by David Kay
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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 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.
Features
NEW! Geometer's Sketchpad Projects. A part of each exercise section, and incorporated into selected examples.
NEW! Glossary.
End of Chapter Material. In addition to the current chapter summary and End of Chapter True/False questions, there are new conceptual exercises to test the students' understanding of the chapter material.
Moments for Discovery. Reinforces chapter material by encouraging students to experiment.
Historical perspective. Appropriate biographies are written throughout the text, to give context to the material that students are learning.
Our Geometric World. Placed throughout each chapter, this feature illustrates the real world application of the material that students are learning.
Author Bio
Kay, David : University of North Carolina-Asheville
(* 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.
Index.
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.
Features
NEW! Geometer's Sketchpad Projects. A part of each exercise section, and incorporated into selected examples.
NEW! Glossary.
End of Chapter Material. In addition to the current chapter summary and End of Chapter True/False questions, there are new conceptual exercises to test the students' understanding of the chapter material.
Moments for Discovery. Reinforces chapter material by encouraging students to experiment.
Historical perspective. Appropriate biographies are written throughout the text, to give context to the material that students are learning.
Our Geometric World. Placed throughout each chapter, this feature illustrates the real world application of the material that students are learning.
Author Bio
Kay, David : University of North Carolina-Asheville
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.
Index.