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South Carolina Holt Geometry Practice Workbook

correlated to

South Carolina Curriculum Standards for Geometry

TO THE TEACHER South Carolina Holt Geometry Practice Workbook is designed to provide additional practice of the skills taught in each lesson of the textbook. On each page students will practice the skills from one particular lesson. There are approximately 10 to 50 practice items on each page. These items include practice of both the basic skills and mathematical applications taught in the lesson. Answers to the questions in this booklet are provided for you in the Geometry Practice Workbook Answer Key. STAFF CREDITS Director of Special Projects: Suzanne Thompson Managing Editor: Joan Marie Lindsay Editor: Tressa Sanders Associate Editor: Brian Howell Editorial Coordinator: John Kendall

Copyright © by Holt, Rinehart and Winston All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Teachers may photocopy pages in sufficient quantity for classroom use only, not for resale. Printed in the United States of America ISBN 0-03-069036-6 123 179 04 03 02

The following is a correlation of the South Carolina Geometry Standards to Holt, Rinehart and Winston's South Carolina Geometry Practice Workbook. Page numbers are given on which treatment of the standards may be found in the practice booklet. In completing the exercises in Holt, Rinehart and South Carolina Geometry Practice Workbook, students will practice the mathematical skills and concepts that are addressed in state standardized tests.

South Carolina Geometry Standards

I. GEOMETRIC STRUCTURE A. Axiomatic Systems 1. Develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. 2. Recognize that the study of geometry was developed for a variety of purposes and that it has historical significance. B. Verification of Conjectures 1. Explore attributes of geometric figures using constructions with straight-edge and compass; paper folding; and dynamic, interactive geometry software. 2. Make and verify conjectures about angles, lines, polygons, circles, and three-dimensional figures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Page References

Practice for the standard is provided throughout the workbook. 9, 54, 69, 73, 80

36, 27, 28, 67, 74

Practice for the standard is provided throughout the workbook.

iii

South Carolina Geometry Standards

C. Logical Reasoning and Proof 1. Determine whether the converse of a conditional statement is true or false. 2. Use logical reasoning to draw conclusions about geometric figures from given assumptions. 3. Construct and judge validity of a logical argument consisting of a set of premises and a conclusion. 4. Use inductive reasoning to formulate a conjecture. 5. Use deductive reasoning to prove a statement. D. Representing Geometric Relationships 1. Select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) to solve a problem. 2. Use dynamic, interactive geometry software to represent geometric relationships and solve problems. 3. Find optimal solutions to problems involving paths, networks, or relationships among a finite number of objects using digraphs or vertex-edge graphs.

Page References

9, 10, 78 812, 16, 2123, 25, 26, 52, 53, 5760

812, 16, 2123, 25, 26, 52, 53, 5760, 7679 8, 9, 12, 21, 23 11, 12, 16, 53, 76, 79

Practice for the standard is provided throughout the workbook. None

1, 4, 5, 8, 27, 34, 37, 42, 67, 70, 74

iv

Copyright © by Holt, Rinehart and Winston. All rights reserved.

South Carolina Geometry Standards

II. GEOMETRIC PATTERNS A. Two- and Three-Dimensional Geometric Figures 1. Use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, triangle inequality, and angle relationships in polygons and circles. 2. Identify and apply patterns from right triangles to solve problems. 3. Draw, examine, and classify cross sections of three-dimensional objects. 4. Construct a three-dimensional object using a two-dimensional diagram such as a blueprint or pattern. 5. Use top, front, side, and corner views of threedimensional objects to create accurate and complete representations and solve problems. 6. Represent a three-dimensional object in two dimensions using graph or dot paper.

Page References

1, 2, 4, 5, 8, 9, 18, 5052, 54, 5759, 6266, 69

32, 33, 6266 3749 42

37, 42

37, 42

Copyright © by Holt, Rinehart and Winston. All rights reserved.

v

South Carolina Geometry Standards

III. GEOMETRY OF LOCATION A. Coordinate Geometry 1. Given geometric figures, utilize a coordinate system to identify and justify conjectures. 2. Use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons. 3. Develop and use formulas including distance and midpoint. 4. Given two ordered pairs, find the distance between them, locate the midpoint, and determine the slope of the line that contains them. B. Transformations 1. Plot coordinates for translations and describe the vertical and horizontal transformational vector(s). 2. Translate, reflect, rotate, and dilate figures on the plane. 3. Analyze the symmetry of objects using the language of transformations. 4. Use transformations and their compositions to make connections between mathematics and applications including tessellations or fractals, in particular with graphing calculators and geometry software.

Page References

7, 20, 34, 40, 41, 49, 50, 75 41

2934, 4348 34, 40

7, 75

6, 7, 28, 49, 50, 68, 75 13, 49 74

Copyright © by Holt, Rinehart and Winston. All rights reserved.

vi

South Carolina Geometry Standards

IV. GEOMETRY OF SIZE A. Measurement 1. Find areas of regular polygons and composite figures. 2. Find areas of sectors and arc lengths of circles using proportional reasoning. 3. Develop, extend, use, and prove the Pythagorean theorem. 4. Use formulas for surface area and volume of three-dimensional objects to solve practical problems. 5. Determine the resulting change in the area and volume of a figure when one or more dimension is changed. B. Properties and Relationships 1. Based on explorations and using concrete models and geometry software, formulate and test conjectures about properties of parallel and perpendicular lines, including two parallel lines cut by a transversal line, properties and attributes of polygons and their component parts, and properties and attributes of circles and the lines that intersect them.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Page References

2931 31, 57, 60 32, 33 4348, 55

4348

1426, 32, 33, 38, 39, 4466, 69

vii

South Carolina Geometry Standards

V. GEOMETRY OF SHAPE A. Similarity 1. Identify, describe, and defend similarity between shapes. 2. Using similarity and transformations, justify conjectures about geometric figures. 3. Utilize ratios to solve problems involving similar figures in a variety of ways, including the use of dynamic, interactive geometry software. 4. Solve applied problems using scale modeling. 5. Develop, apply, and justify triangle similarity relationships. 6. Explore concepts and applications of trigonometry by solving applied problems using right triangle trigonometry. 7. Using graphing calculators, spreadsheets, and dynamic, interactive geometry software, describe the effect on perimeter, area, and volume when length, width, or height of a three-dimensional solid is changed; apply this idea in solving problems. 8. Solve problems using proportion involving similar figures. B. Congruence 1. Use congruence transformations to make conjectures and justify properties of geometric figures. 2. Justify and apply triangle congruence relationships. 3. Identify, describe, and defend congruence between shapes.

Page References

5054 6, 7, 28, 49, 5054, 68, 75 5155, 6266

50 5155 None

2933, 4348

5155

Copyright © by Holt, Rinehart and Winston. All rights reserved.

2124

2124 2124

viii

Table of Contents

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Exploring Geometry Reasoning in Geometry Parallels and Polygons Triangle Congruence Perimeter and Area Shapes in Space Surface Area and Volume Similar Shapes Circles 1 8 13 21 29 37 43 50 56 62 69 76

Chapter 10 Trigonometry Chapter 11 Taxicabs, Fractals, and More Chapter 12 A Closer Look at Proof and Logic

ix

NAME

CLASS

DATE

Practice

1.1

1.

The Building Blocks of Geometry

Q 1

For Exercises 14, refer to the triangle at right.

Name all the segments in the triangle.

2.

Name each of the angles in the triangle by using three different methods.

R

2

3

S

3. 4.

Name the rays that form each of the angles of the triangle. Name the plane that contains the triangle.

State whether each object could best be modeled by a point, line, or plane.

5. 7. 9.

a star a ruler edge a sheet of paper

6. 8. 10.

a notebook cover the tip of a pen a letter opener

Classify each statement as true or false, and explain your reasoning in each false case.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

11.

Two planes intersect in only one point.

12.

A ray starts at one point on a line and goes on forever.

13.

The intersection of two planes is one line.

For Exercises 1418, T is the midpoint of BC . Classify each statement as true or false.

C, T, and B are collinear. 15. RS is the same as RT.

14. 16. 17. 18.

B

S

C, T, and B name a plane. R, T, and C are collinear. Four rays start at T.

R

T C

Geometry

Practice Workbook

1

NAME

CLASS

DATE

Practice

1.2

R 6 1. 3. 5. 5 S 4

Measuring Length

In Exercises 14, find the segment lengths determined by the points on the number line.

T 3 2 1 U 0 1 V 2 2. 4. 3 W 4 5 X 6

RV SW

TX VW

The length of a segment must be a positive number. Explain why the order of the coordinates does not matter when calculating length.

Name all of the congruent segments in each figure.

6. A 3 7. L P 2 B 1 0 C 1 M Q 2 D 3

Copyright © by Holt, Rinehart and Winston. All rights reserved.

N R

O S

Point B is between points Q and R on QR . Sketch a figure for each set of values, and find the missing lengths.

8. 9. 10.

QB QB QB

20; BR 50; BR

10; QR ; QR ; BR 16.9; QR 110 51.5

A x D x E 110 Geometry B 30 F C

Find the indicated values.

11. 12.

AC DF

45; x 135; x

Practice Workbook

2

NAME

CLASS

DATE

Practice

1.3

m SVR m SVQ m SVP m RVQ

Measuring Angles

Q 100°

80

Find the measure of each angle in the diagram at right.

1. 2. 3. 4. 5. 6.

R

70° S 40°

160° P

10 20 3 170 16 0 1 0 4 50 0 14 0

100 1 10 70 0 00 90 80 70 120 6 10 1 13 1 60 0 50 120 50 0 13

90

170 160 150 20 10 0 14 0 30 4

cm

1

2

3

4

5

1

2

3

V

6

7

8

9

1 0

4

5

m PVR Name all sets of congruent angles in the diagram below.

