. P(�1, 2), Q(�9, 6)

5. T(1, �2), U(6, �2) 6. V(�2, 10), W(�4, �3)

Find the slope of each line.

7. AB�� 8. CD��

9. EM�� 10. AE��

11. EH�� 12. BM��

x

y

O

C(–2, 2)

A(–2, –2)

H(–1, –4)

B(0, 4)

M(4, 2)

E(4, –2)

D(0, –2)

x

y

O

(1, 2)

(–2, –2)

(–3, 2)

(2, 0)

ExampleExample

ExercisesExercises


Parallel and Perpendicular Lines If you examine the slopes of pairs of parallel linesand the slopes of pairs of perpendicular lines, where neither line in each pair is vertical, youwill discover the following properties.

Two lines have the same slope if and only if they are parallel.Two lines are perpendicular if and only if the product of their slopes is �1.


Slopes of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3

Find the slope of a lineparallel to the line containing A(�3, 4)and B(2, 5).Find the slope of AB��. Use (�3, 4) for (x1, y1)and use (2, 5) for (x2, y2).

m � �yx

2

2

�

�

yx

1

1�

� �2

5�

�

(�43)

� or �15�

The slope of any line parallel to AB�� must be �

15�.

Find the slope of a lineperpendicular to PQ�� for P(�2, �4) andQ(4, 3).Find the slope of PQ��. Use (�2, �4) for (x1, y1) and use (4, 3) for (x2, y2).

m � �yx

2

2

�

�

yx

1

1�

� �34

�

�

((�

�

42))

� or �76�

Since �76� � ��

67�� 1, the slope of any line

perpendicular to PQ�� must be ��67�.

Example 1Example 1 Example 2Example 2

ExercisesExercises

Determine whether MN�� and RS�� are parallel, perpendicular, or neither.

1. M(0, 3), N(2, 4), R(2, 1), S(8, 4) 2. M(�1, 3), N(0, 5), R(2, 1), S(6, �1)

3. M(�1, 3), N(4, 4), R(3, 1), S(�2, 2) 4. M(0, �3), N(�2, �7), R(2, 1), S(0, �3)

5. M(�2, 2), N(1, �3), R(�2, 1), S(3, 4) 6. M(0, 0), N(2, 4), R(2, 1), S(8, 4)

Find the slope of MN�� and the slope of any line perpendicular to MN��.

7. M(2, �4), N(�2, �1) 8. M(1, 3), N(�1, 5)

9. M(4, �2), N(5, 3) 10. M(2, �3), N(�4, 1)

Skills PracticeSlopes of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3


Less

on

3-3


1. S(�1, 2), W(0, 4) 2. G(�2, 5), H(1, �7)

3. C(0, 1), D(3, 3) 4. J(�5, �2), K(5, �4)


5. NP�� 6. TW��

7. a line parallel to TW�� 8. a line perpendicular to NP��

Determine whether AB�� and MN�� are parallel, perpendicular, or neither.

9. A(0, 3), B(5, �7), M(�6, 7), N(�2, �1) 10. A(�1, 4), B(2, �5), M(�3, 2), N(3, 0)

11. A(�2, �7), B(4, 2), M(�2, 0), N(2, 6) 12. A(�4, �8), B(4, �6), M(�3, 5), N(�1, �3)

Graph the line that satisfies each condition.

13. slope � 3, contains A(0, 1) 14. slope � ��32�, contains R(�4, 5)

15. contains Y(3, 0), parallel to DJ�� 16. contains T(0, �2), perpendicular to CX��

with D(�3, 1) and J(3, 3) with C(0, 3) and X(2, �1)

T(0, –2)

C(0, 3)

X(2, –1)x

y

OD(–3, 1)

J(3, 3)

Y(3, 0)x

y

O

R(–4, 5)

x

y

O

A(0, 1)x

y

O

x

y

ON T

W

P



1. B(�4, 4), R(0, 2) 2. I(�2, �9), P(2, 4)


3. LM�� 4. GR��

5. a line parallel to GR�� 6. a line perpendicular to PS��

Determine whether KM�� and ST�� are parallel, perpendicular, or neither.

7. K(�1, �8), M(1, 6), S(�2, �6), T(2, 10) 8. K(�5, �2), M(5, 4), S(�3, 6), T(3, �4)

9. K(�4, 10), M(2, �8), S(1, 2), T(4, �7) 10. K(�3, �7), M(3, �3), S(0, 4), T(6, �5)

Graph the line that satisfies each condition.

11. slope � ��12�, contains U(2, �2) 12. slope � �

43�, contains P(�3, �3)

13. contains B(�4, 2), parallel to FG�� 14. contains Z(�3, 0), perpendicular to EK��

with F(0, �3) and G(4, �2) with E(�2, 4) and K(2, �2)

15. PROFITS After Take Two began renting DVDs at their video store, business soared.Between 2000 and 2003, profits increased at an average rate of $12,000 per year. Totalprofits in 2003 were $46,000. If profits continue to increase at the same rate, what willthe total profit be in 2009?

K(2, –2)

E(–2, 4)

Z(–3, 0)

x

y

O

F(0, –3)

B(–4, 2)

G(4, –2)x

y

O

P(–3, –3)

x

y

OU(2, –2)

x

y

O

x

y

O

R

G

S

P

M

L

Practice Slopes of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3

Reading to Learn MathematicsSlopes of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3


Less

on

3-3

Pre-Activity How is slope used in transportation?

Read the introduction to Lesson 3-3 at the top of page 139 in your textbook.• If you are driving uphill on a road with a 4% grade, how many feet will

the road rise for every 1000 horizontal feet traveled? • If you are driving downhill on a road with a 7% grade, how many meters

will the road fall for every 500 meters traveled?

Reading the Lesson1. Which expressions can be used to represent the slope of the line containing points (x1, y1)

and (x2, y2)? Assume that no denominator is zero.

A. ��

�

yx� B. �hv

oreirztoicnatlarlirsuen

� C. �yx

2

2

�

�

yx

1

1� D. �

cchh

aann

ggee

iinn

xy�

E. �yx

2

1

�

�

yx

1

2� F. �

yx

1

1

�

�

yx

2

2� G. �

xy

2

2

�

�

xy

1

1� H. �

yy

2

1

�

�

xx

2

1�

2. Match the description of a line from the first column with the description of its slopefrom the second column.

Type of Line Slope

a. a horizontal line i. a negative number

b. a line that rises from left to right ii. 0

c. a vertical line iii. undefined

d. a line that falls from left to right iv. a positive number

3. Find the slope of each line.

a. a line parallel to a line with slope �34�

b. a line perpendicular to the x-axis

c. a line perpendicular to a line with slope 5

d. a line parallel to the x-axis

e. y-axis


4. A good way to remember something is to explain it to someone else. Suppose your friendthinks that perpendicular lines (if neither line is vertical) have slopes that arereciprocals of each other. How could you explain to your friend that this is incorrect andgive her a good way to remember the correct relationship?


Slopes and PolygonsIn coordinate geometry, the slopes of two lines determine if the lines areparallel or perpendicular. This knowledge can be useful when working withpolygons.

1. The coordinates of the vertices of a triangle are A(�6, 4), B(8, 6), and C(4, �4). Graph �ABC.

2. J, K, and L are midpoints of A�B�, B�C�, and A�C�,respectively. Find the coordinates of J, K, and L.Draw �JKL.

3. Which segments appear to be parallel?

4. Show that the segments named in Exercise 3 are parallel by finding the slopes of all six segments.

The coordinates of the vertices of right �PQR are given. Find the slope of eachside of the triangle. Then name the hypotenuse.

5. P(5, 1) Q(1, �1) R(�2, 5) 6. P(�2, �3) Q(5, 1) R(2, 3)

slope of P�Q� � slope of P�Q� �

slope of Q�R� � slope of Q�R� �

slope of P�R� � slope of P�R� �

hypotenuse: hypotenuse:

The coordinates of quadrilateral PQRS are given. Graph quadrilateral PQRS andfind the slopes of the diagonals. State whether the diagonals are perpendicular.

7. P(�2, 6), Q(4, 0), R(1, �4), S(�5, 2) 8. P(0, 6), Q(3, 0), R(�4, �2), S(�5, 4)

S

P

Q

R

x

y

O

S

P

Q

R

x

y

O

A

B

C

K

J

L x

y

O

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3

Study Guide and InterventionEquations of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4


Less

on

3-4

Write Equations of Lines You can write an equation of a line if you are given any ofthe following:• the slope and the y-intercept,• the slope and the coordinates of a point on the line, or• the coordinates of two points on the line.

If m is the slope of a line, b is its y-intercept, and (x1, y1) is a point on the line, then:• the slope-intercept form of the equation is y � mx � b,• the point-slope form of the equation is y � y1 � m(x � x1).

Write an equation inslope-intercept form of the line withslope �2 and y-intercept 4.y � mx � b Slope-intercept form

y � �2x � 4 m � �2, b � 4

The slope-intercept form of the equation ofthe line is y � �2x � 4.

Write an equation inpoint-slope form of the line with slope ��

34� that contains (8, 1).

y � y1 � m(x � x1) Point-slope form

y � 1 � ��34�(x � 8) m � ��

34

�, (x1, y1) � (8, 1)

The point-slope form of the equation of theline is y � 1 � ��

34�(x � 8).


ExercisesExercises

Write an equation in slope-intercept form of the line having the given slope and y-intercept.

1. m: 2, y-intercept: �3 2. m: ��12�, y-intercept: 4

3. m: �14�, y-intercept: 5 4. m: 0, y-intercept: �2

5. m: ��53�, y-intercept: �

13� 6. m: �3, y-intercept: �8

Write an equation in point-slope form of the line having the given slope thatcontains the given point.

7. m � �12�, (3, �1) 8. m � �2, (4, �2)

9. m � �1, (�1, 3) 10. m � �14�, (�3, �2)

11. m � ��52�, (0, �3) 12. m � 0, (�2, 5)


Write Equations to Solve Problems Many real-world situations can be modeledusing linear equations.

Donna offers computer services to small companies in her city. Shecharges $55 per month for maintaining a web site and $45 per hour for eachservice call.


Equations of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4

ExampleExample

a. Write an equation torepresent the totalmonthly cost C formaintaining a web siteand for h hours ofservice calls.

For each hour, the costincreases $45. So the rate ofchange, or slope, is 45. They-intercept is located wherethere are 0 hours, or $55.C � mh � b

� 45h � 55

b. Donna may change her costs to represent themby the equation C � 25h � 125, where $125 is thefixed monthly fee for a web site and the cost perhour is $25. Compare her new plan to the old one if a company has 5 �

12� hours of service calls. Under

which plan would Donna earn more?

First plan

For 5�12� hours of service Donna would earn

C � 45h � 55 � 45�5�12�� 55

� 247.5 � 55 or $302.50

Second Plan

For 5�12� hours of service Donna would earn

C � 25h � 125 � 25(5.5) � 125

� 137.5 � 125 or $262.50Donna would earn more with the first plan.

ExercisesExercises

For Exercises 1–4, use the following information.Jerri’s current satellite television service charges a flat rate of $34.95 per month for the basicchannels and an additional $10 per month for each premium channel. A competing satellitetelevision service charges a flat rate of $39.99 per month for the basic channels and anadditional $8 per month for each premium channel.

1. Write an equation in slope-intercept formthat models the total monthly cost foreach satellite service, where p is thenumber of premium channels.

3. A third satellite company charges a flatrate of $69 for all channels, including thepremium channels. If Jerri wants to adda fourth premium channel, which servicewould be least expensive?

2. If Jerri wants to include three premiumchannels in her package, which servicewould be less, her current service or thecompeting service?

4. Write a description of how the fee for thenumber of premium channels is reflectedin the equation.

Skills PracticeEquations of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4


Less

on

3-4


1. m: �4, y-intercept: 3 2. m: 3, y-intercept: �8

3. m: �37�, (0, 1) 4. m: ��

25�, (0, �6)

Write equations in point-slope form and slope-intercept form of the line havingthe given slope and containing the given point.

5. m: 2, (5, 2) 6. m: �3, (2, �4)

7. m: ��12�, (�2, 5) 8. m: �

13�, (�3, �8)

Write an equation in slope-intercept form for each line.

9. r 10. s

11. t 12. u

13. the line parallel to line r that contains (1, �1)

14. the line perpendicular to line s that contains (0, 0)

Write an equation in slope-intercept form for the line that satisfies the givenconditions.

15. m � 6, y-intercept � �2 16. m � ��53�, y-intercept � 0

17. m � �1, contains (0, �6) 18. m � 4, contains (2, 5)

19. contains (2, 0 ) and (0, 10) 20. x-intercept is �2, y-intercept is �1

x

y

O

r

s

t

u



1. m: �23�, y-intercept: �10 2. m: ��

79�, �0, ��

12�� 3. m: 4.5, (0, 0.25)

Write equations in point-slope form and slope-intercept form of the line havingthe given slope and containing the given point.

4. m: �32�, (4, 6) 5. m: ��

65�, (�5, �2)

6. m: 0.5, (7, �3) 7. m: �1.3, (�4, 4)

Write an equation in slope-intercept form for each line.

8. b 9. c

10. parallel to line b, contains (3, �2)

11. perpendicular to line c, contains (�2, �4)

Write an equation in slope-intercept form for the line that satisfies the givenconditions.

12. m � ��49�, y-intercept � 2 13. m � 3, contains (2, �3)

14. x-intercept is �6, y-intercept is 2 15. x-intercept is 2, y-intercept is �5

16. passes through (2, �4) and (5, 8) 17. contains (�4, 2) and (8, �1)

18. COMMUNITY EDUCATION A local community center offers self-defense classes forteens. A $25 enrollment fee covers supplies and materials and open classes cost $10each. Write an equation to represent the total cost of x self-defense classes at thecommunity center.

x

y

O

c

b

Practice Equations of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4

Reading to Learn MathematicsEquations of Lines

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4


Less

on

3-4

Pre-Activity How can the equation of a line describe the cost of cellulartelephone service?

Read the introduction to Lesson 3-4 at the top of page 145 in your textbook.If the rates for your cellular phone plan are described by the equation inyour textbook, what will be the total charge (excluding taxes and fees) for amonth in which you use 50 minutes of air time?

Reading the Lesson1. Identify what each formula represents.

a. y � y1 � m(x � x1)

b. m � �yx

2

2

�

�

yx

1

1�

c. y � mx � b

2. Write the point-slope form of the equation for each line.

a. line with slope ��12� containing (�2, 5)

b. line containing (�4.5, �6.5) and parallel to a line with slope 0.5

3. Which one of the following correctly describes the y-intercept of a line?

A. the y-coordinate of the point where the line intersects the x-axis

B. the x-coordinate of the point where the line intersects the y-axis

C. the y-coordinate of the point where the line crosses the y-axis

D. the x-coordinate of the point where the line crosses the x-axis

E. the ratio of the change in y-coordinates to the change in x-coordinates

4. Find the slope and y-intercept of each line.

a. y � 2x � 7

b. x � y � 8.5

c. 2.4x � y � 4.8

d. y � 7 � x � 12

e. y � 5 � �2(x � 6)

Helping You Remember5. A good way to remember something new is to relate it to something you already know.

How can the slope formula help you to remember the equation for the point-slope form of a line?


