College Geometry: A Discovery Approach

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  • Edition: 2nd
  • Format: Paperback
  • Copyright: 2001-01-01
  • Publisher: Addison Wesley
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"College Geometry "is 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

Prefacep. ix
To the Studentp. xvii
Exploring Geometryp. 1
Overviewp. 1
Discovery in Geometryp. 2
Variations on Two Familiar Geometric Themesp. 14
Discovery via the Computerp. 27
Steiner's Theoremp. 39
Foundations of Geometry 1: Points, Lines, Segments, Anglesp. 51
Overviewp. 51
An Introduction to Axiomatics and Proofp. 52
The Role of Examples and Modelsp. 62
Incidence Axioms for Geometryp. 70
Distance, Ruler Postulate, Segments, Rays, and Anglesp. 77
Angle Measure and the Protractor Postulatep. 90
Plane Separation, Interior of Angles, Crossbar Theoremp. 103
Chapter Summaryp. 116
Testing Your Knowledgep. 117
Foundations of Geometry 2: Triangles, Quadrilaterals, Circlesp. 119
Overviewp. 119
Triangles, Congruence Relations, SAS Hypothesisp. 120
Taxicab Geometry: Geometry without SAS Congruencep. 127
SAS, ASA, SSS Congruence, and Perpendicular Bisectorsp. 139
Exterior Angle Inequalityp. 152
The Inequality Theoremsp. 166
Additional Congruence Criteriap. 174
Quadrilateralsp. 183
Circlesp. 194
Chapter Summaryp. 208
Testing Your Knowledgep. 209
Euclidean Geometry: Trigonometry, Coordinates and Vectorsp. 211
Overviewp. 211
Euclidean Parallelism, Existence of Rectanglesp. 211
Parallelograms and Trapezoids: Parallel Projectionp. 224
Similar Triangles, Pythagorean Theorem, Trigonometryp. 236
Regular Polygons and Tilingp. 254
The Circle Theoremsp. 269
Euclid's Concept of Area and Volumep. 284
Coordinate Geometry and Vectorsp. 301
Some Modern Geometry of the Trianglep. 315
Chapter Summaryp. 328
Testing Your Knowledgep. 329
Transformations in Geometryp. 331
Overviewp. 331
Euclid's Superposition Proof and Plane Transformationsp. 331
Reflections: Building Blocks for Isometriesp. 341
Translations, Rotations, and Other Isometriesp. 353
Other Linear Transformationsp. 362
Coordinate Characterizationsp. 373
Transformation Groupsp. 389
Using Tranformation Theory in Proofsp. 402
Chapter Summaryp. 418
Testing Your Knowledgep. 419
Alternate Concepts for Parallelism: Non-Euclidean Geometryp. 421
Overviewp. 421
Historical Background of Non-Euclidean Geometryp. 421
An Improbable Logical Casep. 425
Hyperbolic Geometry: Angle Sum Theoremp. 436
Two Models for Hyperbolic Geometryp. 445
Circular Inversion: Proof of SAS Postulate for Half-Plane Modelp. 469
Chapter Summaryp. 489
Testing Your Knowledgep. 490
An Introduction to Three-Dimensional Geometryp. 493
Overviewp. 493
Orthogonality Concepts for Lines and Planesp. 493
Parallelism in Space: Prisms, Pyramids, and the Platonic Solidsp. 503
Cones, Cylinders, and Spheresp. 514
Volume in E[superscript 3]p. 522
Coordinates, Vectors, and Isometries in E[superscript 3]p. 532
Spherical Geometryp. 545
Chapter Summaryp. 559
Testing Your Knowledgep. 560
Bibliographyp. 1
Review of Topics in Secondary School Geometryp. 2
The Geometer's Sketchpad: Brief Instructionsp. 27
Unified Axiom System for the Three Classical Geometriesp. 31
Answers to Selected Problemsp. 35
Symbols, Definitions, Axioms, Theorems, and Corollariesp. 55
Indexp. 1
Special Topics
An Introduction to Projective Geometry
An Introduction to Convexity Theory
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