S 95° P 125°

80

W 65°

10 20 3 170 16 0 1 0 4 50 0 14 0

90 100 110 70 120 100 90 80 70 60 13 110 60 0 50 120 50 0 13

B 30°

14 0

40

170 160 150 20 10 30

170°

Z

Copyright © by Holt, Rinehart and Winston. All rights reserved.

A 0°

cm

1

2

3

4

5

1

2

3

V

6

7

8

9

1 0

4

5

Find the missing angle measures.

7. 8. 9.

B

E M

m BTE m BTE m BTE

40 , m ETM 112 , m ETM

60 , m BTM , m BTM , m ETM 47 , m BTM

(3x

168 92

C

T

L D

In the figure at right, m CED (x 25) . m LED

10. 11.

39 , m CEL

6) , and

What is the value of x? What is m CEL?

E

E D

In the diagram at right, m DSF (45 x) . Find the value of x, and then give each indicated angle measure.

12.

5x + 4 4x + 1 F

S

m DSF

13.

m DSE

14.

m ESF

Geometry

Practice Workbook

3

NAME

CLASS

DATE

Practice

1.4

Exploring Geometry by Using Paper Folding

Construct all of the geometric figures below by folding a sheet of paper.

1.

Describe how to construct a line, , through points C and D.

C

2.

Describe how to construct two lines perpendicular to line from Exercise 1, line m through point D and line p through point C.

D

3.

Describe the relationship between lines m and p.

C

p

4.

Write a conjecture about the number of lines perpendicular to line that can be constructed.

D

m

5.

Describe how to construct a segment bisector, r, of CD.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

6.

Describe line r in relation to CD.

C

p

r

E 7.

m D

Write a conjecture about the number of perpendicular bisectors of CD that can be constructed.

8.

How can you determine whether a given line is the angle bisector of an angle?

4

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

1.5

Special Points in Triangles

In Exercises 14, trace the given triangle on folding paper or sketch it with geometry software. Then construct the indicated geometric figures.

1.

the perpendicular bisectors of the sides of GHI

G H

2.

the angle bisectors of each angle in PQR

R

I 3.

P 4.

Q

the circumscribed circle of JKL

K

the inscribed circle of MNO

M N

J

Copyright © by Holt, Rinehart and Winston. All rights reserved.

L

O

For Exercises 57, draw or fold an acute triangle, with all angles measuring less than 90 ; and a right triangle, with one angle measure of 90 .

5.

Construct the circumcenter of the right triangle and use it to draw its circumscribed circle. Construct the centroid of an acute triangle and label it point C. How many line segments with endpoint C are formed from the medians in Exercise 6? Describe the relationships among these segments.

6. 7.

Complete each statement with always, sometimes, or never.

8.

A median of a triangle contains a vertex and the midpoint of the opposite side.

9. 10.

An altitude is An altitude is

perpendicular to the opposite side. an angle bisector.

Practice Workbook

Geometry

5

NAME

CLASS

DATE

Practice

1.6

1.

Motion in Geometry

Identify each rigid motion as a reflection, translation, or rotation.

2. 3.

image image image

In the diagram at right, point D is shifted 3 cm in the direction shown to form point D .

4.

E

Describe how point E was formed. Given any point of DEF, tell how to locate its image point in D E F .

D

F

5.

E'

D'

Copyright © by Holt, Rinehart and Winston. All rights reserved.

F'

Reflect each figure across the given line.

6. 7. 8.

Reflect the word TOT across each line.

9. 10. 11.

TOT

6

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

1.7

1.

Motion in the Coordinate Plane

Use the given rule to translate each triangle on the grid provided.

H(x, y)

(x, y

y

3)

2.

H(x, y)

(x,

y

y)

1 O 1 x O

1 1 x

3.

H(x, y)

(x

y

4, y)

4.

H(x, y)

(x

y

2, y

1)

1 O

Copyright © by Holt, Rinehart and Winston. All rights reserved.

1 1 x O 1 x

Describe the result of applying each rule below to a figure in a coordinate plane.

5.

G(x, y)

(x

6, y)

6.

F(x, y)

(x, y

1)

7.

P(x, y)

(x,

y)

8.

H(x, y)

( x, y)

9.

T(x, y)

(x

4, y

5)

10.

R(x, y)

(x

2, y

2)

11.

M(x, y)

(x

5, y

2)

12.

N(x, y)

( x,

y)

Geometry

Practice Workbook

7

NAME

CLASS

DATE

Practice

2.1

An Introduction to Proofs

Number of Points 2 3 4 5 6 7 8 9 Number of Line Segments

How many line segments can be drawn between 3, 4, 5, and 6 points? Draw them. Then record your data in the table. (Note: The points are noncollinear--that is, they are not on the same line.)

1. 2.

3.

4.

5. 6.

What is the pattern? If the pattern continues, how many line segments can be drawn

Copyright © by Holt, Rinehart and Winston. All rights reserved.

between 10 noncollinear points?

7.

Write an expression in terms of n for the number of line segments that can be drawn between n points.

Logical arguments that ensure true conclusions are called proofs.

8.

Consider the following conjecture: Opposite sides of a parallelogram are equal in measure. Test the conjecture by measuring the sides of the parallelogram at right. Record your results in the table.

WX ZY WZ XY

Z

W

X

Y

9.

Did the conjecture seem to be true? Explain.

8

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

2.2

Introduction to Logic

Refer to the following statement to answer Exercises 14:

All turtles are reptiles.

1.

Rewrite the statement as a conditional.

2.

Identify the hypothesis and the conclusion of the statement.

3.

Draw an Euler diagram that illustrates the statement.

4.

Write a converse of the statement and construct its Euler diagram. If the converse is false, illustrate this with a counterexample.

5.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Write a conditional statement with the given hypothesis and conclusion, and then write the converse of that statement. Is the original statement true? The converse? If either is false, give a counterexample. hypothesis: EF DG conclusion: The diagonals of a rectangle are equal in length.

D E

F

G

6.

Arrange the three statements below into a logical chain. Then write the conditional statement that follows from the logic. If it is warm, then it is spring. If flowers are blooming, then it is warm. If you see bees, then flowers are blooming.

Geometry

Practice Workbook

9

NAME

CLASS

DATE

Practice

2.3

Definitions

Write the given sentence in the forms requested. State in your conclusion whether the biconditional statement is true. If it is false, state which part makes it false.

1.

I am a high school sophomore in the 10th grade. Conditional statement: Converse: Biconditional: Conclusion:

2.

A diamond can cut glass. Conditional statement: Converse: Biconditional: Conclusion:

3.

A 130° angle is an acute angle. Conditional statement:

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Converse: Biconditional: Conclusion: These figures are hexagons. These figures are not hexagons.

4.

Which of the following are hexagons?

a. b. c. d.

5.

Write a definition for a hexagon.

Practice Workbook Geometry

10

NAME

CLASS

DATE

Practice

2.4

1.

Building a System of Geometry Knowledge

Identify the properties of equality that justify the conclusion.

x x 9 B B

9 9 x C; D CD AB AB

16 16 7 C 9

Given Simplify D Given

2.

3.

AB CD

Given

4.

CD BC BD CD

Given Segment Addition Postulate

AB

BC AC

Complete the proofs below.

5.

Given: m HGK m JGL Prove: m 1 m 3

H 1 2 G J K 3 L

Statements m HGK m JGL

Reasons Given

Copyright © by Holt, Rinehart and Winston. All rights reserved.

6.

Given: Prove:

ABC EFG 1 3 2 4

C E H 2 1 3 4 B F G

Statements ABC EFG (m ABC m EFG) 1 3 (m 1 m 3)

Reasons Given

D

A

Geometry

Practice Workbook

11

NAME

CLASS

DATE

Practice

2.5

1.

Conjectures that Lead to Theorems

In the figure at right, write the angle congruent to each given angle.

CSE

2.

CSF

3.

BSE

F D S

A

Find m ABC.

4. C D 5.

E

C

B

A G C 55° F B 25° D

B 75° A E

E

m ABC

Find the value of x.

6. A B (3x 60)° C D (x + 40)° E 7.

m ABC

Q (120 6x)° R S (10x)°

T U V

Copyright © by Holt, Rinehart and Winston. All rights reserved.

x

Complete the proof below.

x

Given:

B and Prove: A

C, A is a complement of D is a complement of C. D

B,

A B C D

Statements A and B are complementary; C and D are complementary; m B m C. m A m B 90 ; m C m D 90 m A m B m C m D m A m B m B m D Therefore, m A m D ( A D)

Reasons Given

8. 9. 10. 11.

12

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

3.1

1.

Symmetry in Polygons

Draw all of the axes of symmetry for each figure.

2. 3.

Each figure below shows part of a shape with reflectional symmetry. Complete each figure.

4. 5. 6.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Each figure below shows part of a shape with the given rotational symmetry. Complete each figure.

7.

6-fold

8.

4-fold

9.

10-fold

Examine each figure below. Identify the type of polygon. Describe all of its symmetries. If it is regular, find the central angle measure.

10. 11. 12.

Geometry

Practice Workbook

13

NAME

CLASS

DATE

Practice

3.2

Properties of Quadrilaterals

Use your conjectures about quadrilaterals from Activities 14 in the textbook to find the indicated measurements. In parallelogram GRAM, MO 10, MA 75°, and m MRG 35°. m GMA

1. 3. 5.

16,

G

R

m GRA RO m RMA

2. 4. 6.

m MGR GR m GMO

M

O

A

In rhombus ABCD, AB

7. 9. 11. 13.

6, AC

8. 10. 12. 14.

8, and m ABC

30°.

