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Geometry Third Edition cm Harcourt Brace Jovanovich New York Chicago San Francisco Atianta Dallas and London Copyright© 1978. 1972. 1969 by Harcourt Brace Jovanovich. Inc. 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 wntmg from the publisher. Printed in the United States of America Regular Edition: ISBN 0-15-353926·7 With Alternate Answers· ISBN 0-15-353927·5 Curriculum and Writing James F. Ulrich Chairman of Department of Mathematics Arlington High School Arlington Heights, Illinois Fred F. Czarnec Executive Editor, Secondary Mathematics Center for Curriculum Development New York, New York Dorothy L. Guilbault Senior Editor, Secondary Mathematics Center for Curriculum Development New York, New York Picture Credits Cover: Jerry Kresh KEY· (I) top. (c) center. {b) bo\lom p 34. Sl Charles Mar:,ufacturing Co .. p 129, American Telephone and Telegraph Co.; p_ 146 (t), De Wys, Inc.; (b). M White from Monkmeyer. p. 244. courtesy R. Buckmmster Fuller. p 296 tc). Grant Heilman; p 407. Harbrace Art, p. 511 (c). Mt. Wilson and Palomar Observatories. California Institute of Technology Bookstore; (b}. Bauch and Lomb Op11ca1 Co.: p. 533 (I), Westmghouse Research and Development Corporation Cmtents--------- Chapter 1: Introduction to Geometry 1-1 Introduction 2 1-2 Segments 6 1-3 Angles 9 1-4 Triangles 13 1-5 Definitions 18 1-6 Postulates and Theorems 23 1-7 Deductive Reasoning 27 Career: Mathematics and Architecture 34 Chapter Objectives and Review 35 Chapter Test 38 Chapter 2: Segments and Angles 39 2-1 Measure of Segments 40 Constructions 46 2-2 Measuring Angles 48 Constructions 54 2-3 Important Angle Pairs 57 2-4 Equivalence Relations 63 2-5 Two-Column Proofs 69 2-6 Theorems About Perpendicular Lines 75 2-7 Polygons 81 Special Topic: Inductive Reasoning 86 Chapter Objectives and Review 88 Chapter Test 91 Contents iii Chapter 3: Congruent Triangles 93 3-1 Congruence 94 3-2 Congruent Triangles/SAS. 99 Special Topic: Definitions and If and Only If 106 3-3 Congruent Triangles/A.SA and S.S.S. 107 Construction 114 3-4 Isosceles Triangle Theorems 116 3-5 More Isosceles Triangle Theorems 120 Constructions 126 Career: Mathematics and Electronics 129 3-6 Overlapping Triangles 130 3-7 Indirect Proof 135 3-8 Congruence in Right Triangles 140 Special Topic: Contrapositive and Indirect Proof 145 Career. Mathematics and Civil Engineering 146 Chapter Objectives and Review 147 Chapter Test 151 Chapter 4: Parallel Lines 153 4-1 Interior and Exterior Angles 154 4-2 Parallel Lines and Angles 159 4-3 More Angle Relationships 165 4-4 Corresponding Angles of Parallel Lines 172 Construction 177 4-5 The Angles of a Triangle 179 4-6 Other Corollaries 185 4-7 The Angles of a Polygon 190 4-8 The Exterior Angles of a Polygon 194 Special Topic. Drawing Three-Dimensional Figures 198 Chapter Objectives and Review 199 Chapter Test 202 Cumulative Review 203 iv Contents Chapter 5: Quadrilaterals 205 5-1 Parallelograms and their Properties 206 5-2 Conditions Determining a Parallelogram 210 5-3 Rectangle, Rhombus, and Square 214 5-4 Midpoints and Parallel Lines 219 5-5 Parallel Lines and Congruent Segments 223 Constructions 227 5-6 Inequalities in Triangles 230 5-7 More Inequalities in Triangles 234 5-8 Dihedral Angles 239 Career: Mathematics and Urban Planning 244 Chapter Objectives and Review 245 Chapter Test 248 Chapter 6: Similarity 249 6-1 Ratio and Proportion 250 6-2 Other Properties of Proportions 254 6-3 Proportional Segments 258 6-4 More Proportional Segments 262 Construction 266 6-5 Similar Polygons 267 6-6 Similar Triangles 271 6-7 More on Similar Triangles 276 6-8 Similar Right Triangles 281 6-9 The Pythagorean Theorem 286 6-10 Special Right Triangles 291 Career: Mathematics and Agricultural Engineering 296 Chapter Objectives and Review 297 Chapter Test 301 Chapter 7: Circles 303 7-1 The Circle: Related Lines and Segments 304 7-2 Central Angles and Arcs 308 Contents v 7-3 Congruent Chords and Arcs 313 7-4 Chords and Inscribed Polygons 318 Construction 323 7-5 Tangents to a Circle 324 7-6 Inscribed Angles 329 Constructions 334 7-7 More Inscribed Angles 336 Construction 340 7-8 More Angles and Arcs 342 7-9 Angles Formed by Secants and Tangents 346 7-10 Lengths of Segments 351 7-11 The Sphere 356 Special Topic.