Absolute ZeroAll matter is made up of atoms and molecules that are in constantmotion. Temperature is one measure of this motion. Absolute zerois the theoretical temperature limit at which the motion of themolecules and atoms of a substance is the least possible.

Experiments with gaseous substances yield data that allow you toestimate just how cold absolute zero is. For any gas of a constantvolume, the pressure, expressed in a unit called atmospheres,varies linearly as the temperature. That is, the pressure P and thetemperature t are related by an equation of the form P � mt � b,where m and b are real numbers.

1. Sketch a graph for the data in the table.

2. Use the data and your graph to find values for m and b in the equationP � mt � b, which relates temperature to pressure.

3. Estimate absolute zero in degrees Celsius by setting P equal to 0 in the equation above and using the values m and b that you obtained in Exercise 2.

25–25 50 75 100 125

0.5

1.5

1.0

t

P

O

t P(in �C) (in atmospheres)

�25 0.91

0 1.00

25 1.09

100 1.36

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4

Study Guide and InterventionProving Lines Parallel

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5


Less

on

3-5

Identify Parallel Lines If two lines in a plane are cut by a transversal and certainconditions are met, then the lines must be parallel.

If then

• corresponding angles are congruent, the lines are parallel.• alternate exterior angles are congruent,• consecutive interior angles are supplementary,• alternate interior angles are congruent, or• two lines are perpendicular to the same line,

If m�1 � m�2,determine which lines, if any, areparallel.

Since m�1 � m�2, then �1 � �2. �1 and�2 are congruent corresponding angles, so r || s.

n

m

r s

1 2

Find x and m�ABCso that m || n .

We can conclude that m || n if alternateinterior angles are congruent.

m�DAB � m�CDA3x � 10 � 6x � 20

10 � 3x � 2030 � 3x10 � x

m�ABC � 6x � 20� 6(10) � 20 or 40

n

mA

B

C

D

(3x � 10)�

(6x � 20)�


ExercisesExercises

Find x so that � || m .

1. 2. 3.

4. 5. 6.m

n

�

(5x � 20)�

70�

m�

(3x � 20)�2x�

m�

(9x � 1)�(8x � 8)�

m�

(3x � 15)�

m� (4x � 20)�

6x �m�

(5x � 5)�

(6x � 20)�


Prove Lines Parallel You can prove that lines are parallel by using postulates andtheorems about pairs of angles. You also can use slopes of lines to prove that two lines areparallel or perpendicular.


Proving Lines Parallel

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5

ExampleExample

For Exercises 1–6, fill in the blanks.Given: �1 � �5, �15 � �5Prove: � || m, r || s

Statements Reasons

1. �15 � �5 1.

2. �13 � �15 2.

3. �5 � �13 3.

4. r || s 4.

5. 5. Given

6. 6. If corr � are �, then lines ||.

7. Determine whether PQ�� ⊥ TQ��. Explain why or why not.

x

y

O

P Q

T

�

m

r s

1 24 3

5 68 7

9 101112

14131516

a Given: �1 � �2, �1 � �3

Prove: A�B� || D�C�

Statements Reasons1. �1 � �2 1. Given

�1 � �32. �2 � �3 2. Transitive Property of �3. A�B� || D�C� 3. If alt. int. angles are �, then

the lines are ||.

1 2

3A B

CD

b. Which lines are parallel?Which lines are perpendicular?

slope of P�Q� � 0 slope of S�R� � 0

slope of P�S� � �43� slope of Q�R� � �

43�

slope of P�R� � �2 slope of S�Q� � �12�

So P�Q� || S�R�, P�S� || Q�R�, and P�R� ⊥ S�Q�.

x

y

O 4 8–4–8

8

4

–4

–8

P(–2, 4) Q(8, 4)

S(–8, –4) R(2, –4)

ExercisesExercises

Skills PracticeProving Lines Parallel

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5


Less

on

3-5

Given the following information, determine which lines,if any, are parallel. State the postulate or theorem that justifies your answer.

1. �3 � �7 2. �9 � �11

3. �2 � �16 4. m�5 � m�12 � 180


5. 6. 7.

8. PROOF Provide a reason for each statement in the proof of Theorem 3.7.Given:�1 and �2 are complementary.

B�C� ⊥ C�D�Prove: B�A� || C�D�Proof:Statements Reasons1. B�C� ⊥ C�D� 1.2. m�ABC � m�1 � m�2 2.3. �1 and �2 are complementary. 3.4. m�1 � m�2 � 90 4.5. m�ABC � 90 5.6. B�A� ⊥ B�C� 6.7. B�A� || C�D� 7.

Determine whether each pair of lines is parallel. Explain why or why not.

9. 10.

x

y

O

U(4, 2)

T(–4, 0)

S(5, –3)R(0, –4)

x

y

O

A(1, 3)

B(–2, –3)E(0, –3)

F(2, 1)

1

A D

CB 2

m

k�(6x � 4)�

(8x � 8)�mk

�

(4x � 10)�

(3x � 10)�m

k �

(2x � 6)�

130�

m

�

a b1 2 3 4

8 7 6 5

9 10 11 1216 15 14 13


Given the following information, determine which lines,if any, are parallel. State the postulate or theorem that justifies your answer.

1. m�BCG � m�FGC � 180 2. �CBF � �GFH

3. �EFB � �FBC 4. �ACD � �KBF


5. 6. 7.

8. PROOF Write a two-column proof.Given:�2 and �3 are supplementary.Prove: A�B� || C�D�

9. LANDSCAPING The head gardener at a botanical garden wants to plant rosebushes inparallel rows on either side of an existing footpath. How can the gardener ensure thatthe rows are parallel?

1 42 5

3 6AB

CD

(2x � 12)�(5x � 15)�

t

m

�

(5x � 18)�(7x � 24)�

t

m

�

(3x � 6)�

(4x � 6)�

tm

�

A

B CD

E FH

K

GJ

Practice Proving Lines Parallel

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5

Reading to Learn MathematicsProving Lines Parallel

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5


Less

on

3-5

Pre-Activity How do you know that the sides of a parking space are parallel?


How can the workers who are striping the parking spaces in a parking lotcheck to see if the sides of the spaces are parallel?

Reading the Lesson

1. Choose the word or phrase that best completes each sentence.

a. If two coplanar lines are cut by a transversal so that corresponding angles are

congruent, then the lines are (parallel/perpendicular/skew).

b. In a plane, if two lines are perpendicular to the same line, then they are

(perpendicular/parallel/skew).

c. For a line and a point not on the line, there exists

(at least one/exactly one/at most one) line through the point that is parallel to thegiven line.

d. If two coplanar lines are cut by a transversal so that consecutive interior angles are

(complementary/supplementary/congruent), then the lines are

parallel.

e. If two coplanar lines are cut by a transversal so that alternate interior angles are

congruent, then the lines are (perpendicular/parallel/skew).

2. Which of the following conditions verify that p || q ?

A. �6 � �12 B. �2 � �4

C. �8 � �16 D. �11 � �13

E. �6 and �7 are supplementary. F. �1 � �15

G. �7 and �10 are supplementary. H. �4 � �16


3. A good way to remember something new is to draw a picture. How can a sketch help youto remember the Parallel Postulate?

p

t

q

r

1 2

910

1213

14

1516

11

3 46 5

78


Scrambled-Up Proof

The reasons necessary to complete the following proof arescrambled up below. To complete the proof, number thereasons to match the corresponding statements.

Given: C�D� ⊥ B�E�A�B� ⊥ B�E�A�D� � C�E�B�D� � D�E�

Prove: A�D� || C�E�

Proof:

Statements Reasons

1. C�D� ⊥ B�E� Definition of Right Triangle

2. AB ⊥ B�E� Given

3. �3 and �4 are right angles. Given

4. �ABD and �CDE are right triangles. Definition of Perpendicular Lines

5. A�D� � C�E� Given

6. B�D� � D�E� CPCTC

7. �ABD � �CDE In a plane, if two lines are cut by a transversalso that a pair of corresponding angles iscongruent, then the lines are parallel.(Theorem 7-5)

8. �1 � �2 Given

9. A�D� || C�E� HL

A C

B ED

5

3 1 4 2

6

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5

Study Guide and InterventionPerpendiculars and Distance

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3-63-6


Less

on

3-6

Distance From a Point to a Line When a point is not on a line, the distance from the point to the line is the length of the segment that contains the point and isperpendicular to the line.

Draw the segment that represents the distance

from E to AF��.Extend AF��. Draw E�G� ⊥ AF��.E�G� represents the distance from E to AF��.

Draw the segment that represents the distance indicated.

1. C to AB�� 2. D to AB��

3. T to RS�� 4. S to PQ��

5. S to QR�� 6. S to RT��

XR T

S

R

T

S

P

Q

X

RT

S

P QX

U

RS

T

X

A

CD

BXA

C

BX

A F G

B E

A F

B E

Pdistance betweenM and PQ��

Q

M

ExampleExample

ExercisesExercises


Distance Between Parallel Lines The distance between parallel lines is the lengthof a segment that has an endpoint on each line and is perpendicular to them. Parallel linesare everywhere equidistant, which means that all such perpendicular segments have thesame length.

Find the distance between the parallel lines � and m whoseequations are y � 2x � 1 and y � 2x � 4, respectively.


Perpendiculars and Distance

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6

ExampleExample

Draw a line p through (0, 1) that isperpendicular to � and m .

Line p has slope ��12� and y-intercept 1. An

equation of p is y � ��12�x � 1. The point of

intersection for p and � is (0, 1).

x

y

O

mp �

(0, 1)

x

y

O

m� To find the point of intersection of p and m ,solve a system of equations.

Line m : y � 2x � 4Line p: y � ��

12�x � 1

Use substitution.

2x � 4 � ��12�x � 1

4x � 8 � �x � 25x � 10x � 2

Substitute 2 for x to find the y-coordinate.

y � ��12�x � 1

� ��12�(2) � 1 � �1 � 1 � 0

The point of intersection of p and m is (2, 0).

Use the Distance Formula to find thedistance between (0, 1) and (2, 0).

d � �(x2 ��x1)2 �� ( y2 �� y1)2�

� �(2 � 0�)2 � (�0 � 1�)2�� 5�

The distance between � and m is �5� units.

ExercisesExercises

Find the distance between each pair of parallel lines.

1. y � 8 2. y � x � 3 3. y � �2xy � �3 y � x � 1 y � �2x � 5

Skills PracticePerpendiculars and Distance

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6


Less

on

3-6


1. B to AC�� 2. G to EF�� 3. Q to SR��

Construct a line perpendicular to � through K. Then find the distance from K to �.

4. 5.


6. y � 7 7. x � �6 8. y � 3xy � �1 x � 5 y � 3x � 10

9. y � �5x 10. y � x � 9 11. y � �2x � 5y � �5x � 26 y � x � 3 y � �2x � 5

x

y

O

�

K

x

y

O

�

K

S

P Q

RD

E F

GA

B

C



1. O to MN�� 2. A to DC�� 3. T to VU��

Construct a line perpendicular to � through B. Then find the distance from B to �.

4. 5.


6. y � �x 7. y � 2x � 7 8. y � 3x � 12y � �x � 4 y � 2x � 3 y � 3x � 18

9. Graph the line y � �x � 1. Construct a perpendicular segment through the point at (�2, �3). Then find the distance from the point to the line.

10. CANOEING Bronson and a friend are going to carry a canoe across a flat field to thebank of a straight canal. Describe the shortest path they can use.

(–2, –3)

y � �x � 1

x

y

O

x

y

O

�

B

x

y

O

�

B

T

VW

S U

A B

CD

M N

O

Practice Perpendiculars and Distance

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6

Reading to Learn MathematicsPerpendiculars and Distance

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6


Less

on

3-6

Pre-Activity How does the distance between parallel lines relate to hanging new shelves?


Name three examples of situations in home construction where it would beimportant to construct parallel lines.

Reading the Lesson1. Fill in the blank with a word or phrase to complete each sentence.

a. The distance from a line to a point not on the line is the length of the segment

to the line from the point.

b. Two coplanar lines are parallel if they are everywhere .

c. In a plane, if two lines are both equidistant from a third line, then the two lines are

to each other.

d. The distance between two parallel lines measured along a perpendicular to the two

lines is always .

e. To measure the distance between two parallel lines, measure the distance between

one of the lines and any point on the .

2. On each figure, draw the segment that represents the distance indicated.

a. P to � b. D to AB��

c. E to FG�� d. U to RV��

Helping You Remember3. A good way to remember a new word is to relate it to words that use the same root. Use

your dictionary to find the meaning of the Latin root aequus. List three words other thanequal and equidistant that are derived from this root and give the meaning of each.

TV

R S

U

E

F G

A

D C

B

P

�


Parallelism in SpaceIn space geometry, the concept of parallelism must be extended to include two planes and a line and a plane.

Definition: Two planes are parallel if and only if they do not intersect.

Definition: A line and a plane are parallel if and only if they do not intersect.

Thus, in space, two lines can be intersecting, parallel, or skew while two planes or a line and a plane can only beintersecting or parallel. In the figure at the right, t � M,t � P, P || H, and � and n are skew.

The following five statements are theorems about parallel planes.

Theorem: Two planes perpendicular to the same line are parallel.Theorem: Two planes parallel to the same plane are parallel.Theorem: A line perpendicular to one of two parallel planes is

perpendicular to the other.Theorem: A plane perpendicular to one of two parallel planes is

perpendicular to the other.Theorem: If two parallel planes each intersect a third plane, then

the two lines of intersection are parallel.

Use the figure given above for Exercises 1–10. State yes or no to tell whether thestatement is true.

1. M � P 2. � � n 3. M � H 4. � � P 5. � � t

6. n � H 7. � � P 8. t � H 9. M � t 10. t � H

Make a small sketch to show that each statement is false.

11. If two lines are parallel to the same 12. If two planes are parallel, then any line plane, then the lines are parallel. in one plane is parallel to any line in the

other plane.

13. If two lines are parallel, then any plane 14. If two lines are parallel, then any plane containing one of the lines is parallel to containing one of the lines is parallel to any plane containing the other line. the other line.

t

� M

n P

H

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6

Chapter 3 Test, Form 133


Ass

essm

ents

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

For Questions 1–3, refer to the figure at the right.

1. Identify the plane parallel to plane BCD.A. plane ABE B. plane ABFC. plane AEF D. plane DEF

2. Identify a segment parallel to C�D�.A. A�B� B. A�E� C. B�C� D. E�F�

3. Which segment is skew to D�E�?A. A�B� B. B�C� C. B�D� D. C�D�


Identify the special name for each angle pair.

4. �1 and �8A. alternate exterior B. alternate interiorC. consecutive interior D. corresponding


6. Given a || b and m�2 � 65, find m�6.A. 25 B. 65 C. 115 D. 140

7. Given a || b and m�3 � 5x � 10 and m�5 � 3x � 10, find x.A. 110 B. 70 C. 20 D. 2.5


8. Which angle relationship justifies that � || m?A. �1 � �7 B. �3 � �4C. �4 � �5 D. �6 � �8

9. If m�2 � 6x � 8 and m�6 � 8x � 6, find x so that � || m .A. �7 B. 1 C. 7 D. 14

10. Given m�6 � m�7 � 180, which postulate or theorem justifies that � || m ?A. Consecutive Interior Angles TheoremB. Corresponding Angles PostulateC. Alternate Exterior Angles TheoremD. Alternate Interior Angles Theorem

m

�

p

12

34

65

78

b

a 1 23 4

657 8

B

A

DF

C

E

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 3 Test, Form 1 (continued)33

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

For Questions 11–12, determine the slope of the line that contains thegiven points.