A

m ADC BC m BAD CD

m AEB AE m CED EC

C D E B

In rectangle FGHI, FG

15. 17. 19.

8, FI

16. 18. 20.

15, and FH

17.

F

G

Copyright © by Holt, Rinehart and Winston. All rights reserved.

HI GI GJ

GH FJ m FIH

I J

H

In square WXYZ, WX

21. 23.

20 and WY

22. 24.

28.3.

W

X

XY m WVX

XZ

V

m XYV

Z Y

25.

In parallelogram KLMN, m K (3x)° and m L Find x and the measure of each angle in KLMN.

(2x

5)°.

14

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

3.3

1.

Parallel Lines and Transversals

and p are parallel.

1 3 4 2

In the figure at right, lines

List all the angles that are congruent to

1.

5 6 8

2.

List all the angles that are congruent to

2.

7

p

3.

If m 1

115°, find the measure of each angle in the figure.

4.

If m 3 (3x)° and m 7 angle in the figure.

(4x

24)°, find the measure of each

For Exercises 5 8, refer to the diagram below. Lines m and n are parallel. Name all angles congruent to the given angle, and give the theorems or postulates that justify your answer.

5.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

m

n

6

1 3 4 2

6.

8

5 7 6 8

7.

5

8.

7

In KLM, NO ML and measures.

9. 11. 13.

KNO

10. 12. 14.

KON. Find the indicated angle

K 92°

m KNO m NOL m MNL

m KON m LNO m KLN

M 44°

N

O

22° L

Geometry

Practice Workbook

15

NAME

CLASS

DATE

Practice

3.4

If m 5

Proving That Lines Are Parallel

For Exercises 15, refer to the diagram below, and fill in the name of the appropriate theorem or postulate.

1.

m 4, then

m by the converse of the .

1 2 5 6 3 4 7 8

2.

If m 6

m 3, then

m by the converse of the .

3.

If m 1

m 3, then

m by the converse of the .

m

4. 5.

If m 1 If m 6

m 8, then m 7

m by the converse of the m by the converse of the

. .

180°, then

For Exercises 612, use the diagram at right to complete the two-column proof below.

p

q

Given: m 1 pq Prove: m

m 3

1 3

2

Copyright © by Holt, Rinehart and Winston. All rights reserved.

m

Statements pq 1 and m 1 m 1 m 3 3 and m 16

Practice Workbook

Reasons

6.

2 are supplementary. m 2 m 3 m 2 180° 180°

7. 8. 9. 10. 11. 12.

2 are supplementary.

Geometry

NAME

CLASS

DATE

Practice

3.5

1.

The Triangle Sum Theorem

Find the indicated angle measure for each triangle.

2. B 15° F 3. R 80° 55° Q S

A

25°

C

D

130° E

m ABC

m EDF

m RQS

For Exercises 411, refer to the diagram below, in which 45°, and m KLM 35°. NP LM, PL KM , m KNO

4. 6. 8. 10.

m KLP m KON m NOL m MLP

5. 7. 9. 11.

m K m ONM m KOP m P

M N 45°

K

O 35° L

P

Copyright © by Holt, Rinehart and Winston. All rights reserved.

For Exercises 1217, refer to the triangle at right.

m 1

12. 13. 14. 15. 16. 17.

m 2 35°

m 3 60°

m 4

m 1

m 2

R 2

70° 58° 43° 45° 72° 55° 85°

120°

Q

1

3 4 S

What do you notice about the relationship between m 4 and the sum of m 1 m 2? What theorem describes this relationship?

In ABC, m A (3x 10) , m B (2x 5) , and (x 9)°. Find the indicated values. m C

18.

x

19.

m A

20.

m B

21.

m C

Geometry

Practice Workbook

17

NAME

CLASS

DATE

Practice

3.6

1. 115°

Angles in Polygons

Find the unknown angle measures.

65° 2. 110° 3. 80° 80°

120°

?

? 65°

?

For each polygon, determine the measure of an interior angle and the measure of an exterior angle.

4. 6.

a square an equiangular triangle

5. 7.

a regular nonagon an equiangular hexagon

For Exercises 811, an interior angle measure of a regular polygon is given. Find the number of sides of the polygon.

8. 10.

120° 168°

9. 11.

90° 144°

Copyright © by Holt, Rinehart and Winston. All rights reserved.

For Exercises 1215, an exterior angle measure of a regular polygon is given. Find the number of sides of the polygon.

12. 14.

40° 18°

13. 15.

120° 60°

A 6x 4 5x + 1 B

Find each angle measure of trapezoid ABCD.

16. 17. 18. 19.

A B C

3x + 3

4x C

D

D

18

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

3.7

1. C E

Midsegments of Triangles and Trapezoids

Find the indicated measures.

2. X B E F 3. A 7 B

D A

Y F G G D 15 C

16

38

DE

4. A F D 5.

XY

F 60 G 6.

FG

J A 10 S B O D 30 G

C y x

D C

W

C B G

A

B V E

AB WO CD

Copyright © by Holt, Rinehart and Winston. All rights reserved.

FG

AB CD

What special quadrilateral is formed by the shading inside each triangle? Explain.

7. 45° 8. 45°

45°

Geometry

Practice Workbook

19

NAME

CLASS

DATE

Practice

3.8

Analyzing Polygons With Coordinates

In Exercises 14, the endpoints of a segment are given. Determine the slope and midpoint of the segment.

1. 2. 3. 4.

( 1, 1) and (2, 5) (0, 2) and (3, 2) 5)

(4, 3) and (4,

( 6, 1) and ( 3, 0)

In Exercises 58, the endpoints of two segments are given. State whether the segments are parallel, perpendicular, or neither.

5. 6. 7. 8.

(2,

4) and (3, 0); (4,

8) and (6, 0)

( 3, 1) and (1, 2); (5, 2) and (4, 6) (7, 2) and (0, 6); ( 4, 7) and (3, 5) ( 4, 0) and (2, 6); (2, 0) and ( 1, 3)

Graph quadrilateral ABCD with the given vertices on the grid provided. Justify the type of quadrilateral it is.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

9.

A( 2, 3), B(2, 3), C(2,

y

3), D( 2.

3)

10.

D( 1, 5), C(5, 7), A( 1, 0), B(8, 3)

y

O

x

O

x

20

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

4.1

Congruent Polygons

Determine whether the following pairs of figures can be proven to be congruent. Explain your reasoning.

1. 45° 45° 2. 3.

4.

B

E

5.

6.

2 F A C D

2

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Suppose that hexagon ABCDEF

7. 8.

UVWXYZ.

List all pairs of congruent angles. Name the segment that is congruent to each given segment.

a.

BC

b.

XY ADB.

c.

FA

9.

Use the diagram at right to prove that ADC

A

C

D

B

Geometry

Practice Workbook

21

NAME

CLASS

DATE

Practice

4.2

Triangle Congruence

For Exercises 110, some triangle measurements are given. Is there exactly one triangle that can be constructed with those measurements? If so, identify the postulate that applies.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

MNO: MN PQR: m P LJK: JK ABC: AB VWX: VW DBS: DS DEF: DE RST: RS

4, m M 15°, m Q

30°, m N 20°, m R 1°

45° 145°

10, m J 3, BC 4, WX 14, m D 3, EF 13, RT

12°, m K 3, m B 5, m X 10°, m S 3

115° 63° 96°

4, DF

10, m R

70° 120°

GHI: m G STU: TU

20°, m H 5, m T

40°, m I 80°

80°, m U

Determine whether each pair of triangles can be proven congruent by using the SSS, SAS, or ASA Congruence Postulate. If so, identify which postulate is used.

11. M N 12. P Q 13. F

Copyright © by Holt, Rinehart and Winston. All rights reserved.

T

L

O S

R

J

K

L

14.

Draw a pair of triangles, and write the postulate that proves them congruent based upon the markings you drew in your figures.

22

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

4.3

Analyzing Triangle Congruence

For each pair of triangles in Exercises 1 6, is it possible to prove the triangles congruent? If so, write a congruence statement and name the postulate or theorem used.

1. Q T 2. B 3. F G

A P S R U C D J H

4.

N

Y

5.

B

6. W R

X

M OX

Z D

Y E C G A

Copyright © by Holt, Rinehart and Winston. All rights reserved.

For Exercises 79, refer to the diagram at right.

7.

B C A

ABE

CEB by

E

8.

EDC

CBE by

D

9.

EDC

AEB by

10.

Of the three triangles described below, which two are congruent? XYZ: m X ABC: AB KLM: m L 40°, XY 9, m B 9, and m Y 30°, and m A 30° 80° 9

Practice Workbook

30°, m M

110°, and KL

Geometry

23

NAME

CLASS

DATE

Practice

4.4

1. B

Using Triangle Congruence

Find each indicated measure.

2. G 3. Y

A 50° C

15 F H

40° X Z

m B

4. X 5.

FG

K

m Y

6. R 6x 70°

M 3x

S

W

12

Y

L

Q

XY

m K

m S

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Complete the flowchart proof below.

M

N

Given: ML Prove: MP Proof: ML Given MLP Given LP

7.

NP and NL

MLP

NPL

NP

L

P

NPL

MPL

NLP

MP

NL

LP

8.

9.

24

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

4.5

Given MN

Proving Quadrilateral Properties

M N

For Exercises 110, use a parallelogram labeled as shown at right. Find each indicated measure.

1.

2t and SP

(t + 5), find MN.

S P

2.

Given m M

45°, m P

3x°, and NP

x, find NP.

3. 4.

Given m MSP Given m P find NS. Given m M find m M. Given m MNS find m MNS. Given MN Given m P

5x° and m P (x

x°, find m MNP. 5)°, and NS (2x (x (x 10)°, 30)°, 15),

55°, m M (x

5.

