CliffsStudySolver Geometry. By David Alan Herzog

Transcription

1 CliffsStudySolver Geometry By David Alan Herzog

2

3 CliffsStudySolver Geometry

4

5 CliffsStudySolver Geometry By David Alan Herzog

6 Published by: Wiley Publishing, Inc. 111 River Street Hoboken, NJ Copyright 2004 Wiley, Hoboken, NJ Published by Wiley, Hoboken, NJ Library of Congress Cataloging-in-Publication Data Herzog, David Alan. Geometry / by David A. Herzog. 1st ed. p. cm. (CliffsStudySolver) ISBN (pbk.) 1. Geometry Outlines, syllabi, etc. 2. Geometry Problems, exercises, etc. I. Title. II. Series. QA461.H dc ISBN: Printed in the United States of America B/QX/QW/QU/IN No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, , fax Requests to the Publisher for permission should be addressed to the Legal Department, Wiley Publishing, Inc., Crosspoint Blvd., Indianapolis, IN 46256, , or fax THE PUBLISHER AND THE AUTHOR MAKE NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE AC- CURACY OR COMPLETENESS OF THE CONTENTS OF THIS WORK AND SPECIFICALLY DISCLAIM ALL WAR- RANTIES, INCLUDING WITHOUT LIMITATION WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE. NO WARRANTY MAY BE CREATED OR EXTENDED BY SALES OR PROMOTIONAL MATERIALS. THE ADVICE AND STRATEGIES CONTAINED HEREIN MAY NOT BE SUITABLE FOR EVERY SITUATION. THIS WORK IS SOLD WITH THE UNDERSTANDING THAT THE PUBLISHER IS NOT ENGAGED IN RENDERING LEGAL, ACCOUNTING, OR OTHER PROFESSIONAL SERVICES. IF PROFESSIONAL ASSISTANCE IS REQUIRED, THE SERVICES OF A COMPETENT PRO- FESSIONAL PERSON SHOULD BE SOUGHT. NEITHER THE PUBLISHER NOR THE AUTHOR SHALL BE LIABLE FOR DAMAGES ARISING HEREFROM. THE FACT THAT AN ORGANIZATION OR WEBSITE IS REFERRED TO IN THIS WORK AS A CITATION AND/OR A POTENTIAL SOURCE OF FURTHER INFORMATION DOES NOT MEAN THAT THE AUTHOR OR THE PUBLISHER ENDORSES THE INFORMATION THE ORGANIZATION OR WEBSITE MAY PRO- VIDE OR RECOMMENDATIONS IT MAY MAKE. FURTHER, READERS SHOULD BE AWARE THAT INTERNET WEBSITES LISTED IN THIS WORK MAY HAVE CHANGED OR DISAPPEARED BETWEEN WHEN THIS WORK WAS WRITTEN AND WHEN IT IS READ. Trademarks: Wiley, the Wiley Publishing logo, CliffsNotes, the CliffsNotes logo, Cliffs, CliffsAP, CliffsComplete, CliffsQuickReview, CliffsStudySolver, CliffsTestPrep, CliffsNote-a-Day, cliffsnotes.com, and all related trademarks, logos, and trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc. and/or its affiliates. All other trademarks are the property of their respective owners. Wiley Publishing, Inc. is not associated with any product or vendor mentioned in this book. For general information on our other products and services or to obtain technical support, please contact our Customer Care Department within the U.S. at , outside the U.S. at , or fax Wiley also published its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Note: If you purchased this book without a cover, you should be aware that this book is stolen property. It was reported as unsold and destroyed to the publisher, and neither the author nor the publisher has received any payment for this stripped book.

7 This book is dedicated to Francesco, Sebastian, and Gino Bubba, Jakob Cherry, Jeremy Cullen, Hailee Foster, Rocio, Kira, and Myles Herzog, their parents, and Uncles Dylan and Ian, and Karen Herzog.

8

9 About the Author David Alan Herzog is the author of numerous books concerned with test preparation in mathematics and science. Additionally, he has authored over one hundred educational software programs. Prior to devoting his full energies to authoring educational books and software, he taught math education at Fairleigh Dickinson University and William Paterson College, was mathematics coordinator for New Jersey s Rockaway Township Public Schools, and taught in the New York City Public Schools.