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

12. F(�2, �4), G(1, 2)

A. �2 B. ��12� C. �

12� D. 2

13. What is the slope of a line parallel to the line containing (�6, 1) and (3, �2)?

A. �3 B. � �13� C. �

13� D. 3

14. Find the slope of the line perpendicular to the line containing (0, 0) and (�1, 4).

A. � �14� B. �4 C. �

14� D. 4

15. Which is an equation of the line with slope 4 and a y-intercept �3?

A. y � �3x � 4 B. y � �3x � �34� C. y � 4x � 3 D. y � 4x � �

34�

16. Which is an equation of the line with slope 2 that contains (3, 1)?A. y � 1 � 2(x � 3) B. y � 1 � 2(x � 3)C. y � 3 � 2(x � 1) D. y � 3 � (x � 2)

17. Yoga lessons cost $5 per lesson if Kylie enrolls in the health club for a fee of $120 per year. Suppose Kylie joins the health club. Which equationrepresents the yearly cost C of � yoga lessons?A. C � 5� B. C � 5� � 120C. C � 5� � 120 D. C � 5(� � 120)

18. What is the distance from P to n shown in the figure?A. �3B. 1C. 4D. 5

For Questions 19–20, find the distance between each pair of parallel lines.

19. y � 4 and y � 6A. 2 B. 4 C. 6 D. 10

20. y � x and y � x � 2A. 1 B. 1.5 C. �2� D. 2

Bonus What is the slope of a line perpendicular to y � �2?

P

x

y

O

n

B:

NAME DATE PERIOD

Chapter 3 Test, Form 2A33


Ass

essm

ents


For Questions 1 and 2, refer to the figure at the right.

1. Identify the plane parallel to plane PQT.A. plane PQS B. plane PTSC. plane RSV D. plane TUW

2. Which segment is skew to R�V�?A. R�S� B. R�Q� C. S�W� D. S�P�





5. Given p || q and m�3 � 75, find m�5.A. 15 B. 75 C. 105 D. 120

6. Given p || q and m�10 � 3x � 7 and m�13 � 4x � 9, find x.A. �2 B. 2 C. 16 D. 28

7. Given �1 � �5, which postulate or theorem justifies that p || q?A. Corresponding Angles PostulateB. Consecutive Interior Angles TheoremC. Alternate Exterior Angles TheoremD. Alternate Interior Angles Theorem

8. If �12 � �14, which postulate or theorem justifies that p || q ?A. Corresponding Angles PostulateB. Consecutive Interior Angles TheoremC. Alternate Exterior Angles TheoremD. Alternate Interior Angles Theorem

9. If p || q by the Consecutive Interior Angles Theorem, which angle pair must be supplementary?A. �3 and �10 B. �3 and �8 C. �8 and �13 D. �15 and �16

10. If m�4 � 7x � 20 and m�8 � 5x � 18, find x so that p || q.A. �19 B. �1 C. 1 D. 19

r

s

p q

1 234 65

78

9 1011

13 14151612

V

R S

W

U T

Q P 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 3 Test, Form 2A (continued)33

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.


11. P(�6, 3), Q(12, 9)

A. �3 B. ��13� C. �

13� D. 3

12. M(�8, 14), N(2, �11)

A. ��52� B. ��

25� C. �

25� D. �

52�

13. What is the slope of a line parallel to the line containing (�6, �6) and (9, 14)?

A. �34� B. �

43� C. �

130� D. undefined

14. Find the slope of a line perpendicular to the line containing (�8, 10) and (0, 9).

A. �8 B. ��18� C. �

18� D. 8

15. Which is an equation of the line with slope �12� that contains (�4, 7)?

A. y � 7 � �12�(x � 4) B. y � 7 � �

12�(x � 4)

C. y � 7 � �4x � �12� D. y � 7 � �

12�(x � 4)

16. Which is an equation of the line with x-intercept 2 and y-intercept 12?A. y � �6x � 12 B. y � 2x � 12 C. y � 6x � 12 D. y � 12x � 2

17. Which is an equation of the line containing (1, �3) and (7, 15)?A. y � �3x � 8 B. y � 3x C. y � 3x � 6 D. y � 3x � 10

18. Mr. Perugia gives 4 points per question for q questions on English quizzesplus 5 points for a bonus question. Which equation represents the total scoreT a student can receive on a quiz?A. T � 5 � 4q B. T � 4q � 5 C. T � 4(q � 5) D. 4T � q � 5

19. What is the distance from D to t shown in the figure?A. 2B. 3C. 5D. �5�

20. What is the distance between parallel lines whose equations are y � 2x � 7and y � 2x � 3?

A. �2� B. �5� C. 2�5� D. 4�2�

Bonus Suppose Ian reads at the rate of 15 pages an hour. Write an equation to represent the number of pages y Ian will still need to read after reading x hours in a 285-page novel. How long will it take Ian to read the entire novel?

D

x

y

O

t

B:

NAME DATE PERIOD

Chapter 3 Test, Form 2B33


Ass

essm

ents


For Questions 1 and 2, refer to the figure at the right.

1. Identify the plane parallel to plane ACE.A. plane ABG B. plane BCHC. plane EFK D. plane GJF

2. Which segment is skew to J�K�?A. A�B� B. C�D�C. J�H� D. G�F�





5. Given s || t and m�7 � 84, find m�10.A. 6 B. 84 C. 96 D. 104

6. Given s || t and m�1 � 8x � 4 and m�15 � 6x � 24, find x.A. �10 B. 14 C. 20 D. 28

7. Given �6 � �12, which postulate or theorem justifies that s || t?A. Corresponding Angles PostulateB. Consecutive Interior Angles TheoremC. Alternate Exterior Angles TheoremD. Alternate Interior Angles Theorem

8. If �8 � �16, which postulate or theorem justifies that s || t?A. Corresponding Angles PostulateB. Consecutive Interior Angles TheoremC. Alternate Exterior Angles TheoremD. Alternate Interior Angles Theorem

9. If s || t by the Alternate Exterior Angles Theorem, which angle pair must be congruent?A. �1 and �7 B. �1 and �5 C. �1 and �15 D. �1 and �13

10. If m�6 � 10x � 6 and m�11 � 4x � 18, find x so that s || t.A. 2 B. 4 C. 12 D. 32

a bs

t

1 2 3 46 578

9 10 111314151612

B

AE

F

K

J

C H

GD

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 3 Test, Form 2B (continued)33

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.


11. C(1, �7), D(5, 13)

A. �5 B. � �15� C. �

15� D. 5

12. Y(�7, �4), Z(5, 2)

A. �1 B. �12� C. 1 D. 2

13. What is the slope of a line parallel to the line containing (2, 5) and (6, �11)?

A. �13 B. �4 C. ��14� D. 4

14. Find the slope of a line perpendicular to the line containing (�2, �9) and (8, 6).

A. � �23� B. �

25� C. �

23� D. �

32�

15. Which is an equation of the line with slope of ��35� that contains (5, �5)?

A. y � 5 � ��35�(x � 5) B. y � 5 � ��

35�(x � 5)

C. y � 5 � ��35�(x � 5) D. y � 5 � ��

35�x � 5

16. Which is an equation of the line with x-intercept 18 and a y-intercept 3?

A. y � ��16�x � 3 B. y � �

16�x � 3 C. y � 3x � 18 D. y � 6x � 3

17. Which is an equation of the line containing (�3, 13) and (6, �5)?A. y � 2x � 7 B. y � �2x � 12 C. y � �2x � 17 D. y � �2x � 7

18. Mrs. Ekeledo writes computer manuals. She charges $125 to review writingspecifications plus $50 per hour h to write the manual. Which equationrepresents the total fee F that Mrs. Ekeledo earns for writing each computermanual?A. F � 50(h � 125) B. F � 50h � 125C. F � 125 � 50h D. 50F � h � 125

19. What is the distance from R to q shown in the figure?

A. �3�B. 3

C. 3�2�D. �8�

20. What is the distance between parallel lines whose equations are y � �x � 2and y � �x � 8?A. 3�2� B. 2�17� C. 10 D. 2�29�

Bonus Suppose line a is perpendicular to line b and line b is parallel to line c. What is the relationship between line aand line c?

R

x

y

O

q

B:

NAME DATE PERIOD

Chapter 3 Test, Form 2C33


Ass

essm

ents

For Questions 1 and 2, refer to the figure.

1. Identify the intersection of plane SVX and plane STU.

2. Name a segment skew to W�Y�.

For Questions 3–5, identify each pair of angles as alternate interior, alternateexterior, corresponding, or consecutiveinterior angles.

3. �2 and �12

4. �3 and �5

5. �7 and �15

6. Given m || n and m�8 � 86, find m�13.

7. Find x and y given m || n, m�4 � 6x � 5, m�10 � 5x � 8, andm�9 � 3y � 10.

Determine the slope of the line that contains the givenpoints.

8. V(�10, �4), W(5, 5)

9. A(�2, 9), C(2, �15)

10. G(�6, 14), L(�3, 9)

For Questions 11–13, determine whether CS�� and KP�� areparallel, perpendicular, or neither.

11. C(1, �12), S(5, 4), K(1, 9), P(6, �6)

12. C(�5, 6), S(�3, 2), K(�2, 10), P(1, 4)

13. C(�6, �7), S(�3, �5), K(3, 3), P(9, 7)

14. Printer’s Ink charges $1.18 per page p to copy a report plus$12 to bind it. Write an equation that represents the total costC to copy and bind a report. What would be the cost to copyand bind a 50-page report?

m

n

s t

1 234

6578

9 1011

13 141516

12

V

S T

U

Z

X W

Y

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

NAME DATE PERIOD

SCORE


Chapter 3 Test, Form 2C (continued)33

Write an equation in slope-intercept form for the line thatsatisfies the given conditions.

15. m � �9, y-intercept � 3

16. m � 3, contains (�1, 5)

17. x-intercept is 3, y-intercept is �1

18. contains (�7, 9) and (6, �4)

For Questions 19–21, given the following information, determine which lines, if any are parallel. State the postulate or theorem that justifies your answer.

19. �1 � �2

20. �DAB � �EBC

21. m�ADE � m�BED � 180

22. Find x so that a || b.

For Questions 23 and 24, find the distance between eachpair of parallel lines.

23. y � x � 6 and y � x � 8

24. y � �2x � 10 and y � �2x � 5

25. Construct a line perpendicular to � through B(�2, 5). Thenfind the distance from B to �.

Bonus Draw and label a figure you could use to prove thetheorem If two lines in a plane are cut by a transversal sothat a pair of consecutive interior angles is supplementary,then the lines are parallel. State the given and thestatement to be proved.

a

b

(9x � 44)�

(6x � 10)�

1

A

B

C F

E

D

2

NAME DATE PERIOD

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

1

43

ms

�

B

x

y

O

�

Chapter 3 Test, Form 2D33


Ass

essm

ents


1. Identify the intersection of plane ABD andplane CDE.

2. Name a segment skew to B�C�.

Identify each pair of angles as alternate interior, alternate exterior,corresponding, or consecutive interiorangles.

3. �5 and �13

4. �7 and �12

5. �9 and �16

6. Given a || b and m�6 � 89, find m�10.

7. Find x and y given a || b, m�8 � 4x � 10, m�12 � 7x � 17,and m�11 � 3y.


8. P(�5, 11), R(5, 7)

9. B(7, �1), G(14, 0)

10. U(�6, 9), V(�3, 8)

For Questions 11–13, determine whether BT�� and MV�� areparallel, perpendicular, or neither.

11. B(1, �4), T(5, 12), M(�8, 3), V(�4, 2)

12. B(�5, �7), T(10, 17), M(�5, �10), V(5, 6)

13. B(3, �5), T(5, �1), M(�2, 6), V(4, 3)

14. A job hotline service charges $25 plus a 4% commission c onthe first month of earnings if it finds work for a client. Writean equation that represents the total cost T to find a jobthrough the hotline. What is the cost if a client earns $1500the first month?

a b

d

e

12

34

56

78

910

1112

1314

1516

B A

DC

E

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

NAME DATE PERIOD

SCORE


Chapter 3 Test, Form 2D (continued)33

Write an equation in slope-intercept form for the line thatsatisfies the given conditions.

15. m � 7, y-intercept � �8

16. m � �4, contains (�4, 8)

17. x-intercept is �5, y-intercept is 2

18. containing (2, 5) and (7, �5)

Given the following information,determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer.

19. �QSR � �SUT

20. �1 � �2

21. m�RTU � m�TUS � 180

22. Find x so that p || q.


23. y � 3x � 1 and y � 3x � 11

24. y � �5x and y � �5x � 26

25. Construct a line perpendicular to � through Q(2, 3). Then findthe distance from Q to �.

Bonus Draw and label a figure you could use to prove the theoremIf two lines in a plane are cut by a transversal so that a pairof alternate interior angles is congruent, then the lines areparallel. State the given and the statement to be proved.

pq

(4x � 1)�

(12x � 11)�

1

P

R

TU

S

Q

2

NAME DATE PERIOD

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

1

32

mt

�

x

y

O

�

Q

Chapter 3 Test, Form 333


Ass

essm

ents

1. Draw a triangular prism and label the parallel planes ABCand DEF.

2. Identify two planes in your triangular prism that intersect.Name their intersection.

3. Name two lines in your triangular prism that are skew.

Identify each pair of angles as alternate interior, alternate exterior, corresponding,or consecutive interior angles.

4. �9 and �12

5. �2 and �3

6. �4 and �11

7. If m�9 � 110 and m�8 � 30,find m�6.

8. Find x, y, and z in the figure.


9. D(�6, �7), F(12, 23)

10. V(�2, �0.25), U(4, �2.5)

Determine whether QV�� and RM�� are parallel,perpendicular, or neither.

11. Q(�3, �8), V(5, 12), R(�2.5, 1), M(�5, 2)

12. Q(�2, 4.5), V(4, 9), R(�4, �12), M(10, �1.5)

(8x � 7)�

(3x � 11)�(2y � 23)� (3z2 � 5)�

(4y � 8)�

p

q

rst

1 234

56

78

9 101112

1314

f

gh

12

34

56

7

89

10

11

12

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

B

C

F

D

A

E

NAME DATE PERIOD

SCORE


Chapter 3 Test, Form 3 (continued)33

13. Write an equation of the line that is parallel to y � �85�x � 12

and contains (�6, �10).

14. Write an equation of the line that is perpendicular to 3x � 4y � 9 and contains (�4, �5).

15. A car rental company charges $30 per day plus $0.28 per mile mfor each mile over 100 miles to rent a car. Mr. Benitez rented acar for 5 days and drove 255 miles. Write an equation thatrepresents the total cost C to rent a car for 5 days. What wasMr. Benitez’s total cost?

Given the following information,determine which lines, if any areparallel. State the postulate or theorem that justifies your answer.

16. �RNS � �PSN

17. m�MRS � m�RSN � 180

18. Find x so that a || b.

19. Find the distance between parallel lines whose equations are

y � ��14�x � 2 and y � ��

14�x � �

94�.

20. Graph line m whose equation is �6x � 3y � 9. Construct aperpendicular line through P(3, 1). Then find the distancefrom P to m.