20)° and m MNP (5x 10)° and m NSP (40 x), and MS

6.

7. 8.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

3x, SP

2x, find NP.

80°, find m MNP. 120°, m MSP (x 6x°, and NS (x 15),

9.

Given m MNP find NS. Given MS

10.

15, NP

5), and m P

x°, find m M.

Explain whether each pair of triangles can fit together to form a parallelogram without reflecting either triangle.

11. N S T 12. A F 13. Q V X

M

O

R

B

C E

D Z

W

Y

Geometry

Practice Workbook

25

NAME

CLASS

DATE

Practice

4.6

Conditions for Special Quadrilaterals

For Exercises 18, refer to quadrilateral MNOP with diagonals MO and NP intersecting at point Q. For each set of conditions given, state whether the quadrilateral is a parallelogram. If so, give the theorem that justifies your answer.

1.

MN PO and MN

PO

2. 3. 4. 5.

PQ

QN and MQ

QO ON PO NO

MN PO and PQ MN PO and NO MN PO and MP

6. 7. 8.

MP NO and MP MP NO and NO MP NO and MP

PO NQ NO

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Exercises 916 refer to parallelogram CLPK with diagonals CP and LK intersecting at point X. For each condition given below, state whether the parallelogram is a rhombus, a rectangle, or neither. Give the theorem that justifies your answer.

9.

m K

90°

10. 11.

CL

LP m KLP m PCL

m KLC

and m PCK

12. 13. 14. 15. 16.

KL CL

CP KP 90°

m CXL CK CK CL LP

26

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

4.7

1.

Compass and Straightedge Constructions

Construct a figure congruent to each figure below.

2.

3.

4.

Construct the angle bisector of each angle in the triangles below. Using the intersection of the angle bisectors, construct the inscribed circle of each triangle.

5. 6. 7.

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Construct the perpendicular bisector of each side of the triangles below. Using the intersection of the perpendicular bisectors, construct the circumscribed circle of each triangle.

8. 9. 10.

Geometry

Practice Workbook

27

NAME

CLASS

DATE

Practice

4.8

Constructing Transformations

Translate each figure below by the direction and distance of the given translation vector.

1. 2. 3.

4.

5.

6.

State whether each triangle described below is possible. Explain your reasoning.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

7.

EF

12, FG

5, EG

13

8.

SK AL MP

1, KU 104, LD 2.3, PO

1 2 , RS

1, SU

5 51 5.1

9.

53, AD 4.6, MO

3 4 , QS 1

10.

11.

QR KM PQ

18 81 9

12.

64, LM 2 3, QR

5, KL 3 3, PR

13.

28

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

5.1

Perimeter and Area

For Exercises 112, use the figure and measurements below to find the indicated perimeter or area. All measurements are in centimeters.

MS MT XP

1.

15 7 13

US SQ VO

12 20 10

M V

N

U

O

the area of rectangle MNQS

T

X W

P

2.

the perimeter of rectangle MNQS

3.

the area of rectangle MNOU

S R Q

4.

the perimeter of rectangle MNOU

5.

the area of rectangle XPQR

Copyright © by Holt, Rinehart and Winston. All rights reserved.

6.

the perimeter of rectangle XPQR

7.

the perimeter of hexagon SRXWVU the area of PQR

8.

the area of hexagon SRXWVU

9.

10.

If points M and Q were connected by a segment, what would be the area of MQS? If points M and Q were connected by a segment, what would be the perimeter of MQS? If points N and W were connected by a segment, what would be the perimeter of NPW? the perimeter of trapezoid MNWT?

11.

12.

Geometry

Practice Workbook

29

NAME

CLASS

DATE

Practice

5.2

1.

Areas of Triangles, Parallelograms, and Trapezoids

For Exercises 13, find the area of each triangle.

2. 15 10 6 8 3. 17 8

6

For Exercises 46, find the area of each parallelogram.

4. 6 2 3 5. 2 10 1 26 20 6. 10

Copyright © by Holt, Rinehart and Winston. All rights reserved.

For Exercises 79, find the area of each trapezoid.

7. 5 4 11 6 8. 10

5

9.

3 7

18

2

Find the area of the indicated figure.

10. 11. 12. 13.

W

13

X

WVZ parallelogram WXYZ trapezoid WXYV right triangle with hypotenuse XY

V

16

15

17

14

Z 5

Y

30

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

5.3

Circumferences and Areas of Circles

For Exercises 110, find the radius of the circle with the given measurement. Give your answers in terms of exactly and rounded to the nearest tenth.

1.

A

36

2.

C

4

3.

C

3

4.

A

46

5.

A

1 2A 4 5C

212

6.

C

6.2

7.

8

8.

3C

12

9.

17

10.

2A

23

11.

What happens to the area of a circle when the radius is tripled? What happens to the circumference of a circle when the radius is quadrupled? If a 10-inch pizza costs $3 and a 12-inch pizza costs $5, which is the better deal? Explain.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

12.

13.

14.

A solid gold coin with a diameter of 2.9 centimeters sells for $302, and a solid gold coin with diameter 3.1 centimeters sells for $305. If both coins have the same thickness, which is the better deal? Explain.

10

15.

Find the area of the shaded region. The distance from the circle to the center of the shorter base of the trapezoid is 1 unit.

16

Geometry

Practice Workbook

31

NAME

CLASS

DATE

Practice

5.4

The Pythagorean Theorem

For Exercises 18, two side lengths of a right triangle are given. Find the missing side length. Give exact answers.

c a

b 1. 2. 3. 4. 5. 6. 7. 8.

a a a a a a a a

3 2

b b b

7

c c 8 15 12 5

9

c c

6

b b 1 15

c c c

12 13 23

b b b

14

Copyright © by Holt, Rinehart and Winston. All rights reserved.

6

c

Each of the following triples represents the side lengths of a triangle. Determine whether the triangle is right, acute, or obtuse.

9. 10. 11. 12. 13. 14.

7, 4, 6 8, 42, 47 15, 15, 4 339, 565, 452 14, 52, 49 2, 9, 10 What is the length of a diagonal of a square with side lengths of 12? What is the area of an equilateral triangle with side lengths of 4?

15.

16.

32

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

5.5

Special Triangles and Areas of Regular Polygons

For Exercises 15, refer to HGF. For each given length, find the remaining two lengths. Give your answers in simplest radical form.

1. 2. 3. 4. 5.

G

f h g g h

2 3 12 1 5

f

30° h

60°

8

H

g

F

For Exercises 610, refer to XVW. For each given length, find the remaining two lengths. Give your answers in simplest radical form.

6. 7. 8. 9.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

X 45° v

x v w v x

3 9 7 14

45°

w

10.

23

W

x

V

In Exercises 1115, find the area of each figure. Round your answers to the nearest tenth.

11. 12. 4 18 18 7 13.

18 3.5

14. 15.

a square with a diagonal length of 8 2 a square with a diagonal length of 5 feet 33

Geometry

Practice Workbook

NAME

CLASS

DATE

Practice

5.6

The Distance Formula and the Method of Quadrature

Find the distance between each pair of points. Round your answers to the nearest hundredth.

1. 3. 5. 7. 9.

(0, 3) and (2, 1) ( 1, 3) and (2, 5) (7, 3) and (3, 2)

2. 4. 6. 8. 10.

(9, 17) and (4, 5) (4, 10) and (6, 13) (4, 2) and (0, 5)

(1, 1) and (4, 7) (6, 0) and (0, 6)

(3, 8) and (12, 14) ( 7, 11) and ( 1, 1)

Graph each set of points. Use the converse of the Pythagorean Theorem to determine whether the triangle with the given vertices is a right triangle.

11.

(4, 2), (3,

1), and (2, 2)

12.

(1, 7), ( 3, 3), and (5, 3)

Copyright © by Holt, Rinehart and Winston. All rights reserved.

13.

(0, 4), (5, 3), and (7, 9)

14.

(7, 3), ( 2,

2), and (12,

6)

34

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

5.7

Proofs Using Coordinate Geometry

Determine the coordinates of the unknown vertex or vertices of each figure below. Use variables if necessary.

1.

isosceles triangle ABC with AB A(0, 0), B(4, 5), C(?, ?)

y B

CB

2.

parallelogram ABCD A(0, 0), B(a, 0), C(? ?), D(b, c), E(?, ?)

y D C

E A C x A B x

3.

square STUV S(0, a), T(?, ?), U(?, ?), V(0,

y S T x

4.

a)

trapezoid KLMN K(?, ?), L(a, b), M(?, ?), N(0, 0)

y K L

Copyright © by Holt, Rinehart and Winston. All rights reserved.

V

U

x N M

For Exercises 513, refer to the diagram of quadrilateral MNPQ. R, S, T, and U are midpoints. Find the coordinates of each midpoint.

5. 7.

y (2d, 2e) Q U T P(2b, 2c)

S x N(2a, 0)

R T

6. 8.

S

(0, 0) M R

U

Find the slope of each line segment.

9. 13.

RS

10.

TU

11.

RU

12.

ST

Using the results from Exercises 512, draw a conclusion about quadrilateral RSTU.

Geometry

Practice Workbook

35

NAME

CLASS

DATE

Practice

5.8

1.

Geometric Probability

For Exercises 16, refer to the spinner at right.

What is the probability that the spinner will land on 3?

7

8

1 2

2.

What is the probability that the spinner will land on an even number?

6 3 5 4

3.

What is the probability that the spinner will land on a number less than or equal to 3?

4.

What is the probability that the spinner will land on a number larger than 3?

5.

What is the probability that the spinner will land on a prime number?

6.

Add your results from Exercises 3 and 4. What does this result represent in terms of probability?