· Non-Euclidean Geometry 360 Chapter Objectives and Review 361 Chapter Test 364 Cumulative Review 366 Chapter 8: Area 369 8-1 Polygonal Regions 370 8-2 Area/Parallelogram and Triangle 375 8-3 Area/Rhombus and Trapezoid 381 8-4 Circumscribed and Inscribed Polygons 385 Constructions 390 8-5 Areas of Regular Polygons 392 8-6 Similarity and Polygons 395 8-7 Circumference/Length of Arc 398 8-8 Areas of Circles. Sectors. and Segments 402 Career· Mathematics and Cartography 407 Career.· Mathematics and Marine Engineering 408 Chapter Objectives and Review 409 Chapter Test 411 Chapter 9: Surface Area and Volume 413 9-1 The Prism 414 9-2 The Pyramid 419 vi Contents 9-3 Cylinders and Cones 423 9-4 Volume 428 9-5 Surface Area and Volume of a Sphere 436 Speciai Topic Cavalien·s Princ,ple 439 Chapter Objectives and Review 440 Chapter Test d42 Chapter 10: Coordinate Geometry 443 10-1 The Coordlnate Plane 444 10-2 The Midpoint Formula 447 10-3 Slope of a Line 450 10-4 Slope and Linear Equations 453 10-5 Parallel and Perpendicular Lines 456 10-S The Distance Formula 460 10-7 Coordinate Proofs 463 Spec,ai Topic· A Coordlnate System 1n Three-Space 466 Chapter Objectives and Review 467 Chapter Test 469 Cumulat,ve Review 470 Chapter 11: Introduction !o Locus 473 11-1 Drawmg and Describing a Locus 474 11-2 1ntersect1on of Loci 479 11-3 Constructions and Loci .<t.'!2 11-4 Locus in Three D,mens,ons 486 Spec,a' Topic: Area and Constructions 490 Chapter Objectives and Review 492 Chapter Test 493 Chapter 12: Introduction to Trigonometry 495 12-1 The Tangen' Ratio 496 12-2 Using the Tangent Ratio 500 Contents vii 12-3 The Sine and Cosine Ratios 502 12-4 Solving Problems 507 Career: Mathematics and Astronomy 511 Chapter Objectives and Review 512 Chapter Test 514 Chapter 13: Introduction to Transformations 515 13-1 Reflections Over a Line 516 13-2 Special Reflections 520 13-3 Rotations 525 13-4 Dilations 529 Career· Mathematics and Mineralogy 533 Chapter Objectives and Review 534 Chapter Test 536 Chapter 14: Introduction to Vectors 537 14-1 Vectors 538 14-2 Addition of Vectors 541 14-3 Vector Theorems 544 14-4 More Vector Theorems 548 Chapter Objectives and Review 551 Chapter Test 552 Cumulative Review 553 Review: Topics From Algebra 555 Answers to Review: Topics From Algebra One 569 Answers to Try These and Chapter Tests 571 Table of Squares and Square Roots 580 Table of Sines, Cosines, and Tangents 581 Postulates and Theorems 582 Index 595 viii Contents Chapter l Introduction to GeaTetry 1-1 lntrcductim 1-2 Segments 1-3 Angles 1-4 Triarg les 1-5 Definit1ms 1-6 Postulates ard "Theorems 1-7 Deductive f<easonirg Introduction 1-1 You are familiar with sets of points such as lines and angles. Space is the set of all points. Geometry is the study of sets of points in space. Model Name Description Indicates position; has no size. A• Point A Points are iD. or on lines and planes. Line BC Extends without end in two B C or opposites directions. Lines are Line ca iD. or on planes. Plane POR A flat surface that extends or without end in all directions. A Plane I plane is usually represented by a four-sided figure. Points that are on the same line are collinear. D E F For any three distinct (different) points on a line, one is between the other two. Point E is between D and Fon I ine DE. Model Name Description Segment GK Consists of two endpoints, G or and K, and all the points Segment KG between them. Extends without end in one Ray MN direction from its one endpoint, M. 2 Chapter 1