Bonus Suppose line p is perpendicular to line s and line q isperpendicular to line s. Are lines p and q parallel under allcircumstances? Draw a figure to illustrate your answer.

(6x � 13)�

(229 � 4x)�

ab

MN

P

S T

VU

R

NAME DATE PERIOD

13.

14.

15.

16.

17.

18.

19.

20.

B:

sp

q

x

y

OP

m

Chapter 3 Open-Ended Assessment33


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. TOWN PLANNING The lines in the diagram represent the intersection of streets near Todd’s home. ●●A represents the location of Todd’s home, and●●B represents the location of a library 1, 2, 3, 4, and 5 represent the anglesformed at street intersections.

a. Suppose you want to determine whether the streets represented bylines a, b, and c are parallel. What information would you need? Explainyour reasoning.

b. Suppose the streets represented by lines d and e are parallel. If themeasure of �5 is 112, find the measure of �4. Explain how youdetermined the measure.

c. If m�1 � 3x � 7 and m�4 � 2x � 20, find x so that lines a and c areparallel. Explain how you arrived at your answer and describe whythese measures allow you to determine that a and c are parallel.

d. Todd wants to go from the library to his home. If the streets representedby lines a and b are parallel, how could Todd determine the shortestdistance between ●●B and street b on which he lives? Give an explanationand illustrate your answer on a sketch of the above diagram.

2. Draw a line � on a coordinate grid so that the line contains (�3, �1) and(2, 4).

a. Write an equation of the line in slope-intercept form. Provide anexplanation for each of the steps.

b. If you construct a line parallel to �, what is the slope of the line? Howdo you know? Construct a line parallel to � that contains (1, 5).

c. Find the distance between the two lines. Explain how you determinedthe distance.

1 Ba

b

cd e f

2 34 5

A

NAME DATE PERIOD

SCORE


Chapter 3 Vocabulary Test/Review33

Write whether each statement is true or false. If false,replace the underlined word or number to make a truesentence.

For Questions 1–4, refer to the figure.

1. �4 and �5 are .

2. According to the , line ris parallel to line t given �3 � �8.

3. Given r || t, then �4 and �6 are supplementary.

4. Line p is a transversal since it intersects or more lines ina plane at different points.

5. When a linear equation is written in the form y � mx � b, mis the of the line and b is the y-intercept.

6. are located between the lines cut by a transversal.

7. If two lines do not intersect and are everywhere equidistant,the lines are .

8. The Perpendicular Transversal Theorem states that in aplane, if a line is perpendicular to one of two parallel lines,then it is to the other.

9. The equation y � 6 � ��58�(x � 2) is in .

10. The ratio of the rise to the run of a line is called its.

In your own words—

11. Describe what is meant by rate of change.

slope-intercept form

point-slope form

parallel

skew

Interior angles

slope

one

consecutive interior angles

Parallel Postulate

corresponding angles r

t

1 234

5687

p

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

alternate exterior anglesalternate interior anglesconsecutive interior anglescorresponding angles

equidistantnon-Euclidean geometryparallel linesparallel planes

plane Euclidean geometrypoint-slope formrate of changeskew lines

slopeslope-intercept formspherical geometrytransversal

NAME DATE PERIOD

SCORE

Chapter 3 Quiz (Lessons 3–1 and 3–2)

33


Ass

essm

ents

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.


1. Which plane is parallel to plane EGH?

2. Name the intersection of plane ABC and plane EFB.

For Questions 3–8, refer to the figure.

Identify each pair of angles as alternate interior, alternate exterior,corresponding, or consecutive interiorangles.

3. �2 and �10 4. �6 and �12

5. �1 and �5 6. �14 and �15

Given a || b and m�7 � 94, find the measure of thefollowing angles.

7. �10 8. �9

9. Find x and y in the figure.

10. STANDARDIZED TEST PRACTICEWhat is m�UVW?A. 39 B. 42C. 81 D. 138

138�V

U

W39�

(4y � 3)�(5x � 7)�

(3x � 17)�

a

b

1 2 3 4568

9 10 11 1213141516

7

dc

E

F

B A

K

J

HG

LMD

C

Chapter 3 Quiz (Lesson 3–3)

33

1.

2.

3.

4.

5.


1. E(�5, 6), Z(4, �3) 2. Y(�1, �12), P(3, 8)

3. B(�2, 7), N(8, �8) 4. R(�5, �7), Q(5, �5)

Determine whether DC�� and NM�� are parallel,perpendicular, or neither.

5. D(�4, �11), C(2, 7), N(�6, �1), M(3, �4)

NAME DATE PERIOD

SCORE


Chapter 3 Quiz (Lessons 3–4 and 3–5)

33

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

1. Write an equation in point-slope form of the line with slope ��13�

that contains (3, 8).

2. Write an equation in slope-intercept form of the line with slope �

53� and y-intercept of �2.

3. Write an equation in slope-intercept form of the line thatcontains (�1, 7) and (3, �9).

4. A bottled water company charges $8 per month for a watercooler and $5 per bottle b of water. Write an equation thatrepresents the total cost C for monthly water service.

Given the following information,determine which lines, if any, are parallel. State the postulate or theoremthat justifies your answer.

5. �1 � �6 6. �2 � �3

7. �4 � �9 8. m�7 � m�6 � 180

9. Given g || h and m�8 � 64, find m�5.

10. If m�2 � 5x � 17 and m�7 � 3x � 35, find x so that p || q.

12

3 45

67 89

g

h

jp q

NAME DATE PERIOD

SCORE

Chapter 3 Quiz (Lesson 3–6)

33

1.

2.

3.

4.

1. Draw the segment that represents the distance from B to DC��.

2. Construct a line perpendicular to � through H(�1, 3). Thenfind the distance from H to �.


3. y � �8 4. y � �x � 9y � 4 y � �x � 7

x

y

O

�H

BA

CD

NAME DATE PERIOD

SCORE

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

33


Ass

essm

ents


1. Which segment is skew to I�J�?A. G�H� B. A�J�C. H�I� D. A�B�

2. Which plane is parallel to plane CDF?A. BEF B. HIJC. ABE D. ABC

For Questions 3–5, refer to the figure. Identify the special name for each angle pair.

3. �2 and �4A. alternate exterior B. alternate interiorC. corresponding D. consecutive interior

4. �3 and �12A. alternate exterior B. alternate interiorC. corresponding D. consecutive interior

5. Given p || r and m�8 � 119, find m�11.A. 29 B. 61 C. 119 D. 151

p

qr

s

12

34

56

8

910

1112

7

B A

EFI J

C

HG

D

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

Part II

6. Find x and y in the figure.

For Questions 7 and 8, refer to the graph. Find the slope of each line.

7. k

8. m

Determine whether AB�� and DF�� are parallel, perpendicular,or neither.

9. A(�2, �11), B(6, 5), D(�7, �10), F(2, 8)

10. A(�1, 6), B(2, �9), D(�10, �1), F(5, 2)

x

y

O

k

m

(�2, 2)

(�4, �1) (0, �2)

(2, 4)

(5y � 11)�(4x � 6)�

(3x � 22)�

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


Chapter 3 Cumulative Review(Chapters 1–3)

33

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

1. Find y and BC if B is between A and C, AB � 6y, BC � 12y,and AB � 42. (Lesson 1-2)

2. What are the coordinates of D if E(�2, 2) is the midpoint ofD�F�, and F has coordinates (3, 5)? (Lesson 1-3)

For Questions 3 and 4, use the figure at the right. FA�� and FE�� are opposite rays.

3. Classify �AFB. (Lesson 1-4)

4. Find x and �BFC so that B�F� and F�D� areperpendicular. (Lesson 1-5)

5. Determine whether the conjecture is true or false. Give acounterexample for a false conjecture. (Lesson 2-1)

Given: � || mConjecture: The slopes of lines � and m are equal.

6. Identify the hypothesis and conclusion of the statementIf m�1 � m�2 � 180, then �1 and �2 are supplementary.(Lesson 2-3)

7. Determine whether statement (3) follows from statements (1)and (2) by the Law of Detachment or by the Law of Syllogism.Write invalid if neither law applies. (Lesson 2-4)

(1) If you swim, you must use earplugs.(2) Roy swims.(3) Roy must use earplugs.

8. Determine whether the statement If p || q and r ⊥ p, then r ⊥ qis always, sometimes, or never true. (Lesson 2-5)

9. If F(�1, 6), G(5, 3), J(2, �4), and K(6, �6), are FG�� and JK��

parallel, perpendicular, or neither? (Lesson 3-3)

10. Write an equation for a line in point-slope form with a slope of4 that contains the point at (2, 8). (Lesson 3-4)

11. Find the distance from A(�1, 5) to the line whose equation is4x � 5y � 12. (Lesson 3-6)

A

E

C

B

D

F(9x � 25)�

4x�

NAME DATE PERIOD

SCORE

Standardized Test Practice (Chapters 1–3)


For Questions 1–4, use the figure.

1. Points A, B, and D . (Lesson 1-1)

A. are collinear. B. are coplanar.C. lie on AD�� D. contain C.

2. Which segments are congruent? (Lesson 1-2)

E. A�B� and E�D� F. A�C� and E�D�G. A�C� and B�D� H. A�B� and B�D�

3. Find x. (Lesson 1-2)

A. 1 B. 1.5 C. 2 D. 2.7

4. Classify �C. (Lesson 1-4)

E. right angle F. acute angleG. obtuse angle H. straight angle

5. Which statement has the same truth value as 3 � 5? (Lesson 2-2)

A. 3 � x B. AB � 3C. AB � BC D. BC � 3 � x

6. The Law of Syllogism says that [(p → q) � (q → r)] → .(Lesson 2-4)

E. q F. r G. p → q H. p → r

7. State the property that justifies the statement If a � b � 25 and b � c, then a � c � 25. (Lesson 2-6)

A. Reflexive Property B. Symmetric PropertyC. Transitive Property D. Substitution Property

For Questions 8–10, use the figure.

8. What type of angles are �3 and �10? (Lesson 3-1)

E. alternate interior anglesF. alternate exterior anglesG. corresponding anglesH. consecutive interior angles

9. State the transversal that forms �11 and �13. (Lesson 3-1)

A. � B. m C. p D. q

10. If m�1 � 120, find m�8. (Lesson 3-2)

E. 60 F. 110 G. 120 H. 140

2 9 1012

131416

15

1143

mn p q

�

5 687

1

?

A B C

3 x

? 2.7 cm 2.7 cm

4.6 cm

3.5 cm (x � 2) cmC D

BA

E

NAME DATE PERIOD

SCORE 33

Part 1: Multiple Choice

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

1.

2.

3.

4.

5.

6.

7.

8.

9.

10. 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

E F G H

A B C D

Ass

essm

ents


Standardized Test Practice (continued)

11. Find DE. (Lesson 1-2)

12. The measure of the complement of �A is 185less than two times the measure of thesupplement of �A. Find m�A. (Lesson 1-5)

13. The table below shows the number of each typeof question that will be on four quarterlyhistory tests. If Monique prefers the ratio ofmultiple choice questions to essay questions tobe 4:1, which test will she prefer? (Lesson 2-4)

TestNumber of Number of Multiple

Essay Questions Choice Questions

1 4 26

2 8 22

3 6 24

4 5 25

D F

60 � 4a

E

3a � 5 13

NAME DATE PERIOD

33

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.

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

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87654321

87654321

14. Determine the number of line segments that can be drawn connecting this arrangement of points.(Lesson 2-5)

15. The distance between D(�2, �2) and E(a, �2) is 7. Find a.(Lesson 1-3)

16. The line that is perpendicular to y � �x and contains thepoint at (2, 4) is y � x � . (Lesson 3-4)

17. Find the slope of a line parallel to 3y � 6x � 9. (Lesson 3-4)

18. Find x so that � || m . (Lesson 3-5) mn

�(7x � 15)�

25�

?

14.

15.

16.

17.

18.

2 3 8 5

3

Unit 1 Review (Chapter 1–3)

33


Ass

essm

ents

1. Find JK. What is the precision for this measurement?

2. Find the distance between S(5, �7) and T(13, �9). Then findthe coordinates of the midpoint of S�T�.

3. In the figure, QP�� and QT�� are opposite rays. Find m�PQR, m�RQS, and m�SQT. Then classify each angle as right, acute, or obtuse.

4. Find the measures of two complementary angles if thedifference in the measures is 16.

5. Name the polygon by its number of sides. Then classify it as convex or concave and regular orirregular.

6. Lou is roping a boundary for an event at a local carnival. She has 144 feet of rope, and the eventsupervisor has instructed her to rope an area that is three times as long as it is wide. Find the length of each roped side.

7. Make a conjecture about the next letter in the sequence.L M N P Q R T …

8. Construct a truth table for �p � q.

9. Identify the hypothesis and conclusion of the statement. In aplane, if lines � and m are equidistant from line p, then � || m .

10. Cailyn knows that if two angles are vertical, they are congruent.She also knows that if two angles are congruent, then theyhave the same measure. She is given vertical angles 1 and 2,and she concludes that m�1 � m�2. What law of reasoningdoes she use?

11. Determine whether the following statement is always,sometimes, or never true. Points X, Y, and Z determine two lines.

3xx

6x�2x�

P TQ

RS

x�

H J K L

11–8 in. 35–

8 in.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Unit 1 Review (continued)33

For Questions 12–16, complete the proof.

Given: �1 and �2 are complementary.�3 and �4 are complementary.�1 � �3

Prove: �3 and �2 are complementary.

Statements Reasons1. �1 and �2 are complementary. 1.

�3 and �4 are complementary.�1 � �3

2. �2 � �4 2.3. m�2 � m�4 3.4. m�3 � m�4 � 90 4.5. m�3 � m�2 � 90 5.6. �3 and �2 are complementary 6. Def. of comp. �

For Questions 17 and 18, refer to the figure.17. Name the transversal that forms

�3 and �6. Then identify the specialname for the angle pair.

18. If p || q, m�1 � 5b � 23, and m�11 � 2b � 10, find m�2,m�4, m�10, and m�12.

19. Determine whether QR�� and ST�� are parallel, perpendicular, orneither for Q(�4, �4), R(5, 2), S(4, �5), and T(0, 1).

20. Graph the line that contains C(3, �1) and has a slope of �2.Then write an equation for the line in slope-intercept form.

21. Find x so that k || �.

22. Find the distance between two lines that have equations y � 3x � 1 and y � 3x � 19.

�

k(3x � 11)�

(4x � 1)�

m

q

�p 1 2

39 10

11 12

45 67 8

(Question 16)(Question 15)(Question 14)(Question 13)

(Question 12)

1 42 3

NAME DATE PERIOD

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

Standardized Test PracticeStudent Record Sheet (Use with pages 172–173 of the Student Edition.)

33

© Glencoe/McGraw-Hill A1 Glencoe Geometry

An

swer

s

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

1 4 7

2 5 8

3 6 9 DCBADCBADCBA

DCBADCBADCBA

DCBADCBADCBA

NAME DATE PERIOD

Part 1 Multiple ChoicePart 1 Multiple Choice

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

Part 3 Open-EndedPart 3 Open-Ended

Solve the problem and write your answer in the blank.

For Questions 11, 12, and 13, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.

10 11 12 13

11 (grid in)

12 (grid in)

13 (grid in)

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

Record your answers for Questions 14–15 on the back of this paper.