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Find the theoretical probability that a dart tossed at random onto each figure will land in the shaded area.

7. 8. 9. 2 3

5 6

2 3 8 5

10.

What is the probability that a randomly generated point with 4 x 3 and 6 y 1 will lie in a circle centered at ( 1, 2) with radius 3? with radius 1.5?

36

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

6.1

Solid Shapes

For Exercises 14, refer to the isometric drawing at right. Assume that no cubes are hidden from view.

1.

Give the volume in cubic units.

2.

Give the surface area in cubic units.

3.

Draw six orthographic views of the solid in the space at right. Consider the edge with a length of 4 to be the front of the figure.

Left

4.

On the isometric dot paper provided below, draw the solid from a different view.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Each of the three solids at right have a volume of 6 cubic units.

5.

Draw two other solids with a volume of 6 cubic units that are not just different views of solids A, B, or C.

A.

B.

C.

Geometry

Practice Workbook

37

NAME

CLASS

DATE

Practice

6.2

1.

Spatial Relationships

For Exercises 13, refer to the figure at right.

Name a pair of parallel planes.

F

B 2.

Name two segments skew to BF .

E D A C

3.

Name two segments perpendicular to plane BFD.

4.

In the figure at right, plane M and plane N are parallel. What is the relationship between line p and line q?

p

q

5.

In the figure at right, line r is perpendicular to line p, but line q is not. If the paper is folded along the line p, which angle will have the measure of the dihedral angle?

D p

A

B

C q

Copyright © by Holt, Rinehart and Winston. All rights reserved.

E

F r

For Exercises 6 and 7, indicate whether the statements are true or false for a figure in space. Explain your answer with sketches.

6.

If line m is in plane P and line n is in plane Q and m the plane P is perpendicular to plane Q.

n, then

7.

If line p and line q lie in the same plane, and line q and line r lie in the same plane, then there is one plane which contains all three lines.

8.

If line r is parallel to line s, and line s and line t are skew, then the plane containing r and s and the plane containing t never intersect.

38

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

6.3

Prisms

F E

For Exercises 17, refer to the regular right hexagonal prism at right.

1.

Name the two bases.

G L M

D

2.

Name all segments congruent to BC.

H

J

K C

3.

How are the two bases related?

A

B

4. 5. 6. 7.

List all the lateral faces. Name all segments congruent to CD. What type of quadrilateral is FEKJ? In what manner are the lateral faces related?

In Exercises 8 12, find the length of the diagonal of a right rectangular prism with the given dimensions.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

8.

l

6, w

9, h

12

9.

l

4, w

7, h

2.3

h

10.

l

a, w

a, h

2a

d

11.

l

2a, w

3a, h

4a

w l

Find the missing dimensions.

12. 13. 14.

l l w

10 ft, w 16 in, h 9 cm, h

6 ft, d 21 in, d 8 cm, d

19 ft, h 29 in, w 17 cm, l

Geometry

Practice Workbook

39

NAME

CLASS

DATE

Practice

6.4

1. 3. 5. 7.

Coordinates in Three Dimensions

Name the octant, coordinate plane, or axis for each point.

(1, ( 2,

8, 7) 6, 1.7)

2. 4. 6. 8.

(0, 7, 0) (1, 8, 7)

(7, 0, 8) (0, 0, 2)

( 7, 6, 2) ( 2, 3, 4)

For Exercises 912, locate each pair of points in a threedimensional coordinate system. Find the distance between the points, and find the midpoint of the segment connecting them.

9.

(4, 2, 3) and (8, 5, 5)

z

10.

(8, 3, 5) and ( 3,

z

5,

8)

y

y

Copyright © by Holt, Rinehart and Winston. All rights reserved.

x

x

11.

( 3,

1,

5) and ( 1,

z

2,

3)

12.

( 6, 3,

7) and (4,

z

3, 8)

y

y

x

x

40

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

6.5

x y 3t 1 t

y

Lines and Planes in Space

In the coordinate plane provided, plot the line defined by each pair of parametric equations.

1. 2.

x y

t t

1 1

y

x

x

3.

x y

1 2t

y

4.

x y

t 2

y

Copyright © by Holt, Rinehart and Winston. All rights reserved.

x

x

Recall that a trace of a plane is its intersection with the xy-plane. Find the equation of the trace for each plane defined below.

5. 6. 7. 8.

7x 3x 8x x

4y 8y 4y 2y

2 2z 2z 3z 8 7 10

Geometry

Practice Workbook

41

NAME

CLASS

DATE

Practice

6.6

Perspective Drawing

In Exercises 14, locate the vanishing point for the figure and draw the horizon line.

1. 2.

3.

4.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

5.

In the space below, make a one-point perspective drawing of a rectangular solid. Place the vanishing point below the solid.

6.

In the space below, make a two-point perspective drawing of a rectangular solid. Place the vanishing points below the solid.

42

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

7.1

4 4 3

Surface Area and Volume

Determine the surface-area-to-volume ratio for a rectangular prism with the indicated dimensions. Show all of your steps.

1. 2.

80

1

1

3.

24

24

24

4.

7

9

22

5.

4

16

48

6.

25

14

33

Find the surface-area-to-volume ratio for each solid described below. Show all of your steps.

7.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

a cube with a surface area of 150 square units

8.

a cube with a volume of 512 cubic units

9.

a rectangular prism with dimensions 4

4

4

10.

a cube with a volume of 8000 cubic units

11.

a rectangular prism with dimensions 4

1

1

12.

a cube with a volume of 216 cubic inches

13.

a rectangular prism with a diagonal length of 19 and a base of 10

15

Geometry

Practice Workbook

43

NAME

CLASS

DATE

Practice

7.2

1.

Surface Area and Volume of Prisms

Find the volume of a prism with the given dimensions.

B

40 in.2, h 19 cm2, h 14 cm2, h

5 in.

2.

B

16 m2, h 12 ft2, h 16 ft2, h

6m

3.

B

84 cm

4.

B

8.2 ft

5.

B

10 cm

6.

B

8 ft

Find the surface area and volume of a right rectangular prism with the given dimensions.

7.

14, w

2, h

15

8.

3, w

6, h

2.5

9.

10, w

14, h

4

10.

2.5, w

3, h

5.5

Copyright © by Holt, Rinehart and Winston. All rights reserved.

11.

6.5, w

2.5, h

10

12.

15, w

8, h

20

13.

Find the height of a rectangular prism with a surface area of 560 ft2 and a base of 7 ft 8 ft.

14.

Find the surface area of a right rectangular prism with a height of 6 in. The sides of the base measure 2 in.

15.

A leaning stack of playing cards in the shape of an oblique prism has the same volume as an upright stack of the same height. This is an example of .

16.

One right prism has triangular bases with base and altitude lengths 12 and 9 3, respectively. Another oblique prism has regular hexagonal bases with side lengths of 6. If the height of both prisms is 17, do they have equal volumes?

44

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

7.3

Surface Area and Volume of Pyramids

Find the surface area of each regular pyramid with side length s and slant height given below. The number of sides of the base is given by n. Round your answers to the nearest tenth, if necessary.

1.

n

3, s

14,

14

2.

n

4, s

12,

13

3.

n

6, s

5.2,

13

4.

n

3, s

1.4,

19

Find the volume of each rectangular pyramid with height h and w. Round your answers to the nearest base dimensions tenth, if necessary.

5.

h

14,

17.2, w

15.8

6.

h

7,

3.4, w

15

7.

h

10,

3.5, w

3.5

8.

h

40,

16, w

5

9.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

h

3,

5, w

6

10.

h

16,

8, w

2

In Exercises 1113, find the surface area or the volume of the given pyramid. Round your answers to the nearest tenth, if necessary.

11.

Right triangular pyramid: The side of the equilateral triangular base is 8 cm, the altitude of the pyramid is 12 cm, and its slant height is 4 10 cm. Find its surface area. Right rectangular pyramid: The rectangular base is 9 cm 12 cm, and the altitude of the pyramid is 12 cm. Find its volume. Right square pyramid: The base edges measure 10 cm, the altitude is 12 cm, and its slant height is 13 cm. Find its volume. A pyramid has a right triangle as its base, with leg lengths of 10 cm and 20 cm. If the pyramid's volume is 600 cm3, find its altitude.

Practice Workbook

12.

13.

14.

Geometry

45

NAME

CLASS

DATE

Practice

7.4

Surface Area and Volume of Cylinders

For Exercises 1 and 2, find the surface area and volume of each cylinder.

1. 2.

5 2

2 5

Find the unknown value for a right cylinder with radius r, height h, and surface area S. Round your answers to the nearest tenth.

3.

r

26, h ?

16, S

?

4.

r

4, h

18, S ?

?

5.

r

,h ?

14, S

98

6.

r

1.6, h

,S ?

86

7.

r

0.5, h

,S

4

8.

r

15, h

20, S

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Find the unknown value for a right cylinder with radius r, height h, and volume V. Round your answers to the nearest tenth, if necessary.

9.

r

8, h ?

32, V

?

10.

r

12, h

?

,V ?

144

11.

r

,h

16, V

80

12.

r

5.7, h

6.5, V

13.

The limestone pyramid of Khufu weighs about 16 billion pounds. If it were reconstructed as a solid cylinder, how large could the cylinder be? Give the largest possible heights, to the nearest ten feet, for diameters of 600 feet, 800 feet, and 1000 feet. Note: limestone weighs 167 pounds per cubic foot.

46

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

7.5

Surface Area and Volume of Cones

In Exercises 1 and 2, find the surface area and volume of each right cone. Express your answers in terms of .

1. 2.

12

15 40 41 9 9

Find the surface area of each right cone with radius r, height h, and slant height . Express your answers in terms of .