10 Publisher s Acknowledgments Editorial Project Editor: Suzanne Snyder Acquisitions Editor: Greg Tubach Technical Editor: Tom Page Editorial Assistant: Amanda Harbin Composition Project Coordinator: Kristie Rees Indexer: Sherry Massey Proofreader: Arielle Mennelle Wiley Publishing, Inc. Composition Services

11 Table of Contents Geometry Pretest 1 Answers 28 Chapter 1: Basic Geometric Ideas 37 Naming Basic Forms 37 Points 37 Lines 37 Planes 38 Postulates and Theorems 38 Finding Segments, Midpoints, and Rays 41 Line Segments 41 Segment Addition and Midpoint 42 Rays 43 Angles and Angle Pairs 45 Forming and Naming Angles 45 The Protractor Postulate and Addition of Angles 46 Angle Bisector 47 Right Angles 47 Acute Angles 48 Obtuse Angles 48 Straight Angles 48 Reflex Angles 48 Special Angle Pairs 50 Adjacent Angles 50 Vertical Angles 50 Complementary Angles 51 Supplementary Angles 51 Special Lines and Segments 53 Intersecting Lines and Segments 53 Perpendicular Lines and Segments 54 Parallel Lines and Segments 54 Chapter Problems 55 Problems 55 Answers and Solutions 56

12 x CliffsStudySolver Geometry Supplemental Chapter Problems 57 Problems 57 Answers 58 Chapter 2: Parallel Lines 59 Angles Created by Lines and a Transversal 59 Angles Created by Parallel Lines and a Transversal 60 Proving Lines Parallel 61 Chapter Problems 64 Problems 64 Answers and Solutions 65 Supplemental Chapter Problems 67 Problems 67 Answers 69 Chapter 3: Triangles 71 Sum of Angle Measure 71 Exterior Angles 72 Classifying Triangles by Angles 75 Classifying Triangles by Sides 76 Specially Named Sides and Angles 78 Segments Inside and Outside Triangles 80 Base and Altitude 80 Median 81 Angle Bisector 81 Congruent Triangles 83 Proofs of Congruence 83 Corresponding Parts (CPCTC) 89 Isosceles Triangles 91 Triangle Inequality Theorems 92 Chapter Problems 96 Problems 96 Answers and Solutions 99 Supplemental Chapter Problems 103 Problems 103 Answers 106 Chapter 4: Polygons 107 Types of Polygons 107 Naming Polygons Parts 108 Number of Sides and Angles 108 Regular Polygons 109

13 Table of Contents xi Angle Sums 109 Quadrilaterals 112 Trapezoids 112 Parallelograms 113 Proofs of Parallelograms 115 Special Parallelograms 118 Rectangle 118 Rhombus 119 Square 119 Special Trapezoids 121 The Midpoint Theorem 122 Chapter Problems 124 Problems 124 Answers and Solutions 125 Supplemental Problems 128 Problems 128 Answers 129 Chapter 5: Perimeter and Area 131 Squares and Rectangles 131 Finding the Perimeter 131 Finding the Area 132 Triangles 135 Finding the Perimeter 135 Finding the Area 135 Parallelograms 137 Finding the Perimeter 137 Finding the Area 138 Trapezoids 140 Finding the Perimeter 141 Finding the Area 141 Regular Polygons 142 Special Parts of Regular Polygons 143 Finding the Perimeter 143 Finding the Area 143 Circles 144 Finding Circumference 145 Finding the Area 146

14 xii CliffsStudySolver Geometry Chapter Problems 148 Problems 148 Answers and Solutions 149 Supplemental Chapter Problems 152 Problems 152 Answers 153 Chapter 6: Similar Figures 155 Ratio and Proportion 155 Ratio 155 Proportions 156 Means and Extremes 156 Properties of Proportions 157 Similar Polygons 160 Similar Triangles 161 Proportional Parts of Triangles 164 Proportional Parts of Similar Triangles 169 Perimeter and Areas of Similar Triangles 171 Chapter Problems 174 Problems 174 Answers and Solutions 176 Supplemental Chapter Problems 180 Problems 180 Answers 182 Chapter 7: Right Triangles 185 Geometric Mean 185 Altitude to the Hypotenuse 186 The Pythagorean Theorem 189 Pythagorean Triples 191 Outgrowths of the Pythagorean Theorem 194 Special Right Triangles 195 Isosceles Right Triangle Right Triangle 197 Chapter Problems 199 Problems 199 Answers and Solutions 200 Supplemental Chapter Problems 203 Problems 203 Answers 205