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3-1

3-1

Exam

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Exam

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Exer

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Exer

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Answers (Lesson 3-1)


An

swer

s

Skil

ls P

ract

ice

Par

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l Lin

es a

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ls

NA

ME

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3-1

3-1

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Lesson 3-1

For

Exe

rcis

es 1

–4,r

efer

to

the

figu

re a

t th

e ri

ght.

1.N

ame

all

plan

es t

hat

are

par

alle

l to

pla

ne

DE

H.

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G

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ts o

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nes

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h g

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lin

e is

a

tran

sver

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san

d t

,san

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,san

d z

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d z

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ran

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d z

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d z

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ran

d s

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d z

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tify

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air

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es a

s a

lter

na

te i

nte

rior

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ern

ate

ex

teri

or,c

orre

spon

din

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teri

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din

g

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ing

k;

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r

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sver

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and

g,f

and

h,f

and

i,g

and

h,g

and

i,h

and

i

6.h e

and

f,e

and

g,e

and

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and

g,f

and

i,g

and

i

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tify

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air

of a

ngl

es a

s a

lter

na

te i

nte

rior

,alt

ern

ate

ex

teri

or,c

orre

spon

din

g,or

con

secu

tive

in

teri

oran

gles

.

7.�

9 an

d �

138.

�6

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�16

corr

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on

din

gal

tern

ate

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e th

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the

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3-1

3-1



Readin

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3-1

3-1

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Lesson 3-1

Pre-

Act

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ow a

re p

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lin

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pla

nes

use

d i

n a

rch

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ture

?

Rea

d th

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trod

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to

Les

son

3-1

at

the

top

of p

age

126

in y

our

text

book

.

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ive

an e

xam

ple

of p

aral

lel

lin

es t

hat

can

be

fou

nd

in y

our

clas

sroo

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Sam

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ges

of

flo

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vert

ical

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r•

Giv

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e of

par

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l pl

anes

th

at c

an b

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un

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r cl

assr

oom

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amp

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nsw

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ceili

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Less

on

1.W

rite

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eom

etri

cal

term

th

at m

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ach

def

init

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.

a.tw

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lin

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at i

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or m

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lin

es i

n a

pla

ne

at d

iffe

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sver

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pair

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angl

es d

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ed b

y tw

o li

nes

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d a

tran

sver

sal

con

sist

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n i

nte

rior

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e an

d an

ext

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gle

that

hav

e di

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ent

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ices

an

d th

at l

ie o

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sam

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th

e tr

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nd

ing

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gle

s

2.R

efer

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the

figu

re a

t th

e ri

ght.

Giv

e th

e sp

ecia

l n

ame

for

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gle

pair

.

a.�

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mat

hem

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to

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3.

4.T

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ings

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the

cube

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ind

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ple

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per

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tive

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a n

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ME

____

____

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ER

IOD

____

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3-1

3-1



An

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s

Stu

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nte

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An

gle

s an

d P

aral

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3-2

3-2

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Lesson 3-2

Para

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an

d A

ng

le P

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en t

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para

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es a

re c

ut

by a

tra

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e fo

llow

ing

pair

s of

an

gles

are

con

gru

ent.

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rres

pon

din

g an

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d t

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ind

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Alg

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nd

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n v

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es i

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ed b

y a

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sver

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llel

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�15

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2 �

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d m

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fin

d x

and

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p||q

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m�

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rres

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uid

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)

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gle

s an

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ines

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3-2

Exam

ple

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m�

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y

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cises

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d x

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re.

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z �

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) �( 5y

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) �( 6

y �

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s an

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Lesson 3-2

In t

he

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�70

.Fin

d t

he

mea

sure

of

each

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gle.

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ind

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e m

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an

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ind

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e m

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d m

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ure

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21.

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In t

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d m

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ure

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9.10

.

130

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11.P

RO

OF

Wri

te a

par

agra

ph p

roof

of T

heo

rem

3.3

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iven

: �|| m

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ve:�

1 �

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Sam

ple

pro

of:

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giv

en t

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�|| m

,so

�1

��

8 by

th

e A

lter

nat

eE

xter

ior

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gle

s T

heo

rem

.Sin

ce it

is g

iven

th

at m

|| n,

�8

��

12 b

y th

e C

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esp

on

din

g A

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re,�

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on

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ENC

ING

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tren

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e fe

nce

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d pr

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rom

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akes

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ind

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3-2

3-2



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csA

ng

les

and

Par

alle

l Lin

es

NA

ME

____

____

____

____

____

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3-2

3-2

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Lesson 3-2

Pre-

Act

ivit

yH

ow c

an a

ngl

es a

nd

lin

es b

e u

sed

in

art

?

Rea

d th

e in

trod

uct

ion

to

Les

son

3-2

at

the

top

of p

age

133

in y

our

text

book

.•

You

r te

xtbo

ok s

how

s a

pain

tin

g th

at c

onta

ins

two

para

llel

lin

es a

nd

atr

ansv

ersa

l.W

hat

is t

he n

ame

for

�1

and

�2?

corr

esp

on

din

g a

ng

les

•W

hat

is

the

rela

tion

ship

bet

wee

n t

hes

e tw

o an

gles

?T

hey

are

co

ng

ruen

t.

Rea

din

g t

he

Less

on

1.C

hoo

se t

he

corr

ect

wor

d to

com

plet

e ea

ch s

ente

nce

.

a.If

tw

o pa

rall

el l

ines

are

cu

t by

a t

ran

sver

sal,

then

alt

ern

ate

exte

rior

an

gles

are

(con

gru

ent/

com

plem

enta

ry/s

upp

lem

enta

ry).

b.

If t

wo

para

llel

lin

es a

re c

ut

by a

tra

nsv

ersa

l,th

en c

orre

spon

din

g an

gles

are

(con

gru

ent/

com

plem

enta

ry/s

upp

lem

enta

ry).

c.If

par

alle

l li

nes

are

cu

t by

a t

ran

sver

sal,

then

con

secu

tive

in

teri

or a

ngl

es a

re

(con

gru

ent/

com

plem

enta

ry/s

upp

lem

enta

ry).

d.

In a

pla

ne,

if a

lin

e is

per

pen

dicu

lar

to o

ne

of t

wo

para

llel

lin

es,t

hen

it

is

(par

alle

l/per

pen

dicu

lar/

skew

) to

th

e ot

her

.

Use

th

e fi

gure

for

Exe

rcis

es 2

an

d 3

.

2.a.

Nam

e fo

ur

pair

s of

ver

tica

l an

gles

.�

1 an

d �

3,�

2 an

d �

4,�

5 an

d �

7,�

6 an

d �

8b

.N

ame

all

angl

es t

hat

for

m a

lin

ear

pair

wit

h �

7.�

6,�

8c.

Nam

e al

l an

gles

th

at a

re c

ongr

uen

t to

�1.

�3,

�6,

�8

d.

Nam

e al

l an

gles

th

at a

re c

ongr

uen

t to

�4.

�2,

�5,

�7

e.N

ame

all

angl

es t

hat

are

su

pple

men

tary

to

�3.

�2,

�4,

�5,

�7

f.N

ame

all

angl

es t

hat

are

su

pple

men

tary

to

�2.

�1,

�3,

�6,

�8

3.W

hic

h c

oncl

usi

on(s

) co

uld

you

mak

e ab

out

lin

es u

and

vif

m�

4 �

m�

1?B

,DA

.t

|| uB

.t

⊥u

C.

v⊥

uD

.v

⊥t

E.

v|| t

Hel

pin

g Y

ou

Rem

emb

er4.

How

can

you

use

an

eve

ryda

y m

ean

ing

of t

he

adje

ctiv

e al

tern

ate

to h

elp

you

rem

embe

rth

e ty

pes

of a

ngl

e pa

irs

for

two

lin

es a

nd

a tr

ansv

ersa

l?

Sam

ple

an

swer

:O

ne

mea

nin

g o

f al

tern

ate

is “

ob

tain

ed b

y sw

itch

ing

bac

kan

d f

ort

h f

rom

on

e th

ing

to

an

oth

er.”

Th

e an

gle

pai

rs in

th

is le

sso

n a

llu

se a

ng

les

wit

h d

iffe

ren

t ve

rtic

es,a

nd

th

ose

wh

ose

nam

es c

on

tain

th

ead

ject

ive

alte

rnat

eca

n b

e lo

cate

d in

a f

igu

re b

y sw

itch

ing

fro

m o

ne

sid

eo

f th

e tr

ansv

ersa

l to

th

e o

ther

.Th

e p

airs

wh

ose

nam

es d

o n

ot

incl

ud

e th

ew

ord

alt

ern

ate

are

fou

nd

on

th

e sa

me

sid

e o

f th

e tr

ansv

ersa

l.

t

u

v

1 234

6578

per

pen

dic

ula

r

sup

ple

men

tary

con

gru

ent

con

gru

ent

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Mo

re O

pti

cal I

llusi

on

sIn

dra

win

gs,d

iago

nal

lin

es m

ay c

reat

e th

e il

lusi

on o

f de

pth

.F

or e

xam

ple,

the

figu

re a

t th

e ri

ght

can

be

thou

ght

of a

s pi

ctu

rin

g a

flat

fig

ure

or

a cu

be.T

he

opti

cal

illu

sion

s on

th

is

page

in

volv

e de

pth

per

cept

ion

.

An

swer

eac

h q

ues

tion

.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-2

3-2

1.H

ow m

any

cube

s do

you

see

in

th

edr

awin

g?5

or

6

3.D

oes

the

draw

ing

show

a v

iew

fro

m

the

top

or t

he

bott

om o

f th

e st

airs

?

An

swer

s m

ay v

ary.

2.C

an t

his

fig

ure

sh

ow a

n a

ctu

al

obje

ct?

no

4.W

hic

h l

ine

segm

ent

is l

onge

r,A�

B�or

C �D�

? M

easu

re t

o ch

eck

you

r an

swer

.

A�B�

A

BC

D

5.W

hic

h p

erso

n i

n t

he

draw

ing

at t

he

righ

t ap

pear

s to

be

tall

est?

Mea

sure

to

chec

k yo

ur

answ

er.

Th

e p

erso

n a

t th

e ri

gh

t ap

pea

rs t

alle

st,

but

all a

re t

he

sam

e si

ze.

6.D

raw

tw

o m

ore

obje

cts

the

sam

e si

ze o

n t

he

figu

re

at t

he

righ

t.D

oes

one

appe

ar l

arge

r th

an t

he

oth

er?

An

swer

s w

ill v

ary.



Stu

dy G

uid

e a

nd I

nte

rven

tion

Slo

pes

of

Lin

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-3

3-3

©G

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lenc

oe G

eom

etry

Lesson 3-3

Slo

pe

of

a Li

ne

Th

e sl

ope

mof

a l

ine

con

tain

ing

two

poin

ts w

ith

coo

rdin

ates

(x 1

,y1)

and

(x2,

y 2)

is g

iven

by

the

form

ula

m�

�y x2 2

� �

y x1 1�

,wh

ere

x 1�

x 2.

Fin

d t

he

slop

e of

eac

h l

ine.

For

lin

e p,

let

(x1,

y 1)

be (

1,2)

an

d (x

2,y 2

) be

(�

2,�

2).

m�

�y x2 2

� �

y x1 1�

��

2 2� �

2 1�

or �4 3�

For

lin

e q,

let

(x1,

y 1)

be (

2,0)

an

d (x

2,y 2

) be

(�

3,2)

.

m�

�y x2 2

� �

y x1 1�

�� 2 3� �

0 2�

or �

�2 5�

Det

erm

ine

the

slop

e of

th

e li

ne

that

con

tain

s th

e gi

ven

poi

nts

.

1.J

(0,0

),K

(�2,

8)�

42.

R(�

2,�

3),S

(3,�

5)�

�2 5�

3.L

(1,�

2),N

(�6,

3)�

�5 7�4.

P(�

1,2)

,Q(�

9,6)

��1 2�

5.T

(1,�

2),U

(6,�

2)0

6.V

(�2,

10),

W(�

4,�

3)�1 23 �

Fin

d t

he

slop

e of

eac

h l

ine.

7.A

B��

�3

8.C

D��

��

2

9.E

M��

�u

nd

efin

ed10

.AE

��

0

11.E

H��

��2 5�

12.B

M��

��

�1 2�

x

y

O

C( –

2, 2

)

A( –

2, –

2) H

( –1,

–4)

B( 0

, 4)

M( 4

, 2)

E( 4

, –2)

D( 0

, –2)

x

y

O

( 1, 2

)

( –2,

–2)( –

3, 2

)

( 2, 0

)

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill13

8G

lenc

oe G

eom

etry

Para

llel a

nd

Per

pen

dic

ula

r Li

nes

If y

ou e

xam

ine

the

slop

es o

f pa

irs

of p

aral

lel

lin

esan

d th

e sl

opes

of

pair

s of

per

pen

dicu

lar

lin

es,w

her

e n

eith

er l

ine

in e

ach

pai

r is

ver

tica

l,yo

uw

ill

disc

over

th

e fo

llow

ing

prop

erti

es.

Tw

o li

nes

hav

e th

e sa

me

slop

e if

an

d on

ly i

f th

ey a

re p

aral

lel.

Tw

o li

nes

are

per

pen

dicu

lar

if a

nd

only

if

the

prod

uct

of

thei

r sl

opes

is

�1.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Slo

pes

of

Lin

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-3

3-3

Fin

d t

he

slop

e of

a l

ine

par

alle

l to

th

e li

ne

con

tain

ing

A(�

3,4)

and

B(2

,5).

Fin

d th

e sl

ope

of A

B� �

� .U

se (

�3,

4) f

or (

x 1,y

1)an

d u

se (

2,5)

for

(x 2

,y2)

.

m�

�y x2 2

� �

y x1 1�

�� 2

5 �

� (�4 3)

�or

�1 5�

Th

e sl

ope

of a

ny

lin

e pa

rall

el t

o A

B��

�m

ust

be

�1 5� .

Fin

d t

he

slop

e of

a l

ine

per

pen

dic

ula

r to

PQ

� ��

for

P(�

2,�

4) a

nd

Q(4

,3).

Fin

d th

e sl

ope

of P

Q� �

� .U

se (

�2,

�4)

for

(x

1,y 1

) an

d u

se (

4,3)

for

(x 2

,y2)

.

m�

�y x2 2

� �

y x1 1�

��3 4

� �

( (� �

4 2) )�

or �7 6�

Sin

ce �7 6�

��

�6 7� ��

1,th

e sl

ope

of a

ny

lin

e

perp

endi

cula

r to

PQ

�� m

ust

be

��6 7� .

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Det

erm

ine

wh

eth

er M

N��

�an

d R

S��

� are

pa

rall

el,p

erp

end

icu

lar,

or n

eith

er.

1.M

(0,3

),N

(2,4

),R

(2,1

),S

(8,4

)2.

M(�

1,3)

,N(0

,5),

R(2

,1),

S(6

,�1)

par

alle

lp

erp

end

icu

lar

3.M

(�1,

3),N

(4,4

),R

(3,1

),S

(�2,

2)4.

M(0

,�3)

,N(�

2,�

7),R

(2,1

),S

(0,�

3)

nei

ther

par

alle

l

5.M

(�2,

2),N

(1,�

3),R

(�2,

1),S

(3,4

)6.