3.

r

5.1, h

2,

5.5

4.

r

13, h

17,

21.4

Copyright © by Holt, Rinehart and Winston. All rights reserved.

5.

r

4.2, h

3.8,

5.7

6.

r

1.1, h

3,

3.2

Find the volume of each right cone with radius r and height h. Express your answers in terms of .

7.

r

0.5, h

4

8.

r

15, h

20

9.

r

24, h

30

10.

r

8.2, h

9

11.

The volume of a right cone is 1680 cm3. The radius of its base is 12 cm. What is the height of the cone?

12.

The surface area of a right cone is 300 cm2. Its slant height and the diameter of its base are equal. Find its radius.

Geometry

Practice Workbook

47

NAME

CLASS

DATE

Practice

7.6

r 32

Surface Area and Volume of Spheres

Find the surface area and volume of each sphere, with radius r or diameter d. Round your answers to the nearest hundredth.

1. 2.

d

22

3.

r

3.8

4.

d

6.2

5.

r

6

6.

d

18

In Exercises 710, find the surface area and volume of each sphere with radius r or diameter d. Give exact answers in terms of and a variable.

7.

r

86y

8.

d

5.2x

9.

r

(x

1)

10.

r

2.5y

Copyright © by Holt, Rinehart and Winston. All rights reserved.

For Exercises 1114, consider a sphere with a radius of 16 cm.

11.

Find the volume and surface area of the the sphere.

12.

If the radius of the sphere is doubled, what happens to the surface area?

13.

If the radius of the sphere is doubled, what happens to the volume?

14.

If the radius of the sphere is halved, what happens to the surface area and volume?

15.

If the volume is doubled, how much longer is the radius? (Hint: the inverse of the power of 3 is the power of 3 .)

1

16.

If the volume is halved, how much shorter is the radius?

48

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

7.7

Three-Dimensional Symmetry

What are the coordinates of the image when each point below is reflected across the xy-plane in a three-dimensional coordinate system?

1. 3.

(4, 5,

2)

2. 4.

(7, (4, 3,

5, 6) 2)

( 1, 5, 6)

What are the coordinates of the image when each point below is reflected across the yz-plane in a three-dimensional coordinate system?

5. 7.

( 14, 6, 8) ( 2, 1, 3)

6. 8.

(8,

6, 0) 10, 10)

( 10,

What are the coordinates of the image when each point below is reflected across the xz-plane in a three-dimensional coordinate system?

9. 11.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

( 2, 0, 12) ( 6, 6, 0)

10. 12.

( 7, 3, (15, 22,

9) 11)

In Exercises 13 and 14, plot the segment with the given endpoints on the three-dimensional coordinate system provided. Transform each segment by multiplying each x-coordinate by the given number.

13.

A(3, 3, 4) and B(2,

z

1, 2); 3

14.

Q( 3, 4, 1) and R(3,

z

2, 1);

2

y

y

x

x

Geometry

Practice Workbook

49

NAME

CLASS

DATE

Practice

8.1

Dilations and Scale Factors

In Exercises 14, the dashed figures represent the preimages of dilations, and the solid figures represent the images. Find the scale factor of each dilation.

1. y 2. y

O

x x

O

3.

y (5, 4)

4.

y

(2, 1) x (2.5, 2) (3, 1.5)

x

Copyright © by Holt, Rinehart and Winston. All rights reserved.

For Exercises 510, given a point and a scale factor, find the line that passes through the preimage and image, and show that it contains the origin.

5.

(2, 3); n (1, 4); n ( 1, 2); n (3, 4); n ( 4, 3); n 3, 3); n

3 2

1 3

6.

7.

8.

1 2 1

Geometry

9.

10.(

50

Practice Workbook

NAME

CLASS

DATE

Practice

8.2

Similar Polygons

In Exercises 14, determine whether the polygons are similar. Explain your reasoning. If the polygons are similar, write a similarity statement.

1. Q 2. B 17 17 8 P 12 R S 6 T 8 U A 20 C 16 12 24 D 15 F 18 E

3. M

5

N W 2 X

4.

A

D 1.8 E

1.4 3.5 4.2 F 1.2

15

6 Z Y

Copyright © by Holt, Rinehart and Winston. All rights reserved.

P

O

C

3.0

B

Solve each proportion for x.

5.

x 18

1 4

22 12

5 8

6.

22 x

2 18

7.

16

x

8.

8 x

x 50 2x 1 21

9.

3 x 3 x

5 1

10.

3x 7 15

Geometry

Practice Workbook

51

NAME

CLASS

DATE

Practice

8.3

Triangle Similarity

Determine whether each pair of triangles can be proven similar by using AA, SSS, or SAS. If so, write a similarity statement, and identify the postulate or theorem used.

1. E B 2 A 2 C D 3 F 1.5 2.25 G 3 I 30° H M 60° L 2. K

3.

O Q 64°

4.

T

W 1.5

3 N 61° 61° P S 58°

X

1.25

Y

R U 2.5 V

Copyright © by Holt, Rinehart and Winston. All rights reserved.

In Exercises 5 and 6, indicate which figures are similar. Explain your reasoning.

5. 12 12 10 I G 10 K 1.5 L Q 65° R H C D 7.5 B 9 4 5 E 3 F 3 25° N P 1.5 O 6. J 2 M

A

52

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

8.4

1. 5 x 10 18

The Side-Splitting Theorem

Use the Side-Splitting Theorem to find x.

2. 12 8 24 x

3.

2.5 7.5

4. x1

x+2 x

x 12

x2

5.

4+x x+1

6.

2x + 1

2x + 3

Copyright © by Holt, Rinehart and Winston. All rights reserved.

3 x

x+2 9

Name all similar triangles in each figure. State the postulate or theorem that justifies each similarity.

7.

AB BC, BD AC

A D

8.

EDA DAC

B

E

D

B

C

A

C

Geometry

Practice Workbook

53

NAME

CLASS

DATE

Practice

8.5

Indirect Measurement and Additional Similarity Theorems

In Exercises 14, use the diagrams to find the height of each building.

1. 2. 30 ft 45 ft

36 ft 30 ft 40 ft

16 ft

3.

20 ft 24 ft 18 ft

4.

16 ft 12 ft 24 ft

In Exercises 58, the triangles are similar. Find x.

5. 6. 6.4 8 x 10 x 6 4.8 9

Copyright © by Holt, Rinehart and Winston. All rights reserved.

7. 8 6 3.6 x

8. 5.2 x 3.9 4

54

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

8.6

1.

Area and Volume Ratios

A D B E C

In ABC, D and E are midpoints. What fraction of the area of ABC is ADE?

The ratio of the corresponding sides of two similar triangles is 5 . Find the ratio of the following:

2.

3

their altitudes

3.

their perimeters

4.

their areas

The side lengths of two squares are 4 cm and 9 cm. Find the ratio of the following:

5.

their diagonals

6.

their perimeters

7.

their areas

Two spheres have radii of 6 cm and 8 cm. Find the ratio of the following:

8.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

the circumferences of their great circles

9.

their surface areas

10.

their volumes

The ratio of the base areas of two similar cones is 25 . Find the ratio of the following:

11.

16

the circumference of their bases

12.

their heights

13.

their volumes

14.

Two cubes have volumes of 3375 and 1331. What is the ratio of their heights? Suppose that the triangles from Exercise 1 are bases of two prisms with the same height. What is the ratio of the volume of the prism with ADE as a base to the volume of the prism with ABC as a base?

15.

Geometry

Practice Workbook

55

NAME

CLASS

DATE

Practice

9.1

Chords and Arcs

Determine the length of an arc with the given central angle measure, m W, in a circle with radius r. Round your answers to the nearest hundredth.

1. 3. 5. 7. 9. 11.

m W m W m W m W m W m W

20°; r 70°; r 45°; r 15°; r 25°; r 53°; r

1 24 4 7 12 7

2. 4. 6. 8. 10. 12.

m W m W m W m W m W m W

26°; r 110°; r 10°; r 25°; r 14°; r 123°; r

18 6 5 6 13 18

Determine the degree measure of an arc with the given length, L, in a circle with radius r. Round your answers to the nearest hundredth.

13. 15. 17. 19. 21. 23.

L L L L L L

15; r 3; r 12; r 23; r 12; r 7; r 4

14 10 15 14 6

14. 16. 18. 20. 22. 24.

L L L L L L

22; r 25; r 33; r 6; r 3; r 2; r 4 2 1

50 20 13

Copyright © by Holt, Rinehart and Winston. All rights reserved.

For Exercises 2528, find the degree measure of each arc by using the central angle measures given in circle F.

25. 26. 27. 28.

A B 110° E

mAB mAEC mEC mBDE

F

60° 45° C

125°

D

56

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

9.2

Refer to

1. 2.

Tangents to Circles

K, in which KG MH at F, for Exercises 18.

M

MF KG is congruent to which two segments?

G 3.

If KH 2 and KF What is FH?

1, what is MF?

F

K

H 4.

If KH 12 and KF What is FH?

3, what is MF?

5.

If KM 16 and KF What is FH?

10, what is MF?

6.

If KM 64 and KF What is FH?

20, what is MF?

Copyright © by Holt, Rinehart and Winston. All rights reserved.

7.

If KG 14 and FG What is MH?

11, what is MF?

8.

If KG 20 and FG What is MH?

12, what is MF?

9.

In the diagram of O, AB CD at E. If CD 8 and OE 3, find the length of the radius.

A O C E B D

10.

In the diagram of P, the radius is 10, and the distance from the center P to chord RS is 6. Find RS.

R T P S

Geometry

Practice Workbook

57

NAME

CLASS

DATE

Practice

9.3

1.

Inscribed Angles and Arcs

2.

