euclidean and non euclidean geometry solutions chapter 1

NEUTRAL GEOMETRY lIS Geometry without the parallel axiom 115 Alternate interior angle theorem 116 Exterior angle theorem 118 Measure of angles and segments 122 Saccheri-Legendre theorem 124 Equivalence of parallel postulates 128 Angle sum of a triangle 130 Review exercise 134 Exercises 136 Major exercises 143 Projects 146 III Review exercise 103 Major exercises 111 HILBERT'S AXIOMS 70 Flaws in Euclid 70 Axioms of betweenness 7Z Axioms of congruence 82 Axioms of continuity 93 Axiom of parallelism 102 LOGIC AND INCIDENCE GEOMETRY 38 Informal logic 38 Theorems and proofs 40 RAA proofs 42 Negation 44 Quantifiers 45 Implication 48 Law of excluded middle and proof by cases 49 Incidence geometry 50 Models 52 Isomorphism of models 56 Projective and affine planes 58 Review exercise 62 Exercises 63 Major exercises 65 Projects 68 The moral ofthis book is: Check your premises.ĮUCLID'S GEOMETRY 6 The origins of geometry 6 The axiomatic method 10 Undefined terms 11 Euclid's first four postulates 14 The parallel postulate 18 Attempts to prove the parallel postulate 21 The danger in diagrams 23 The power of diagrams 25 Review exercise 26 Exercises 27 Major exercises 31 Projects 35 And to the memory ofStanford Professor Karel De Leeuw, inspiring teacher and outrageous friend. To Moses Ma, President of Velocity Development, without whose technical and emotional support this third edition could not have been written. Printed in the United States of America 2

Freeman and Company No part of this book may be reproduced by any mechanical, photographic, or electronic process, or in the form of a phonographic recording, nor may it be stored in a retrieval system, transmitted, or otherwise copied for public or private use, without written permission from the publisher. Includes bibliographical references and indexes. Euclidean and non-Euclidean geometries: development and history I Marvin Jay Greenberg. Library of Congress Cataloging-in-Publication Data Greenberg, Marvin J. Back cover image: This tessellation was generated by Douglas Dunham (Department of Computer Science, University of Minnesota, Duluth, MN 55812-2496). It is easy to modify this latter tessellation to obtain Circle Limit IV (which appears on the front cover). triangles, which can also be alternately colored.In fact, by combining pairs of triangles that share long sides, one obtains a different tessellation by 45 Escher to create his (hyperbolic) 'Circle Limit' patterns. Coxeter (Figure 7 in 'Crystal Symmetry and Its Generalizations,' Transactions of the Royal Society ofCanada 3 (51), 1957, pp. triangles, which appeared in a paper by H.Jay Greenberg University of California, Santa CruzĬover image: This tessellation of the hyperbolic plane by alternately colored 30 EUCLIDEAN AND NON-EUCLIDEAN GEOMETRIES Development and History Third Edition

Euclidean and non euclidean geometry solutions chapter 1