M(0

,0),

N(2

,4),

R(2

,1),

S(8

,4)

per

pen

dic

ula

rn

eith

er

Fin

d t

he

slop

e of

MN

��

and

th

e sl

ope

of a

ny

lin

e p

erp

end

icu

lar

to M

N��

� .

7.M

(2,�

4),N

(�2,

�1)

8.M

(1,3

),N

(�1,

5)

��3 4� ;

�4 3��

1;1

9.M

(4,�

2),N

(5,3

)10

.M(2

,�3)

,N(�

4,1)

5;�

�1 5��

�2 3� ;�3 2�



An

swer

s

Skil

ls P

ract

ice

Slo

pes

of

Lin

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-3

3-3

©G

lenc

oe/M

cGra

w-H

ill13

9G

lenc

oe G

eom

etry

Lesson 3-3

Det

erm

ine

the

slop

e of

th

e li

ne

that

con

tain

s th

e gi

ven

poi

nts

.

1.S

(�1,

2),W

(0,4

)2

2.G

(�2,

5),H

(1,�

7)�

4

3.C

(0,1

),D

(3,3

)�2 3�

4.J

(�5,

�2)

,K(5

,�4)

��1 5�

Fin

d t

he

slop

e of

eac

h l

ine.

5.N

P��

�6.

TW

��

�3 4��

2

7.a

lin

e pa

rall

el t

o T

W��

�8.

a li

ne

perp

endi

cula

r to

NP

��

�2

��4 3�

Det

erm

ine

wh

eth

er A

B��

�an

d M

N��

�ar

e p

ara

llel

,per

pen

dic

ula

r,or

nei

ther

.

9.A

(0,3

),B

(5,�

7),M

(�6,

7),N

(�2,

�1)

10.A

(�1,

4),B

(2,�

5),M

(�3,

2),N

(3,0

)p

aral

lel

nei

ther

11.A

(�2,

�7)

,B(4

,2),

M(�

2,0)

,N(2

,6)

12.A

(�4,

�8)

,B(4

,�6)

,M(�

3,5)

,N(�

1,�

3)p

aral

lel

per

pen

dic

ula

r

Gra

ph

th

e li

ne

that

sat

isfi

es e

ach

con

dit

ion

.

13.s

lope

�3,

con

tain

s A

(0,1

)14

.slo

pe �

��3 2� ,

con

tain

s R

(�4,

5)

15.c

onta

ins

Y(3

,0),

para

llel

to

DJ

��

16.c

onta

ins

T(0

,�2)

,per

pen

dicu

lar

to C

X��

�

wit

h D

(�3,

1) a

nd

J(3

,3)

wit

h C

(0,3

) an

d X

(2,�

1)

T( 0

, –2)

C( 0

, 3)

X( 2

, –1)x

y OD

( –3,

1)

J(3,

3)

Y( 3

, 0)

x

y O

R( –

4, 5

)

x

y O

A( 0

, 1)

x

y O

x

y ON

TW

P

©G

lenc

oe/M

cGra

w-H

ill14

0G

lenc

oe G

eom

etry

Det

erm

ine

the

slop

e of

th

e li

ne

that

con

tain

s th

e gi

ven

poi

nts

.

1.B

(�4,

4),R

(0,2

)�

�1 2�2.

I(�

2,�

9),P

(2,4

)�1 43 �

Fin

d t

he

slop

e of

eac

h l

ine.

3.L

M��

�4.

GR

��

�2 3��

�2 5�

5.a

lin

e pa

rall

el t

o G

R��

�6.

a li

ne

perp

endi

cula

r to

PS

��

��2 5�

��1 2�

Det

erm

ine

wh

eth

er K

M��

�an

d S

T��

�ar

e p

ara

llel

,per

pen

dic

ula

r,or

nei

ther

.

7.K

(�1,

�8)

,M(1

,6),

S(�

2,�

6),T

(2,1

0)8.

K(�

5,�

2),M

(5,4

),S

(�3,

6),T

(3,�

4)n

eith

erp

erp

end

icu

lar

9.K

(�4,

10),

M(2

,�8)

,S(1

,2),

T(4

,�7)

10.K

(�3,

�7)

,M(3

,�3)

,S(0

,4),

T(6

,�5)

par

alle

lp

erp

end

icu

lar

Gra

ph

th

e li

ne

that

sat

isfi

es e

ach

con

dit

ion

.

11.s

lope

��

�1 2� ,co

nta

ins

U(2

,�2)

12.s

lope

��4 3� ,

con

tain

s P

(�3,

�3)

13.c

onta

ins

B(�

4,2)

,par

alle

l to

FG

��

14.c

onta

ins

Z(�

3,0)

,per

pen

dicu

lar

to E

K��

�

wit

h F

(0,�

3) a

nd

G(4

,�2)

wit

h E

(�2,

4) a

nd

K(2

,�2)

15. P

RO

FITS

Aft

er T

ake

Tw

o be

gan

ren

tin

g D

VD

s at

th

eir

vide

o st

ore,

busi

nes

s so

ared

.B

etw

een

200

0 an

d 20

03,p

rofi

ts i

ncr

ease

d at

an

ave

rage

rat

e of

$12

,000

per

yea

r.T

otal

prof

its

in 2

003

wer

e $4

6,00

0.If

pro

fits

con

tin

ue

to i

ncr

ease

at

the

sam

e ra

te,w

hat

wil

lth

e to

tal

prof

it b

e in

200

9?$1

18,0

00

K( 2

, –2)

E( –

2, 4

)

Z( –

3, 0

)

x

y

O

F( 0

, –3)

B( –

4, 2

)

G( 4

, –2)

x

y

O

P( –

3, –

3)

x

y OU

( 2, –

2)x

y O

x

y O

R

G

S

P

M

L

Pra

ctic

e (

Ave

rag

e)

Slo

pes

of

Lin

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-3

3-3



Readin

g t

o L

earn

Math

em

ati

csS

lop

es o

f L

ines

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-3

3-3

©G

lenc

oe/M

cGra

w-H

ill14

1G

lenc

oe G

eom

etry

Lesson 3-3

Pre-

Act

ivit

yH

ow i

s sl

ope

use

d i

n t

ran

spor

tati

on?

Rea

d th

e in

trod

uct

ion

to

Les

son

3-3

at

the

top

of p

age

139

in y

our

text

book

.•

If y

ou a

re d

rivi

ng

uph

ill

on a

roa

d w

ith

a 4

% g

rade

,how

man

y fe

et w

ill

the

road

ris

e fo

r ev

ery

1000

hor

izon

tal

feet

tra

vele

d? 4

0 ft

•If

you

are

dri

vin

g do

wn

hil

l on

a r

oad

wit

h a

7%

gra

de,h

ow m

any

met

ers

wil

l th

e ro

ad f

all

for

ever

y 50

0 m

eter

s tr

avel

ed?

35 m

Rea

din

g t

he

Less

on

1.W

hic

h e

xpre

ssio

ns

can

be

use

d to

rep

rese

nt

the

slop

e of

th

e li

ne

con

tain

ing

poin

ts (

x 1,y

1)an

d (x

2,y 2

)? A

ssu

me

that

no

den

omin

ator

is

zero

.A

,C,F

A.

��

y x�B

.�h vor ei rz to icn at la rl ir su en

�C

.�y x2 2

� �

y x1 1�

D.�c ch h

a an ng ge e

i in nx y

�

E.�y x2 1

� �

y x1 2�

F.�y x1 1

� �

y x2 2�

G.

�x y2 2

� �

x y1 1�

H.�y y2 1

� �

x x2 1�

2.M

atch

th

e de

scri

ptio

n o

f a

lin

e fr

om t

he

firs

t co

lum

n w

ith

th

e de

scri

ptio

n o

f it

s sl

ope

from

th

e se

con

d co

lum

n.

Typ

e of

Lin

eS

lop

e

a.a

hor

izon

tal

lin

eii

i.a

neg

ativ

e n

um

ber

b.

a li

ne

that

ris

es f

rom

lef

t to

rig

ht

ivii

.0

c.a

vert

ical

lin

eiii

iii.

un

defi

ned

d.

a li

ne

that

fal

ls f

rom

lef

t to

rig

ht

iiv

.a

posi

tive

nu

mbe

r

3.F

ind

the

slop

e of

eac

h l

ine.

a.a

lin

e pa

rall

el t

o a

lin

e w

ith

slo

pe �3 4�

�3 4�

b.

a li

ne

perp

endi

cula

r to

th

e x-

axis

un

def

ined

slo

pe

c.a

lin

e pe

rpen

dicu

lar

to a

lin

e w

ith

slo

pe 5

��1 5�

d.

a li

ne

para

llel

to

the

x-ax

is0

e.y-

axis

un

def

ined

slo

pe

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

you

r fr

ien

dth

inks

th

at p

erpe

ndi

cula

r li

nes

(if

nei

ther

lin

e is

ver

tica

l) h

ave

slop

es t

hat

are

reci

proc

als

of e

ach

oth

er.H

ow c

ould

you

exp

lain

to

you

r fr

ien

d th

at t

his

is

inco

rrec

t an

dgi

ve h

er a

goo

d w

ay t

o re

mem

ber

the

corr

ect

rela

tion

ship

?

Sam

ple

an

swer

:In

ord

er f

or

two

lin

es (

nei

ther

on

e ve

rtic

al)

to m

eet

atri

gh

t an

gle

s,o

ne

mu

st g

o u

pw

ard

fro

m le

ft t

o r

igh

t an

d t

he

oth

er m

ust

go

do

wn

war

d,s

o t

hei

r sl

op

es m

ust

hav

e o

pp

osi

te s

ign

s.R

ecip

roca

ls h

ave

the

sam

e si

gn

.Th

e p

rod

uct

of

the

slo

pes

mu

st b

e �

1,n

ot

1.R

emem

ber

:T

he

slo

pes

are

op

po

site

reci

pro

cals

.

©G

lenc

oe/M

cGra

w-H

ill14

2G

lenc

oe G

eom

etry

Slo

pes

an

d P

oly

go

ns

In c

oord

inat

e ge

omet

ry,t

he

slop

es o

f tw

o li

nes

det

erm

ine

if t

he

lin

es a

repa

rall

el o

r pe

rpen

dicu

lar.

Th

is k

now

ledg

e ca

n b

e u

sefu

l w

hen

wor

kin

g w

ith

poly

gon

s.

1.T

he

coor

din

ates

of

the

vert

ices

of

a tr

ian

gle

are

A(�

6,4)

,B(8

,6),

and

C(4

,�4)

.Gra

ph �

AB

C.

2.J,

K,a

nd

Lar

e m

idpo

ints

of

A�B�

,B�C�

,an

d A�

C�,

resp

ecti

vely

.Fin

d th

e co

ordi

nat

es o

f J,

K,a

nd

L.

Dra

w �

JK

L.

J(1,

5),K

(6,1

),L

(�1,

0)

3.W

hic

h s

egm

ents

app

ear

to b

e pa

rall

el?

A�B�

and

L�K�

,B�C�

and

J�L�,

A�C�

and

J�K�

4.S

how

th

at t

he

segm

ents

nam

ed i

n E

xerc

ise

3 ar

e pa

rall

el b

y fi

ndi

ng

the

slop

es o

f al

l si

x se

gmen

ts.

A�B�

:�1 7� ;

L�K�:

�1 7� ;B�

C�:�

5 2� ;J�L�

:�5 2� ;

A�C�

:�

�4 5� ;J�K�

:��4 5�

Th

e co

ord

inat

es o

f th

e ve

rtic

es o

f ri

ght

�P

QR

are

give

n.F

ind

th

e sl

ope

of e

ach

sid

e of

th

e tr

ian

gle.

Th

en n

ame

the

hyp

oten

use

.

5.P

(5,1

)Q

(1,�

1)R

(�2,

5)6.

P(�

2,�

3)Q

(5,1

)R

(2,3

)

slop

e of

P�Q�

��1 2�

slop

e of

P�Q�

��4 7�

slop

e of

Q�R�

��

2sl

ope

of Q�

R��

��2 3�

slop

e of

P�R�

��

�4 7�sl

ope

of P�

R��

�3 2�

hyp

oten

use

:P�R�

hyp

oten

use

:P�Q�

Th

e co

ord

inat

es o

f q

uad

rila

tera

l P

QR

Sar

e gi

ven

.Gra

ph

qu

adri

late

ral

PQ

RS

and

fin

d t

he

slop

es o

f th

e d

iago

nal

s.S

tate

wh

eth

er t

he

dia

gon

als

are

per

pen

dic

ula

r.

7.P

(�2,

6),Q

(4,0

),R

(1,�

4),S

(�5,

2)8.

P(0

,6),

Q(3

,0),

R(�

4,�

2),S

(�5,

4)

P�R�

:�

�1 30 �;

S�Q�

:�

�2 9� ;n

oP�

R�:

2;S�

Q�:

��1 2� ;

yes

S

P

Q

R

x

y

O

S

P

Q

R

x

y

O

A

B

C

K

J

Lx

y

O

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-3

3-3



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Eq

uat

ion

s o

f L

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____

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3-4

3-4

©G

lenc

oe/M

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w-H

ill14

3G

lenc

oe G

eom

etry

Lesson 3-4

Wri

te E

qu

atio

ns

of

Lin

esYo

u c

an w

rite

an

equ

atio

n o

f a

lin

e if

you

are

giv

en a

ny

ofth

e fo

llow

ing:

•th

e sl

ope

and

the

y-in

terc

ept,

•th

e sl

ope

and

the

coor

din

ates

of

a po

int

on t

he

lin

e,or

•th

e co

ordi

nat

es o

f tw

o po

ints

on

th

e li

ne.

If m

is t

he

slop

e of

a l

ine,

bis

its

y-i

nte

rcep

t,an

d (x

1,y 1

) is

a p

oin

t on

th

e li

ne,

then

:•

the

slop

e-in

terc

ept

form

of t

he

equ

atio

n i

s y

�m

x�

b,•

the

poi

nt-

slop

e fo

rmof

th

e eq

uat

ion

is

y�

y 1�

m(x

�x 1

).

Wri

te a

n e

qu

atio

n i

nsl

ope-

inte

rcep

t fo

rm o

f th

e li

ne

wit

hsl

ope

�2

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y-i

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y�

mx

�b

Slo

pe-in

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�4

m�

�2,

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4

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e sl

ope-

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rm o

f th

e eq

uat

ion

of

the

lin

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y�

�2x

�4.

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te a

n e

qu

atio

n i

np

oin

t-sl

ope

form

of

the

lin

e w

ith

slo

pe

��3 4�

that

con

tain

s (8

,1).

y�

y 1�

m(x

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orm

y�

1 �

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x�

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��

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x 1,

y 1)

�(8

, 1)

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e po

int-

slop

e fo

rm o

f th

e eq

uat

ion

of

the

lin

e is

y�

1 �

��3 4�

(x�

8).

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

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te a

n e

qu

atio

n i

n s

lop

e-in

terc

ept

form

of

the

lin

e h

avin

g th

e gi

ven

slo

pe

and

y-

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1.m

:2,y

-in

terc

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4

y�

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4

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2

y�

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8

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,�3)

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(�2,

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y�

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Wri

te E

qu

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olv

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ob

lem

sM

any

real

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itu

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mod

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usi

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lin

ear

equ

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rvic

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o sm

all

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in h

er c

ity.