XYZ is inscribed in the circle. If Y Z and mYZ 100o, find m Z.

X

OA, OB, CA, and CB are drawn in the circle with center O. If m AOB 60°, find m ACB.

A

O C Y Z

3.

PQR is inscribed in the circle. If m P 70° and mPR 120°, find m R.

Q

4.

Quadrilateral KLMN is inscribed in the circle. If mKL 124° and mLM 78o, find m KNM.

L

P

R

K

M

N

Copyright © by Holt, Rinehart and Winston. All rights reserved.

In T, m XTU 35°, m VWU 50°, and WU is a diameter. Find the following:

5. 6. 7. 8. 9. 10. 11. 12.

X

m XWT m UTV m WUV m XTW mWXU m WTV m UVT mXU

V T W U

58

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

9.4

In to

1. 2. 3.

Angles Formed by Secants and Tangents

50°. NH is tangent

L, mIK 15°, mKM 180°, and mHM L at H. Find each of the following:

m JHN m NJH m JNH

J I H K L M N

In the figure, QR is tangent to

4.

S at Q.

Q S P

If mQP

105°, find m RQP.

5.

If m RQP

110°, find mQP .

R

6.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

If m RQP

90°, find mQP .

7.

If mQP

x

35, find m RQP.

8.

If mQP

2x

17, find m RQP.

In the figure at right, PA and PD are secants to the circle, chords AC and BD intersect at E, BA CD , mBC 40°, 60°. Find the following: and m ABD

9. 10. 11. 12. 13.

m AB m ACD m AEB m BDP m P

A B E C D P

Geometry

Practice Workbook

59

NAME

CLASS

DATE

Practice

9.5

Segments of Tangents, Secants, and Chords

H, the radius of H is 4, and ED 10.

ED and EG are tangent to Find the following:

1. 2. 3. 4. 5.

EG DH HE FE Name a right angle.

F D H G

6.

Name a pair of complementary angles.

E

7.

Name an angle congruent to Name an arc congruent to DF .

DEH.

8.

For Exercises 9 and 10, refer to the figure at right. Chords AB and CD intersect inside the circle at E.

9.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

If AE If AE

6, EB 10, EB

8, and CE 9, and CE

4, find ED.

A

D

10.

6, find ED.

C

E B

For Exercises 1113, refer to the figure at right. PC is a tangent and PB is a secant to the circle.

11.

If PC If BP If BA

8 and PA 16 and PA 5 and PA

4, find BA. 4, find PC. 4, find PC.

B

C P

12.

A

13.

60

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

9.6

1.

Circles in the Coordinate Plane

Find the x- and y- intercepts for the graph of each equation.

x2

y2

144

2.

x2

y2

24

3.

x2

y2

25

4.

(x

1)2

y2

4

5.

(x

5)2

(y

5)2

25

6.

(x

14)2

(y

10)2

16

Find the center and radius of each circle.

7.

x2

y2

169

8.

x2

y2

63

Copyright © by Holt, Rinehart and Winston. All rights reserved.

9.

(x + 12)2

y2

225

10.

(x

3)2

y2

81

Write an equation for the circle with the given center and radius.

11.

center: (3, 12); radius

5

12.

center: ( 2, 6); radius

7

13.

center: (0, 0); radius

5

14.

center: (2,

3); radius

7

Geometry

Practice Workbook

61

NAME

CLASS

DATE

Practice

10.1

1. A

Tangent Ratios

Find tan A for each triangle below.

2. 2.4 9 10.82 6 A 6.46

6

3.

A

4.

A 4.39 1.8

3.5 1.8

3

4

5.

6. 1.5 32 A A 2

Copyright © by Holt, Rinehart and Winston. All rights reserved.

2.5

For Exercises 79, use the definition of tangent ratio to write an equation involving x. Find the tangent of the given angle with a calculator, and solve the equation to find the unknown side of the triangle. Round your answers to the nearest hundredth.

7. x 41° 14 x 17.7 8. 68° 15 9. x 12 48°

62

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

10.2

Sines and Cosines

ABC.

B

For Exercises 110, refer to Find each of the following:

1. 2. 3. 4. 5. 6. 7. 8. 9.

sin A sin B cos A cos B tan A tan B m A m B (sin A)2 (cos A)2

10. A 4 C 5 3

(sin B)2

(cos B)2

Copyright © by Holt, Rinehart and Winston. All rights reserved.

For Exercises 1116, refer to Find each of the following:

11.

DEF.

5

E 1

sin cos cos tan

1 5 1 5 2 5 1 2

14. D

12.

2

F

13.

sin tan

2 5

15.

16.

2

Use a scientific or graphics calculator for Exercises 1725. Round your answers to the nearest hundredth.

17. 20. 23.

sin 22° cos 33° cos 54°

18. 21. 24.

cos 78° sin 18° sin 82°

19. 22. 25.

tan 12° tan 2° tan 76°

Geometry

Practice Workbook

63

NAME

CLASS

DATE

Practice

10.3

sin 76° cos 76° sin 95° cos 95° sin 58°

Extending the Trigonometric Ratios

Use a calculator to find each of the following, rounded to four decimal places:

1. 4. 7. 10. 13. 2. 5. 8. 11. 14.

sin 129° cos 129° sin 183° cos 183° cos 58°

3. 6. 9. 12. 15.

sin 307° cos 307° sin 359° cos 359° sin 32°

In Exercises 1624, use a calculator to find the sine and cosine of each angle. Round your answers to four decimal places, and write these values as x- and y-coordinates of a point at the given angle on the unit circle.

16.

10°

17.

50°

18.

130°

19.

230°

20.

250°

21.

310°

22.

25°

23.

115°

24.

205°

Copyright © by Holt, Rinehart and Winston. All rights reserved.

In Exercises 2536, give two values of angle between 0° and 180° for the given value of sin . Round your answers to the nearest degree.

25.

0.9511

26.

0.6691

27.

0.3584

28.

0.5150

29.

0.7880

30.

0.8005

31.

0.2122

32.

0.7194

33.

0.0914

34.

0.2588

35.

0.5878

36.

0.9511

64

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

10.4

The Law of Sines

B a c C A b

In Exercises 19, find the indicated measures. Assume that all angles are acute. It may be helpful to sketch the triangle roughly to scale. Round your answers to the nearest tenth.

1. 2. 3. 4. 5. 6. 7. 8. 9.

m A m A m B m B m B m A m A m A m A

48° 37° 78° 78° 25° 40° 41° 72° 35°

m B m B

73° 80°

b a b b a c a c b

1.7 cm 3.4 cm 2.63 cm 2.63 cm 5.2 cm 3.62 cm 14 cm 15 cm 12 cm

a c m A c b a b a m B

? ? ? ? ? ? ? ? ?

P q

a = 1.45 cm a 1.45 cm 80° 64° 58° 40°

m C m B m B m B a

8 cm

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Find all unknown sides and angles for each triangle described below. If the triangle is ambiguous, give both possible angles. It may be helpful to sketch the triangle roughly to scale. r

Q 10.

R p

m P

25°, m Q

55°, q

10

11.

m Q

30°, m R

70°, r

8

12.

m P

42°, m R

34°, q

9

13.

m R

48°, p

3, r

2.5

14.

m P

35°, m R

41°, q

23

15.

m Q

53°, m R

72°, p

26

Geometry

Practice Workbook

65

NAME

CLASS

DATE

Practice

10.5

The Law of Cosines

B a c C A b

In Exercises 15, find the indicated measures. It may be helpful to sketch the triangle roughly to scale. Round your answers to the nearest tenth.

1. 2. 3. 4. 5.

a a a a a

19 8 6 5 3

b b b b b

20 9 6 6 4

c c c c c 10 9 8

?

m C m C m B m A

50° ? ? ? ?

33

m C

Solve each triangle.

6. D 2.1 3.9 F I 39° g 41° H

Copyright © by Holt, Rinehart and Winston. All rights reserved.

3.5

E

7. 2.5

G i

8. 3 J 55° 3

L j K

9.

M n 35° 3

2 N

O

10. 32°

B

11. z 25° x X 22° 5 Z

Y

2.5 45° b

a

A

C

66

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

10.6

Vectors in Geometry

Draw the vector sum a b by using the head-to-tail method. You may need to translate one of the vectors.

1. a b 2. a b

3.

4.

a b

a b

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Draw the vector sum a b by using the parallelogram method. You may need to translate the vectors.

5. a 6.

b

b a

7. b a

8. b

a

Geometry

Practice Workbook

67

NAME

CLASS

DATE

Practice

10.7

P(0, 2); P(3, 4); P( 3, 2); P(5, 3); P( 2, 30° 180°

Rotations in the Coordinate Plane

For Exercises 110, a point and an angle of rotation are given. Determine the coordinates of the image, P . Round your answers to the nearest tenth.

1. 3. 5. 7. 9. 2. 4. 6. 8.

P( 2, 2); P(7, 0); P( 3, P(3,

45° 200° 275° 132° 400°

12° 38° 10°

2); 5);

4);

10.

P( 3, 1);

Find the rotation matrix for each angle of rotation below by filling in the sine and the cosine values. Round your answers to the nearest hundredth.

11.

15° matrix

12.

90° matrix

13.

150° matrix

14.

170° matrix

Copyright © by Holt, Rinehart and Winston. All rights reserved.

15.

210° matrix

16.

310° matrix

17.

50° matrix

18.

200° matrix

19.

10° matrix

20.

215° matrix

68

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

11.1

Golden Connections

Determine the indicated side length of each golden rectangle. Round your answers to the nearest hundredth.

1. 14 2. ?

?

20

3. 6 ?

4. 2

?

2 ?

5.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

6. 13 ?

?

11

7.

8.

7

20

?

?

