Kay, David : University of North Carolina-Asheville
Summary
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 Moments 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
(* Indicates optional section.)
Preface.
To the Student.
1. Exploring Geometry.
Discovery in Geometry.
Variations on Two Familiar Geometric Themes.
*Ptolemy, Brahmagupta and the Quadrilateral.
A Glimpse at Modern Classical Geometry.
*Discovery via the Computer.
2. Foundations of Geometry I: Points, Lines, Segments, and Angles.
*An Introduction to Axiomatics and Proof.
*The Role of Examples and Models.
*An Excursion: Euclid's Concept of Area and Volume.
Incidence Axioms for Geometry.
Metric Betweeness, Segments, Rays, and Angles.
Plan Separation, Postulate of Pasch, Interiors of Angles.
Angle Measure and the Ruler, Protractor Postulates.
Crossbar Theorem, Linear Pair Axiom, Perpendicularity.
Chapter Summary.
3. Foundations of Geometry II: Triangles, Quadrilaterals, and Circles.
Triangles, Congruence Relations, SAS Hypothesis.
*Taxicab Geometry: Geometry Without SAS Congruence.
SAS, ASA, SSS Congruence, and Perpendicular Bisectors.
Exterior Angle Inequality.
The Inequality Theorems.
Additional Congruence Criteria.
Quadrilaterals.
Circles.
Chapter Summary.
4. Euclidean Geometry: Trigonometry, Coordinates, and Vectors.
Euclidean Parallelism, Existence of Rectangles.
Parallelograms and Trapezoids: Parallel Projection.
Similar Triangles, Pythagorean Theorem, Trigonometry.
*Regular Polygons, Tiling.
The Circle Theorems.
Coordinate Geometry and Vectors.
*Families of Orthogonal Circles, Circular Inversion.
*Some Modern Geometry of the Triangle.
Chapter Summary.
5. Transformations in Geometry.
Euclid's Superposition Proof and Plan Transformation.
Reflections: Building Blocks for Isometrics.
Translations, Rotations and Other Euclidean Motions.
Other Transformations.
Coordinate Characterization of Linear Transformations.
*Transformation Groups.
*Using Transformation Theory in Proofs.
Chapter Summary.
6. Alternative Concepts for Parallelism: Non-Euclidean.
*Historical Background of Non-Euclidean Geometry.
An Improbable Logical Case.
The Beltrami-Poincaré Half-Plan Model.
Hyperbolic Geometry 1: Angle Sum Theorem.
*Hyperbolic Geometry 2: Asymptotic Triangles.
*Hyperbolic Geometry 3: Theory of Parallels.
Models for Hyperbolic Geometry: Relative Consistency.
*Axioms for a Bounded Metric: Elliptic Geometry.
Chapter Summary.
7. An Introduction to Three Dimensional Geometry.
Orthogonality Concepts for Lines and Plans.
Parallelism in Space, Prisms, Pyramids, and Boxes.
Cones, Cylinders, and Spheres.
Volume in E^3.
Coordinates, Vectors, and Isometries in E^3.^
Spherical Geometry.
Appendixes.
A. Bibliography.
B. Solutions for Selected Problems.
C. Symbols, Axioms, Theorems.