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ech

arge

s $5

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onth

for

mai

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g a

web

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for

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nte

rven

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Eq

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____

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3-4

3-4

Exam

ple

Exam

ple

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rite

an

eq

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to

rep

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the

tota

lm

onth

ly c

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Cfo

rm

ain

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ite

and

for

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ours

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call

s.

For

eac

h h

our,

the

cost

incr

ease

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5.S

o th

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te o

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,or

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45.

Th

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omp

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For

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atio

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Jerr

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anad

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$8

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mon

th f

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ach

prem

ium

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nnel

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1.W

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an

equ

atio

n i

n s

lope

-in

terc

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onth

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for

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If J

erri

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ts t

o ad

da

fou

rth

pre

miu

m c

han

nel

,wh

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ser

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wou

ld b

e le

ast

expe

nsi

ve?

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thir

d c

om

pan

y

2.If

Jer

ri w

ants

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incl

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,wh

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serv

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4.W

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th

e eq

uat

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.T

he

fee

for

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nu

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er o

f p

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ch

ann

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rep

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nts

th

e ra

te o

f ch

ang

e,o

r sl

op

e,o

f th

e eq

uat

ion

.



Skil

ls P

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Eq

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s o

f L

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NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-4

3-4

©G

lenc

oe/M

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w-H

ill14

5G

lenc

oe G

eom

etry

Lesson 3-4

Wri

te a

n e

qu

atio

n i

n s

lop

e-in

terc

ept

form

of

the

lin

e h

avin

g th

e gi

ven

slo

pe

and

y-

inte

rcep

t.

1.m

:�4,

y-in

terc

ept:

32.

m:3

,y-i

nte

rcep

t:�

8

y�

�4x

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y�

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8

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:�3 7� ,

(0,1

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m:�

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,�6)

y�

�3 7� x�

1y

��

�2 5� x�

6

Wri

te e

qu

atio

ns

in p

oin

t-sl

ope

form

an

d s

lop

e-in

terc

ept

form

of

the

lin

e h

avin

gth

e gi

ven

slo

pe

and

con

tain

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the

give

n p

oin

t.

5.m

:2,(

5,2)

6.m

:�3,

(2,�

4)

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2 �

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,y�

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8y

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��

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,y�

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:��1 2� ,

(�2,

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m:�

1 3� ,(�

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8)

y�

5 �

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x�

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��

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4y

�8

��1 3� (

x�

3),y

��1 3� x

�7

Wri

te a

n e

qu

atio

n i

n s

lop

e-in

terc

ept

form

for

eac

h l

ine.

9.r

y�

x�

310

.sy

��

2x�

2

11.t

y�

3x�

312

.uy

��1 3� x

�5

13.t

he

lin

e pa

rall

el t

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ne

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at c

onta

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(1,�

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�x

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14.t

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lin

e pe

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dicu

lar

to l

ine

sth

at c

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(0,0

)y

��1 2� x

Wri

te a

n e

qu

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n i

n s

lop

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form

for

th

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es t

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give

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nd

itio

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(0,�

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)

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3

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nd

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1

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Wri

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form

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e h

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g th

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lop

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form

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____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-4

3-4



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csE

qu

atio

ns

of

Lin

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-4

3-4

©G

lenc

oe/M

cGra

w-H

ill14

7G

lenc

oe G

eom

etry

Lesson 3-4

Pre-

Act

ivit

yH

ow c

an t

he

equ

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n o

f a

lin

e d

escr

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the

cost

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cell

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e?

Rea

d th

e in

trod

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to

Les

son

3-4

at

the

top

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age

145

in y

our

text

book

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th

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llu

lar

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an a

re d

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ur

text

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,wh

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l ch

arge

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xes

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fees

) fo

r a

mon

th i

n w

hic

h y

ou u

se 5

0 m

inu

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of a

ir t

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$23

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Less

on

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ich

on

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th

e fo

llow

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corr

ectl

y de

scri

bes

the

y-in

terc

ept

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lin

e?C

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the

y-co

ordi

nat

e of

th

e po

int

wh

ere

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lin

e in

ters

ects

th

e x-

axis

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he

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nat

e of

th

e po

int

wh

ere

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lin

e in

ters

ects

th

e y-

axis

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the

y-co

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nat

e of

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wh

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lin

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osse

s th

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x-co

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e of

th

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int

wh

ere

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e cr

osse

s th

e x-

axis

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the

rati

o of

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e ch

ange

in

y-c

oord

inat

es t

o th

e ch

ange

in

x-c

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inat

es

4.F

ind

the

slop

e an

d y-

inte

rcep

t of

eac

h l

ine.

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pe

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y-in

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y�

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t �

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y�

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y-in

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ou

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s to

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it t

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met

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u a

lrea

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now

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an t

he

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rmu

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elp

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to

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uat

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slop

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rm

of a

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amp

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nsw

er:T

he

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pe

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a lin

e th

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gh

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ltip

ly e

ach

sid

e o

f th

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�x 1

.

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e re

sult

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be

the

po

int-

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.

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Ab

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cal

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pera

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quat

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the

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se t

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ur

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nd

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lan

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t by

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ran

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cons

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rior

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uppl

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alte

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r an

gles

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pend

icul

ar t

o th

e sa

me

line,

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erm

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ich

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f an

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d x

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m|| n

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.You

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lope

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lin

es t

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tinued

)

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TQ��

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If a

lt.i

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|| .

123

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CD

b.

Wh

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lin

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Lesson 3-5

Giv

en t

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owin

g in

form

atio

n,d

eter

min

e w

hic

h l

ines

,if

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te t

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9 �

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t.in

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;co

rr.�

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;al

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6.20

7.6

8.PR

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rovi

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th

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oof

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3.7

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An

gle

Ad

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Po

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late

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init

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gle

s5.

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siti

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of

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ity

6.B�

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B�C�

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itio

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lin

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re ⊥

to t

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wh

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ach

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lin

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s p

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lain

wh

y or

wh

y n

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9.10

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Yes;

the

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B��

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on

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th.H

ow c

an t

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gard

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en

sure

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atth

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para

llel

?

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ple

an

swer

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th

e g

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ener

dig

s ea

ch r

ow

at

a 90

�an

gle

to

th

efo

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,eac

h r

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will

be

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pen

dic

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r to

th

e fo

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.If

each

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the

row

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pen

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th

e fo

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,th

en t

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e p

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42

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____

____

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____

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____

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ER

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____

_

3-5

3-5



Readin

g t

o L

earn

Math

em

ati

csP

rovi

ng

Lin

es P

aral

lel

NA

ME

____

____

____

____

____

____

____

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____

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____

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ER

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____

_

3-5

3-5

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Lesson 3-5

Pre-

Act

ivit

yH

ow d

o yo

u k

now

th

at t

he

sid

es o

f a

par

kin

g sp

ace

are

par

alle

l?

Rea

d th

e in

trod

uct

ion

to

Les

son

3-5

at

the

top

of p

age

151

in y

our

text

book

.

How

can

th

e w

orke

rs w

ho

are

stri

pin

g th

e pa

rkin

g sp

aces

in

a p

arki

ng

lot

chec

k to

see

if

the

side

s of

th

e sp

aces

are

par

alle

l?S

amp

le a

nsw

er:

Use

a T-

squ

are

or

oth

er d

evic

e fo

r fo

rmin

g r

igh

t an

gle

s to

lay

ou

tp

erp

end

icu

lar

seg

men

ts c

ut

fro

m s

trin

g o

r ro

pe

con

nec

tin

gth

e tw

o s

ides

of

a p

arki

ng

sp

ace

at b

oth

en

ds

and

in t

he

mid

dle

.Mea

sure

th

e th

ree

per

pen

dic

ula

r se

gm

ents

.If

they

are

all t

he

sam

e le

ng

th,t

he

sid

es o

f th

e p

arki

ng

sp

ace

are

par

alle

l.

Rea

din

g t

he

Less

on

1.C

hoo

se t

he

wor

d or

ph

rase

th

at b

est

com

plet

es e

ach

sen

ten

ce.

a.If

tw

o co

plan

ar l

ines

are

cu

t by

a t

ran

sver

sal

so t

hat

cor

resp

ondi

ng

angl

es a

re

con

gru

ent,

then

th

e li

nes

are

(p

aral

lel/p

erpe

ndi

cula

r/sk

ew).

b.

In a

pla

ne,

if t

wo

lin

es a

re p

erpe

ndi

cula

r to

th

e sa

me

lin

e,th

en t

hey

are

(per

pen

dicu

lar/

para

llel

/ske

w).

c.F

or a

lin

e an

d a

poin

t n

ot o

n t

he

lin

e,th

ere

exis

ts

(at

leas

t on

e/ex

actl

y on

e/at

mos

t on

e) l

ine

thro

ugh

th

e po

int

that

is

para

llel

to

the

give

n l

ine.

d.

If t

wo

copl

anar

lin

es a

re c

ut

by a

tra

nsv

ersa

l so

th

at c

onse

cuti

ve i

nte

rior

an

gles

are

(com

plem

enta

ry/s

upp

lem

enta

ry/c

ongr

uen

t),t

hen

th

e li

nes

are

para

llel

.

e.If

tw

o co

plan

ar l

ines

are

cu

t by

a t

ran

sver

sal

so t

hat

alt

ern

ate

inte

rior

an

gles

are

con

gru

ent,

then

th

e li

nes

are

(p

erpe

ndi

cula

r/pa

rall

el/s

kew

).

2.W

hic

h o

f th

e fo

llow

ing

con

diti

ons

veri

fy t

hat

p|| q

?A

,C,F

,GA

.�

6 �

�12

B.�

2 �

�4

C.

�8

��

16D

.�11

��

13

E.

�6

and

�7

are

supp

lem

enta

ry.

F.�

1 �

�15

G.�

7 an

d �

10 a

re s

upp

lem

enta

ry.

H.�

4 �

�16

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

som

eth

ing

new

is

to d

raw

a p

ictu

re.H

ow c

an a

ske

tch

hel

p yo

uto

rem

embe

r th

e P

aral

lel

Pos

tula

te?

Sam

ple

an

swer

:D

raw

a li

ne

wit

h a

ru

ler

or

stra

igh

ted

ge

and

ch

oo

se a

po

int

no

t o

n t

he

line.

Try

to d

raw

lin

es t

hro

ug

h t

he

po

int

that

are

par

alle

lto

th

e lin

e yo

u o

rig

inal

ly d

rew

.Yo

u w

ill s

ee t

hat

th

ere

is e

xact

ly o

ne

way

to d

o t

his

.

p

t

q r

12 9

10

1213

14

1516

11

34 6

5

78

par

alle

l

sup

ple

men

tary

exac

tly

on

ep

aral

lel

par

alle

l

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lenc

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eom

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Scr

ambl

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p P

roo

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Th

e re

ason

s n

eces

sary

to

com

ple

te t

he

foll

owin

g p

roof

are

scra

mb

led

up

bel

ow.T

o co

mp

lete

th

e p

roof

,nu

mb

er t

he

reas

ons

to m

atch

th

e co

rres

pon

din

g st

atem

ents

.

Giv

en:C �

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A�B�

⊥B�

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B�D�

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of:

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tem

ents

Rea

son

s

1.C�

D�⊥

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Def

init

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of

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ht

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angl

e4

2.A

B⊥

B�E�

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en1

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d �

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e ri

ght

angl

es.

Giv

en2

4.�

AB

Dan

d �

CD

Ear

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ght

tria

ngl

es.

Def

init

ion

of

Per

pen

dicu

lar

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es3

5.A�

D��

C�E�

Giv

en5

6.B�

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D�E�

CP

CT

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7.�

AB

D�

�C

DE

In a

pla

ne,

if t

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lin

es a

re c

ut

by a

tra

nsv

ersa

lso

th

at a

pai

r of

cor

resp

ondi

ng

angl

es i

sco

ngr

uen

t,th

en t

he

lin

es a

re p

aral

lel.

(Th

eore

m 7

-5)

9

8.�

1 �

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Giv

en6

9.A�

D�|| C�

E�H

L7

AC

BE

D

5 31

42

6

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-5

3-5



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Per

pen

dic

ula

rs a

nd

Dis

tan

ce

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-6

3-6

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Lesson 3-6

Dis

tan

ce F

rom

a P

oin

t to

a L

ine

Wh

en a

poi

nt

is

not

on

a l

ine,

the

dist

ance

fro

m t

he

poin

t to

th

e li

ne

is t

he

len

gth

of

the

segm

ent

that

con

tain

s th

e po

int

and

ispe

rpen

dicu

lar

to t

he

lin

e.

Dra

w t

he

segm

ent

that

rep

rese

nts

th

e d

ista

nce

from

Eto

AF

�� .

Ext

end

AF

� �� .

Dra

w E�

G�⊥

AF

�� .

E �G�

repr

esen

ts t

he

dist

ance

fro

m E

to A

F��

� .

Dra

w t

he

segm

ent

that

rep

rese

nts

th

e d

ista

nce

in

dic

ated

.

1.C

to A

B� �

�2.

Dto

AB

��

3.T

to R

S��

�4.

Sto

PQ

��

5.S

to Q

R��

�6.

Sto

RT

��

XR

T

S

R

T

S

P

Q

X

RT

S

PQ

X

URS

T

X

A

CD

BX

A

C

BX

AF

G

BE

AF

BE

Pdi

stan

ce b

etw

een

M a

nd P

Q��

�

Q

M

Exam

ple

Exam

ple

Exer

cises

Exer

cises

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w-H

ill15

6G

lenc

oe G

eom

etry

Dis

tan

ce B

etw

een

Par

alle

l Lin

esT

he

dist

ance

bet

wee

n p

aral

lel

lin

es i

s th

e le

ngt

hof

a s

egm

ent

that

has

an

en

dpoi

nt

on e

ach

lin

e an

d is

per

pen

dicu

lar

to t

hem

.Par

alle

l li

nes

are

ever

ywh

ere

equ

idis

tan

t,w

hic

h m

ean

s th

at a

ll s

uch

per

pen

dicu

lar

segm

ents

hav

e th

esa

me

len

gth

.

Fin

d t

he

dis

tan

ce b

etw

een

th

e p

aral

lel

lin

es �

and

mw

hos

eeq

uat

ion

s ar

e y

�2x

�1

and

y�

2x�

4,re

spec

tive

ly.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Per

pen

dic

ula

rs a

nd

Dis

tan

ce

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-6

3-6

Exam

ple

Exam

ple

Dra

w a

lin

e p

thro

ugh

(0,

1) t

hat

is

perp

endi

cula

r to

�an

d m

.

Lin

e p

has

slo

pe �

�1 2�an

d y-

inte

rcep

t 1.

An

equ

atio

n o

f p

is y

��

�1 2� x�

1.T

he

poin

t of

inte

rsec

tion

for

pan

d �

is (

0,1)

.

x

y O

mp

�

( 0, 1

)

x

y O

m�

To

fin

d th

e po

int

of i

nte

rsec

tion

of

pan

d m

,so

lve

a sy

stem

of

equ

atio

ns.

Lin

e m

:y

�2x

�4

Lin

e p:

y�

��1 2� x

�1

Use

su

bsti

tuti

on.

2x�

4 �

��1 2� x

�1

4x�

8 �

�x

�2

5x�

10x

�2

Su

bsti

tute

2 f

or x

to f

ind

the

y-co

ordi

nat

e.

y�

��1 2� x

�1

��

�1 2� (2)

�1

��

1 �

1 �

0

Th

e po

int

of i

nte

rsec

tion

of

pan

d m

is (

2,0)

.