?

9.

The golden ratio is equal to the fraction and approximately equal to the decimal .

Geometry

Practice Workbook

69

NAME

CLASS

DATE

Practice

11.2

1.

Taxicab Geometry

Find the taxidistance between each pair of points.

(2, 4) and (4, 7)

2.

( 3, 6) and (5, 4)

3.

( 1,

4) and ( 3,

6)

4.

(100, 82) and (82, 100)

5.

(7,

2) and ( 6, 4)

6.

( 2, 3) and (8, 0)

Find the number of points on the taxicab circle with the given radius.

7. 9.

r r

6 21

8. 10.

r r

14 12

Find the circumference of the taxicab circle with the given radius.

11. 13.

r r

40 17

12. 14.

r r

13 28

Copyright © by Holt, Rinehart and Winston. All rights reserved.

In Exercises 15 and 16, plot the taxicab circle described onto the grid and find its circumference.

15.

center O at (0, 0); radius of 4 units

y

16.

center C at ( 2, 3); radius of 5 units

y

O

x

O

x

70

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

11.3

Graph Theory

Determine whether the graphs below contain an Euler path, an Euler circuit, or neither.

1. A 2. B

A B C C

3. A

B

4.

A

B D

Copyright © by Holt, Rinehart and Winston. All rights reserved.

C

C

5.

6. D

C

A

B

C

D A

E B

7.

A

8.

B

C B C D D

A

Geometry

Practice Workbook

71

NAME

CLASS

DATE

Practice

11.4

Topology: Twisted Geometry

In Exercises 13, determine the number of regions into which the plane is divided by the curve.

1. 2. 3.

Which of the shapes below are topologically equivalent?

4. 5.

A

B

C

For Exercises 6 9, refer to the simple closed curve at right.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

6.

Is point J on the inside or the outside of the curve? Is point M inside or outside?

J

7.

Can you draw a line connecting points J and M that does not intersect the curve? What theorem justifies your answer?

M

8.

Into how many regions does the curve divide the plane?

9.

Draw a point and connect it to J or M with a line that does not intersect the curve.

72

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

11.5

Euclid Unparalleled

For Exercises 15, refer to the figure that shows line , points K and P, and line m on the surface of Poincaré's model of hyperbolic geometry.

1.

Are perpendicular lines possible in Poincaré's model?

P

2.

Are rectangles possible in this model?

K m

3.

Are lines m and parallel?

4.

How many lines are possible through point K that are parallel to line ?

5.

Is there a line through point P that is parallel to line ? If so, how many are possible?

Copyright © by Holt, Rinehart and Winston. All rights reserved.

6.

Is there a line through point P that is parallel to line m? If so, how many are possible?

For Exercises 79, refer to the figure that shows line surface of a sphere.

7.

on the

Are perpendicular lines possible in the spherical geometry?

8.

Do two points determine a line in spherical geometry?

9.

Given that the radius of the sphere is 1, what is the length of any line?

Geometry

Practice Workbook

73

NAME

CLASS

DATE

Practice

11.6

Fractal Geometry

Here are the first two levels in the construction of a fractal. Level 0: Begin with a line segment. Level 1: Divide the segment into thirds. Replace the middle third with two segments whose lengths are each one-third the length of the original segment.

1.

In the space provided, construct the next level of the fractal by replacing each segment of the figure with a one-third replica of level 1.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

2.

Follow these instructions to create a fractal. Level 0: Begin with a long strip of paper. Level 1: Fold the strip's right end onto its left end, and crease it. Open the strip so that it forms a right angle. Lay the strip on its edge in an L-shape on your desk. Look down at the open strip and sketch this view. Level 2: Start over. Refold the strip as in Level 1. Now fold the right end onto the left end a second time. Open the strip to form right angles. Lay the strip on your desk, look down at it, and sketch this view. Level 3: Start over again. Refold levels 1 and 2. Add level 3 by folding again. Open the strip and sketch as before. Level 4: Start over. Refold levels 1, 2, and 3. Add level 4 by folding again. Open the strip and make the final sketch.

74

Practice Workbook

Geometry

NAME

CLASS

DATE

Practice

11.7

Other Transformations: Projective Geometry

For Exercises 1 and 2, sketch the preimage and image for each affine transformation on the given set of axes.

1.

square: O(0, 0); P( 2, 0); Q( 2, 2); R(0, 2) T(x, y) (2x, 2y)

y

2.

triangle: A(2, 2); B(6, 0); C(8, 4) T(x, y)

y

( 1 x, y) 2

O x

x

Copyright © by Holt, Rinehart and Winston. All rights reserved.

For Exercises 38, use the figure below.

If point P is the center of projection, then

3. 4.

F

the projective rays are the projection of A onto line

2 is

.

E P

.

5.

3

the projection of B onto line

2

is .

1 2

A

B

If point Q is the center of projection, then

6. 7.

C

the projective rays are the projection of A onto line

3

. is

D Q

.

8.

the projection of B onto line

3

is

.

Geometry

Practice Workbook

75

NAME

CLASS

DATE

Practice

12.1

1.

Truth and Validity in Logical Arguments

In Exercises 16, write a valid conclusion from the given premises. Identify the form of the argument.

If it is a weekend, then José is not at work. It is a weekend.

2.

If it is a weekday, then José is at work. José is not at work.

3.

If José is not at work, then he is with Anna. José is not with Anna.

4.

If José is at work, then he is not with Anna. José is with Anna.

5.

If it is a weekday, then José is at work. If José is at work, then he is wearing a tie. It is a weekday.

6.

If it is a weekend, then José is not at work. If José is not at work, then he is wearing jeans. It is a weekend.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

You are given the following premises: If you exercise, then you are energized. Ian exercised. Jon did not exercise. Jean was energized. Johan was not energized.

Which of the following conclusions are valid? When possible, name the argument form.

7. 8. 9. 10. 11. 12. 13. 14.

Ian was energized. Jon was energized. Jean had exercised. Johan had exercised. Ian was not energized. Jon was not energized. Jean had not exercised. Johan had not exercised.

Practice Workbook Geometry

76

NAME

CLASS

DATE

Practice

12.2

And, Or, and Not in Logical Arguments

Write the conjunction of each pair of statements. Determine whether the conjunction is true or false.

1.

A cat is a mammal. France is a country.

2.

Butterflies have wings. 3 is an even integer.

Write the disjunction of each pair of statements. Determine whether the disjunction is true or false.

3.

A ray is of finite length. A triangle has 3 sides.

4.

Dogs have 4 legs. Birds have 2 legs.

In Exercises 512, write the statement expressed by the symbols, where p, q, r, and s represent the statements shown below.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

p: 3 is an odd integer. q: 2 is prime. r: 7 is an even integer. s: 26 is a perfect square.

5.

p AND q

6.

(q OR s)

7.

q AND

s

8.

q OR

p

9.

p

10.

p AND (q OR r)

11.

(p AND s) OR

q

12.

q OR r

Geometry

Practice Workbook

77

NAME

CLASS

DATE

Practice

12.3

A Closer Look at If-Then Statements

For each conditional in Exercises 13, explain why it is true or false. Then write the converse, inverse, and contrapositive, and explain why each is true or false.

1.

Conditional: If Aja lives in Texas, she lives in the United States. Converse:

Inverse:

Contrapositive:

2. Conditional:

If 2

2

3, then 7

8

56.

Converse:

Inverse:

Contrapositive:

3.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Conditional: If an angle lies in the first quadrant, then the sine of the angle is positive. Converse:

Inverse:

Contrapositive:

For Exercises 46, write each statement in if-then form.

4.

Cindy does not eat meat if she is a vegetarian.

5. 6.

3 is an odd number. I will eat when I am hungry.

Practice Workbook Geometry

78

NAME

CLASS

DATE

Practice

12.4

Indirect Proof

For Exercises 13, determine whether the given argument is an example of indirect reasoning. Explain why or why not.

1.

If Tom joined the army, then his hair was cut short. If his hair was cut short, then it does not cover his ears. But Tom's hair does cover his ears. Therefore, Tom did not join the army.

2.

If I am sleepy, then I yawn. I am yawning. Therefore, I am sleepy.

3.

If it were raining, then I would be holding an umbrella. I am not holding an umbrella. Therefore, it is not raining.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Complete the indirect proof below. In Euclidean geometry, a triangle cannot have two right angles.

Given: ABC Prove: A and

B cannot both be right angles. .

Proof: Suppose that 4.

It is a property of triangles in Euclidean geometry that the sum of the measures of the three angles is equal to 5.

6. 8.

. Thus m A A and B are 7. m C

m B ,

10.

m C

. Since

9. 11.

.

Thus, m C

12. 13.

. This is a contradiction because . Therefore, .

Practice Workbook

Geometry

79

NAME

CLASS

DATE

Practice

12.5

1.

Computer Logic

Use the logic gates below to answer each question.

If p

p q

1 and q

NOT

0, what is the output?

AND

2.

If p

p q s

0, q

1, and s

OR

0, what is the output?

AND

In Exercises 318, complete the input-output table for each network of logic gates.

p

3. 4. 5. 6.

q 1 0 1 0 q 1 0 1 0 q 1 1 0 0 1 1 0 0

p

p AND q

p OR (p AND q)

p p q

NOT OR AND

1 1 0 0 p

q

p AND

q

(p AND

q)

p q AND NOT NOT

Copyright © by Holt, Rinehart and Winston. All rights reserved.

7. 8. 9. 10.

1 1 0 0 p

r 1 0 1 0 1 0 1 0

q OR r

p AND (p OR r)

p q r OR AND

11. 12. 13. 14. 15. 16. 17. 18.

1 1 1 1 0 0 0 0

80

Practice Workbook

Geometry

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