Foundations of Geometry/Gerard A
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For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116, fax your request to 617-671-3447, or e-mail at http://www.pearsoned.com/legal/permissions.htm. Library of Congress Cataloging-in-Publication Data Venema, Gerard. The foundations of geometry/Gerard A. Venema. -- 2nd ed. p. cm. Includes bibliographical references index. ISBN 0-13-602058-5 1. Geometry—Foundations--Textbooks. I. Title. QA445.V455 2012 2012 514--dc22 2011002700 CIP 10 9 8 7 6 5 4 3 2 1 www.pearsonhighered.com ISBN-10: 0-13-602058-5 ISBN-13: 978-0-13-602058-5 Second Edition Foundations of Geometry Gerard A. Venema Department of Mathematics and Statistics Calvin College Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo This page intentionally left blank To my family: Patricia, Sara, Emily, Daniel, David, Christian, and Ian. This page intentionally left blank Contents Preface ix 1 Prologue: Euclid’s Elements 1 1.1 Geometry before Euclid ............................. 1 1.2 The logical structure of Euclid’s Elements ................... 2 1.3 The historical significance of Euclid’s Elements ................ 3 1.4 A look at Book I of the Elements ........................ 4 1.5 A critique of Euclid’s Elements ......................... 7 1.6 Final observations about the Elements ..................... 9 2 Axiomatic Systems and Incidence Geometry 14 2.1 The structure of an axiomatic system ...................... 14 2.2 An example: Incidence geometry ........................ 16 2.3 The parallel postulates in incidence geometry ................. 20 2.4 Axiomatic systems and the real world ...................... 22 2.5 Theorems, proofs, and logic ........................... 24 2.6 Some theorems from incidence geometry .................... 32 3 Axioms for Plane Geometry 35 3.1 The undefined terms and two fundamental axioms .............. 36 3.2 Distance and the Ruler Postulate ........................ 37 3.3 Plane separation .................................. 46 3.4 Angle measure and the Protractor Postulate .................. 51 3.5 The Crossbar Theorem and the Linear Pair Theorem ............. 55 3.6 The Side-Angle-Side Postulate .......................... 62 3.7 The parallel postulates and models ....................... 66 4 Neutral Geometry 69 4.1 The Exterior Angle Theorem and existence of perpendiculars . ...... 70 4.2 Triangle congruence conditions ......................... 74 4.3 Three inequalities for triangles .......................... 77 4.4 The Alternate Interior Angles Theorem .................... 82 4.5 The Saccheri-Legendre Theorem ........................ 84 4.6 Quadrilaterals ................................... 88 4.7 Statements equivalent to the Euclidean Parallel Postulate .......... 91 4.8 Rectangles and defect ............................... 97 4.9 The Universal Hyperbolic Theorem .......................104 5 Euclidean Geometry 106 5.1 Basic theorems of Euclidean geometry .....................107 5.2 The Parallel Projection Theorem ........................110 5.3 Similar triangles ..................................112 5.4 The Pythagorean Theorem ............................114 5.5 Trigonometry ...................................116 5.6 Exploring the Euclidean geometry of the triangle ...............118 v vi Contents 6 Hyperbolic Geometry 131 6.1 Basic theorems of hyperbolic geometry .....................133 6.2 Common perpendiculars .............................138 6.3 The angle of parallelism .............................140 6.4 Limiting parallel rays ...............................144 6.5 Asymptotic triangles ...............................149 6.6 The classification of parallels ...........................152 6.7 Properties of the critical function ........................155 6.8 The defect of a triangle ..............................160 6.9 Is the real world hyperbolic? ...........................164 7 Area 167 7.1 The Neutral Area Postulate ...........................167 7.2 Area in Euclidean geometry ...........................173 7.3 Dissection theory .................................180 7.4 Proof of the dissection theorem in Euclidean geometry ............181 7.5 The associated Saccheri quadrilateral ......................185 7.6 Area and defect in hyperbolic geometry ....................190 8 Circles 195 8.1 Circles and lines in neutral geometry ......................195 8.2 Circles and triangles in neutral geometry ....................200 8.3 Circles in Euclidean geometry ..........................206 8.4 Circular continuity .................................212 8.5 Circumference and area of Euclidean circles ..................215 8.6 Exploring Euclidean circles ............................222 9 Constructions 228 9.1 Compass and straightedge constructions ....................228 9.2 Neutral constructions ...............................231 9.3 Euclidean constructions ..............................234 9.4 Construction of regular polygons ........................235 9.5 Area constructions ................................239 9.6 Three impossible constructions .........................242 10 Transformations 245 10.1 Properties of isometries ..............................246 10.2 Rotations, translations, and glide reflections ..................252 10.3 Classification of Euclidean motions .......................260 10.4 Classification of hyperbolic motions .......................263 10.5 A transformational approach to the foundations ................265 10.6 Similarity transformations in Euclidean geometry ...............270 10.7 Euclidean inversions in circles ..........................275 11 Models 287 11.1 The Cartesian model for Euclidean geometry .................289 11.2 The Poincare´ disk model for hyperbolic geometry ...............290 11.3 Other models for hyperbolic geometry .....................296 11.4 Models for elliptic geometry ...........................301 12 Polygonal Models and the Geometry of Space 303 12.1 Curved surfaces ..................................304 Contents vii 12.2 Approximate models for the hyperbolic plane .................315 12.3 Geometric surfaces ................................321 12.4 The geometry of the universe ..........................326 12.5 Conclusion .....................................331 12.6 Further study ....................................331 12.7 Templates .....................................336 APPENDICES A Euclid’s Book I 344 A.1 Definitions .....................................344 A.2 Postulates ......................................346 A.3 Common Notions .................................346 A.4 Propositions ....................................346 B Systems of Axioms for Geometry 350 B.1 Hilbert’s axioms ..................................351 B.2 Birkhoff’s axioms .................................353 B.3 MacLane’s axioms .................................354 B.4 SMSG axioms ...................................355 B.5 UCSMP axioms ..................................357 C The Postulates Used in this Book 360 C.1 Criteria used in selecting the postulates .....................360 C.2 Statements of the postulates ...........................362 C.3 Logical relationships ...............................363 D The van Hiele Model 365 E Set Notation and the Real Numbers 366 E.1 Some elementary set theory ...........................366 E.2 Axioms for the real numbers ...........................367 E.3 Properties of the real numbers ..........................368 E.4 One-to-one and onto functions ..........................370 E.5 Continuous functions ...............................371 F Hints for Selected Exercises 372 Bibliography 379 Index 382 This page intentionally left blank Preface This is a textbook for an undergraduate course in axiomatic geometry. The text is targeted at mathematics students who have completed the calculus sequence and perhaps a first course in linear algebra, but who have not yet encountered such upper-level mathematics courses as real analysis and abstract algebra. A course based on this book will enrich the education of all mathematics majors and will ease their transition into more advanced mathematics courses. The book also includes emphases that make it especially appropriate as the textbook for a geometry course taken by future high school mathematics teachers. WHAT’S NEW IN THE SECOND EDITION? For the benefit of those who have used the first edition of this book, here is a quick summary of what has changed in the second edition. • The first part