Use

th

e D

ista

nce

For

mu

la t

o fi

nd

the

dist

ance

bet

wee

n (

0,1)

an

d (2

,0).

d�

�(x

2�

�x 1

)2�

�(y

2�

�y 1

)2�

��

(2 �

0�

)2�

(�

0 �

1�

)2 ��

�5�

Th

e di

stan

ce b

etw

een

�an

d m

is �

5�u

nit

s.

Exer

cises

Exer

cises

Fin

d t

he

dis

tan

ce b

etw

een

eac

h p

air

of p

aral

lel

lin

es.

1.y

�8

2.y

�x

�3

3.y

��

2xy

��

3y

�x

�1

y�

�2x

�5

11�

8��

5�



Skil

ls P

ract

ice

Per

pen

dic

ula

rs a

nd

Dis

tan

ce

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-6

3-6

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lenc

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eom

etry

Lesson 3-6

Dra

w t

he

segm

ent

that

rep

rese

nts

th

e d

ista

nce

in

dic

ated

.

1.B

to A

C��

�2.

Gto

EF

��

3.Q

to S

R��

�

Con

stru

ct a

lin

e p

erp

end

icu

lar

to �

thro

ugh

K.T

hen

fin

d t

he

dis

tan

ce f

rom

Kto

�.

4.5.

�13�

3�2�

Fin

d t

he

dis

tan

ce b

etw

een

eac

h p

air

of p

aral

lel

lin

es.

6.y

�7

7.x

��

68.

y�

3xy

��

1x

�5

y�

3x�

10

811

�10�

9.y

��

5x10

.y�

x�

911

.y�

�2x

�5

y�

�5x

�26

y�

x�

3y

��

2x�

5

�26�

3�2�

2�5�x

y O

�

K

x

y

O

�

K

S

PQ

RD

EF

GA

B

C

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lenc

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ill15

8G

lenc

oe G

eom

etry

Dra

w t

he

segm

ent

that

rep

rese

nts

th

e d

ista

nce

in

dic

ated

.

1.O

to M

N��

�2.

Ato

DC

��

3.T

to V

U��

�

Con

stru

ct a

lin

e p

erp

end

icu

lar

to �

thro

ugh

B.T

hen

fin

d t

he

dis

tan

ce f

rom

Bto

�.

4.5.

4�2�

�13�

Fin

d t

he

dis

tan

ce b

etw

een

eac

h p

air

of p

aral

lel

lin

es.

6.y

��

x7.

y�

2x�

78.

y�

3x�

12y

��

x�

4y

�2x

�3

y�

3x�

18

2�2�

2�5�

3�10�

9.G

raph

th

e li

ne

y�

�x

�1.

Con

stru

ct a

per

pen

dicu

lar

segm

ent

thro

ugh

th

e po

int

at (

�2,

�3)

.Th

en f

ind

the

dist

ance

fro

m t

he

poin

t to

th

e li

ne.

3 �2 �

10.C

AN

OEI

NG

Bro

nso

n a

nd

a fr

ien

d ar

e go

ing

to c

arry

a c

anoe

acr

oss

a fl

at f

ield

to

the

ban

k of

a s

trai

ght

can

al.D

escr

ibe

the

shor

test

pat

h t

hey

can

use

.

Sam

ple

an

swer

:Th

e sh

ort

est

pat

h w

ou

ld b

e a

per

pen

dic

ula

r se

gm

ent

fro

m w

her

e th

ey a

re t

o t

he

ban

k o

f th

e ca

nal

.

( –2,

–3)

y �

�x

� 1

x

y

O

x

y O

�B

x

y O

� B

T

VW

SU

AB

CD

MN

O

Pra

ctic

e (

Ave

rag

e)

Per

pen

dic

ula

rs a

nd

Dis

tan

ce

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-6

3-6



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csP

erp

end

icu

lars

an

d D

ista

nce

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-6

3-6

©G

lenc

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ill15

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lenc

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eom

etry

Lesson 3-6

Pre-

Act

ivit

yH

ow d

oes

the

dis

tan

ce b

etw

een

par

alle

l li

nes

rel

ate

to h

angi

ng

new

sh

elve

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

3-6

at

the

top

of p

age

159

in y

our

text

book

.

Nam

e th

ree

exam

ples

of

situ

atio

ns

in h

ome

con

stru

ctio

n w

her

e it

wou

ld b

eim

port

ant

to c

onst

ruct

par

alle

l li

nes

.S

amp

le a

nsw

er:

op

po

site

wal

ls o

f a

roo

m,p

lan

ks o

f h

ard

wo

od

flo

ori

ng

,to

ps

and

bo

tto

ms

of

cab

inet

s

Rea

din

g t

he

Less

on

1.F

ill

in t

he

blan

k w

ith

a w

ord

or p

hra

se t

o co

mpl

ete

each

sen

ten

ce.

a.T

he

dist

ance

fro

m a

lin

e to

a p

oin

t n

ot o

n t

he

lin

e is

th

e le

ngt

h o

f th

e se

gmen

t

to t

he

lin

e fr

om t

he

poin

t.

b.

Tw

o co

plan

ar l

ines

are

par

alle

l if

th

ey a

re e

very

wh

ere

.

c.In

a p

lan

e,if

tw

o li

nes

are

bot

h e

quid

ista

nt

from

a t

hir

d li

ne,

then

th

e tw

o li

nes

are

to e

ach

oth

er.

d.

Th

e di

stan

ce b

etw

een

tw

o pa

rall

el l

ines

mea

sure

d al

ong

a pe

rpen

dicu

lar

to t

he

two

lin

es i

s al

way

s .

e.T

o m

easu

re t

he

dist

ance

bet

wee

n t

wo

para

llel

lin

es,m

easu

re t

he

dist

ance

bet

wee

n

one

of t

he

lin

es a

nd

any

poin

t on

th

e .

2.O

n e

ach

fig

ure

,dra

w t

he

segm

ent

that

rep

rese

nts

th

e di

stan

ce i

ndi

cate

d.

a.P

to �

b.D

to A

B��

�

c.E

to F

G� �

�d

.Uto

RV

� ��

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r a

new

wor

d is

to

rela

te i

t to

wor

ds t

hat

use

th

e sa

me

root

.Use

you

r di

ctio

nar

y to

fin

d th

e m

ean

ing

of t

he

Lat

in r

oot

aequ

us.

Lis

t th

ree

wor

ds o

ther

th

aneq

ual

an

d eq

uid

ista

nt

that

are

der

ived

fro

m t

his

roo

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Sam

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plan

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and

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init

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wo

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es a

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aral

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if a

nd

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if

they

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in

ters

ect.

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init

ion

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e an

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f an

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ly

if t

hey

do

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us,

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,tw

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or

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and

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____

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3-6

3-6



Chapter 3 Assessment Answer Key Form 1 Form 2APage 161 Page 162 Page 163


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

C

D

B

A

D

B

C

D

C

A

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

A

D

B

C

C

A

B

D

A

C

undefined

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

C

D

C

D

B

D

A

C

B

D


Chapter 3 Assessment Answer KeyForm 2A (continued) Form 2BPage 164 Page 165 Page 166

An

swer

s

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

C

A

B

D

A

A

C

B

D

C

Sample answer:y � 285 � 15x;

19 h

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

D

A

A

B

C

B

D

A

C

C

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

D

B

B

A

C

A

D

B

C

A

a ⊥ c


Chapter 3 Assessment Answer KeyForm 2CPage 167 Page 168

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

SV��

Sample answer:

V�Z�

alternate exterior

alternate interior

corresponding

94

x � 13, y � 39

�35

�

�6

� �53

�

neither

parallel

parallel

C � 1.18p � 12;$71

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

y � �9x � 3

y � 3x � 8

y � �13

�x � 1

y � �x � 2

AC�� || DF��; alt. int. �

AD�� || BE��; corr. �

AD�� || BE��; cons. int. �

18

�98� or 7�2��45� or 3�5��20� or 2�5�

Sample answer:Given: �3 and �4are supplementary

Prove: � || m

1

43

ms

�

B

x

y

O

�


Chapter 3 Assessment Answer KeyForm 2DPage 169 Page 170

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

CD��

Sample answer:A�E�

corresponding

alternate interior

alternate exterior

91

x � 17; y � 26

��25

�

�17

�

��13

�

perpendicular

parallel

perpendicular

T � 0.04c � 25;$85

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

y � 7x � 8

y � �4x � 8

y � �25

�x � 2

y � �2x � 9

RS�� || TU��; corr. �

PT�� || QU��; alt. int. �

PT�� || QU��; cons. int. �

12

�10��26�

�32� or 4�2�

Sample answer:Given: �2 � �3

Prove: � || m

1

32

mt

�

x

y

O

�

Q


Chapter 3 Assessment Answer KeyForm 3Page 171 Page 172

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

Sample answer:

Sample answer:Plane ABC

intersects planeBCE at BC��.

Sample answer: AD��

is skew to EF��.

corresponding

consecutive interior

alternate exterior

40

x � 18, y � 7.5,and z � �7

�53

�

��38

�

perpendicular

parallel

B

C

F

D

A

E

13.

14.

15.

16.

17.

18.

19.

20.

B:

y � 10 � �85

�(x � 6)

or y � �85

�x � �25

�

y � ��43

�x � �331�

C � 0.28(m � 100)� 150; $193.40

NU�� || PV��; alt. int. �

MR�� || NS��; cons. int. �

24.2

�17�

�20� or 2�5�

Sample answer: No, if

line p and line q lie indifferent planes, thelines could be skew

and not parallel.

sp

q

x

y

OP

m

Chapter 3 Assessment Answer KeyPage 173, Open-Ended Assessment

Scoring Rubric


Score General Description Specific Criteria

• Shows thorough understanding of using the relationshipsbetween lines and their properties, proving lines parallel,identifying and determining angle relationships, findingslope, and graphing parallel and perpendicular lines.

• 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 using the relationshipsbetween lines and their properties, proving lines parallel,identifying and determining angle relationships, findingslope, and graphing parallel and perpendicular lines.

• 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 a partial understanding of using the relationshipsbetween lines and their properties, proving lines parallel,identifying and determining angle relationships, findingslope, and graphing parallel and perpendicular lines.

• 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 is 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 using the relationshipsbetween lines and their properties, proving lines parallel,identifying and determining angle relationships, findingslope, and graphing parallel and perpendicular lines.

• 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

An

swer

s


Chapter 3 Assessment Answer KeyPage 173, Open-Ended Assessment

Sample Answers

1a. If you know the measure of one or moreangles formed by the intersections of thestreets, you can determine congruence ofangles. If certain angles are congruent,the streets are parallel. For instance,lines a, b, and c are cut by transversal d.If you know the measure of �1 and it iscongruent to �2, then the streetsrepresented by a and b are parallel bythe Alternate Exterior Angles Theorem.If �2 is congruent to �4, b and c areparallel by the Alternate Interior AnglesTheorem.

b. m�4 � 68. Given lines d and e areparallel, �5 and �4 are consecutiveinterior angles. Since consecutiveinterior angles are supplementary,180 � m�5 will give the measure of 4:180 � 112 � 68.

c. x � 27; �1 and �4 are correspondingangles. If lines a and c are parallel, then�1 � �4. Therefore, m�1 � m�4.Substitute the measures for the anglesand solve for x and then verify that theangles are congruent. Since m�1 � 3x � 7 or 74 and m�4 � 2x � 20 or 74; �1 � �4 andlines a and c are parallel.

d. Student should draw a line through ●●Bperpendicular to line b. The perpendicularline segment should include a right anglesymbol. To find the shortest distancefrom the library to his street, Todd couldmeasure this perpendicular segmentsince a perpendicular line segment fromany point to a line is the shortestdistance between the point and the line.

2a. The student first draws the line, thenwrites the equation y � x � 2.

m � �yx

2

2

�

�

yx

1

1� Use the slope formula to find the slope

of the line.

� �42

��

13� Substitute coordinates into the formula.

� �55� or 1

Then use one of the pair of coordinates towrite the equation in point-slope form.

y � y1 � m(x � x1) Point-slope form

y � 1 � 1(x �3) Substitute (�3, �1).

y � x � 3 � 1 Simplify and subtract 1 from

each side.

y � x � 2

b. The slope of a line parallel to y � x � 2is 1, since parallel lines have the sameslope.

c. �2�; the student should draw a lineperpendicular to the two lines, and thenuse the points of intersections and theDistance Formula to find the distance.

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


Chapter 3 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 174 Page 175 Page 176

An

swer

s

Quiz 2Page 175

Quiz 4Page 176

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

false; alternateinterior angles

false; Corr. �Postulate

true

false; two

true

true

false; parallel

false;perpendicular

true

false; slope

Sample answer:Rate of change

describes how aquantity is changing

over time.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Sample answer:plane ABM

BC��

corresponding

alternate interior

alternate exterior

consecutive interior

86

94

x � 12, y � 31

C

1.

2.

3.

4.

5.

�1

5

��32

�

�15

�

perpendicular

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

y � 8 � ��13

�(x � 3)

y � �53

�x � 2

y � �4x � 3

C � 5b � 8

g || h; corr. �

p || q; alt. int. �

g || h; alt. ext. �

p || q; cons. int. �

116

26

1.

2.

3.

4.

�32� or 4�2�

12

�2�

x

y

O

�H

BA

CD


Chapter 3 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 177 Page 178

Part I

Part II

6.

7.

8.

9.

10.

x � 28, y � 19

�12

�

��14

�

parallel

perpendicular

1.

2.

3.

4.

5.

D

C

C

A

B

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

7; 84

(�7, �1)

acute

5; 20

true

H: m�1 � m�2 �180; C: �1 and �2are supplementary

Law of Detachment

always

parallel

y � 8 � 4(x � 2)

�41� or � 6.4


An

swer

s

Chapter 3 Assessment Answer KeyStandardized Test Practice

Page 179 Page 180

1.

2.

3.

4.

5.

6.

7.

8.

9.

10. 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

E F G H

A B C D

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

14.

15.

16.

17.

18.

10

5 or �9

2

2

20

2 3 8 5

3

© Glencoe/McGraw-Hill A30 Geometry

Chapter 3 Assessment Answer KeyUnit 1 Test/Review

Page 181 Page 182

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

2�12

� in.; �14

� in.

�68� or about8.25; (9, �8)

m�PQR � 120,obtuse;

m�RQS � 40, acute;m�SQT � 20, acute

53 and 37

pentagon,convex, irregular

54 ft long by 18 ft wide

U

hypothesis: in aplane, lines � andm are equidistant

from line p;conclusion: � || m

Law of Syllogism

sometimes

p q �p �p � q

T T F FT F F FF T T TF F T F

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

Given

Angles comp. to� � are �.

Def. of � �

Def. of comp. �

Substitution

p; alternateexterior angles

m�2 � 52, m�4 �128, m�10 � 52,

m�12 � 128

perpendicular

y � �2x � 5

12

�40� or about6.32 units

x

y

O

Documents

Chapter 3 Resource Masters - Math Problem Solving · ©Glencoe/McGraw-Hill iv Glencoe Geometry Teacher’s Guide to Using the Chapter 3 Resource Masters The Fast FileChapter Resource