College Geometry A Discovery Approach

by Kay, David

• ISBN13: 9780321046246
• ISBN10: 0321046242
• Edition: 2nd
• Format: Paperback
• Copyright: 2019-12-11
• Publisher: Pearson

Summary

College Geometryis an approachable text, covering both Euclidean and Non-Euclidean geometry. This text is directed at the one semester course at the college level, for both pure mathematics majors and prospective teachers. A primary focus is on student participation, which is promoted in two ways: (1) Each section of the book contains one or two units, called M oments for Discovery,that use drawing, computational, or reasoning experiments to guide students to an often surprising conclusion related to section concepts; and (2) More than 650 problems were carefully designed to maintain student interest.

Table of Contents

Preface	ix
To the Student	xvii
Exploring Geometry	1	(50)
Overview	1	(1)
Discovery in Geometry	2	(12)
Variations on Two Familiar Geometric Themes	14	(13)
Discovery via the Computer	27	(12)
Steiner's Theorem	39	(12)
Foundations of Geometry 1: Points, Lines, Segments, Angles	51	(68)
Overview	51	(1)
An Introduction to Axiomatics and Proof	52	(10)
The Role of Examples and Models	62	(8)
Incidence Axioms for Geometry	70	(7)
Distance, Ruler Postulate, Segments, Rays, and Angles	77	(13)
Angle Measure and the Protractor Postulate	90	(13)
Plane Separation, Interior of Angles, Crossbar Theorem	103	(16)
Chapter Summary	116	(1)
Testing Your Knowledge	117	(2)
Foundations of Geometry 2: Triangles, Quadrilaterals, Circles	119	(92)
Overview	119	(1)
Triangles, Congruence Relations, SAS Hypothesis	120	(7)
Taxicab Geometry: Geometry without SAS Congruence	127	(12)
SAS, ASA, SSS Congruence, and Perpendicular Bisectors	139	(13)
Exterior Angle Inequality	152	(14)
The Inequality Theorems	166	(8)
Additional Congruence Criteria	174	(9)
Quadrilaterals	183	(11)
Circles	194	(17)
Chapter Summary	208	(1)
Testing Your Knowledge	209	(2)
Euclidean Geometry: Trigonometry, Coordinates and Vectors	211	(120)
Overview	211	(1)
Euclidean Parallelism, Existence of Rectangles	211	(13)
Parallelograms and Trapezoids: Parallel Projection	224	(12)
Similar Triangles, Pythagorean Theorem, Trigonometry	236	(18)
Regular Polygons and Tiling	254	(15)
The Circle Theorems	269	(15)
Euclid's Concept of Area and Volume	284	(17)
Coordinate Geometry and Vectors	301	(14)
Some Modern Geometry of the Triangle	315	(16)
Chapter Summary	328	(1)
Testing Your Knowledge	329	(2)
Transformations in Geometry	331	(90)
Overview	331	(1)
Euclid's Superposition Proof and Plane Transformations	331	(10)
Reflections: Building Blocks for Isometries	341	(12)
Translations, Rotations, and Other Isometries	353	(9)
Other Linear Transformations	362	(11)
Coordinate Characterizations	373	(16)
Transformation Groups	389	(13)
Using Transformation Theory in Proofs	402	(19)
Chapter Summary	418	(1)
Testing Your Knowledge	419	(2)
Alternate Concepts for Parallelism: Non-Euclidean Geometry	421	(72)
Overview	421	(1)
Historical Background of Non-Euclidean Geometry	421	(4)
An Improbable Logical Case	425	(11)
Hyperbolic Geometry: Angle Sum Theorem	436	(9)
Two Models for Hyperbolic Geometry	445	(24)
Circular Inversion: Proof of SAS Postulate for Half-Plane Model	469	(24)
Chapter Summary	489	(1)
Testing Your Knowledge	490	(3)
An Introduction to Three-Dimensional Geometry	493	(67)
Overview	493	(1)
Orthogonality Concepts for Lines and Planes	493	(10)
Parallelism in Space: Prisms, Pyramids, and the Platonic Solids	503	(11)
Cones, Cylinders, and Spheres	514	(8)
Volume in E3	522	(10)
Coordinates, Vectors, and Isometries in E3	532	(13)
Spherical Geometry	545	(15)
Chapter Summary	559	(1)
Testing Your Knowledge	560
Appendixes
Appendix A Bibliography	A-1
Appendix B Review of Topics in Secondary School Geometry	A-2
Appendix C The Geometer's Sketchpad: Brief Instructions	A-27
Appendix D Unified Axiom System for the Three Classical Geometries	A-31
Appendix E Answers to Selected Problems	A-35
Appendix F Symbols, Definitions, Axioms, Theorems, and Corollaries	A-55
Index	I-1