15 Table of Contents xiii Chapter 8: Circles 207 Parts of a Circle 207 Central Angles and Arcs 209 Central Angles 210 Arcs 210 Some Common Sense Stuff 211 Arcs and Inscribed Angles 212 Angles Formed by Chords, Secants, and Tangents 215 Arcs and Chords 219 Segments of Chords, Secants, and Tangents 222 Arc Lengths and Sectors 225 Arc Length 226 Sectors of a Circle 226 Chapter Problems 230 Problems 230 Answers and Solutions 232 Supplemental Chapter Problems 235 Problems 235 Answers 237 Chapter 9: Solid Geometry 239 Prisms 239 Oblique vs. Right Prisms 239 Volume of a Prism 244 Right Circular Cylinders 245 Pyramids 249 Right Circular Cones 251 Spheres 253 Chapter Problems 256 Problems 256 Answers and Solutions 258 Supplemental Chapter Problems 262 Problems 262 Answers 264 Chapter 10: Coordinate Geometry 265 Locating Points on Coordinate Axes 265 The Distance Formula 267

16 xiv CliffsStudySolver Geometry The Midpoint Formula 269 Slope of a Line 272 Slopes of Parallel and Perpendicular Lines 274 Equations of Lines 275 Standard Form of a Linear Equation 276 Point-Slope Form of a Linear Equation 277 Slope-y-intercept Form of a Linear Equation 278 Chapter Problems 281 Problems 281 Answers and Solutions 282 Supplemental Chapter Problems 287 Problems 287 Answers 288 Customized Full-Length Exam 291 Problems 291 Appendix: Postulates and Theorems 313 Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Index 321

17 Geometry Pretest Directions: Questions 1 through 132. Circle the letter of the appropriate answer. 1. A single letter, V, may be used to represent which of the following? a. a ray b. a point c. a line segment d. a line 2. Which of these extends infinitely far in two directions? a. a plane b. a ray c. a line d. a line segment 3. Which of these is defined by a minimum of three points, one of which is outside the line formed by the other two? a. a ray b. a line c. a line segment d. a plane 4. If a point lies outside a line then which of the following must be true? a. The point and the line may or may not be in the same plane. b. The line must be in one plane and the point in a second one. c. Exactly one plane contains the line and the point. d. There may be up to three separate planes involved. 5. If two lines intersect, at how many points may they intersect? a. one and only one b. one or two c. no more than three d. an infinite number 1

18 2 CliffsStudySolver Geometry 6. What name is given to a portion of a line bounded by two endpoints? a. a ray b. a segment c. a section d. a sector 7. Which is the correct name of B A a. segment BA b. segment AB c. ray AB d. ray BA 8. Points A, B, C, D, E, F, and G are evenly spaced on the following segment. Which is the midpoint? AG a. halfway between C and D b. D c. halfway between D and E d. C 9. Angle FDG is formed by the intersection of the following rays. Where must the vertex of the angle be? DF DG a. either F or G b. F c. G d. D

19 Geometry Pretest Select the one correct letter-name for the angle formed by adding 3 and 4. E F C D I G H a. EFG b. EIH c. DIG d. EIG 11. Which of the following rays bisects KON? K O L M N a. b. c. d. OL OM LO KO In the following three questions, m A means the degree measure of angle A. 12. Which of the following describes a right angle? a. m A = 45 b. m A = 60 c. m A = 90 d. m A = Which of the following is not an acute angle? a. m A = 45 b. m A = 60 c. m A = 75 d. m A = 100

20 4 CliffsStudySolver Geometry 14. Which of the following is an obtuse angle? a. m A = 45 b. m A = 60 c. m A = 90 d. m A = What is the degree measure of a straight angle? a. 80 b. 150 c. 180 d A reflex angle must have a degree measure between which two values? a. 180 and 360 b. 170 and 270 c. > 180 and < 270 d. > 180 and < In the following figure, which answer-choice names a pair of adjacent angles? E F D G a. CID and DIC b. CID and DIE c. CID and EIF d. CID and GIH C I H 18. Where lines l and m cross in the following figure, which answer choice names a pair of vertical angles? l a. 1 and 3 b. 2 and 3 c. 4 and 3 d. 1 and 4 m