Preface 

ix  
To the Student 

xvii  


1  (50) 


1  (1) 


2  (12) 

Variations on Two Familiar Geometric Themes 


14  (13) 

Discovery via the Computer 


27  (12) 


39  (12) 

Foundations of Geometry 1: Points, Lines, Segments, Angles 


51  (68) 


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) 


116  (1) 


117  (2) 

Foundations of Geometry 2: Triangles, Quadrilaterals, Circles 


119  (92) 


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) 


166  (8) 

Additional Congruence Criteria 


174  (9) 


183  (11) 


194  (17) 


208  (1) 


209  (2) 

Euclidean Geometry: Trigonometry, Coordinates and Vectors 


211  (120) 


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) 


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) 


328  (1) 


329  (2) 

Transformations in Geometry 


331  (90) 


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) 


389  (13) 

Using Transformation Theory in Proofs 


402  (19) 


418  (1) 


419  (2) 

Alternate Concepts for Parallelism: NonEuclidean Geometry 


421  (72) 


421  (1) 

Historical Background of NonEuclidean 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 HalfPlane Model 


469  (24) 


489  (1) 


490  (3) 

An Introduction to ThreeDimensional Geometry 


493  (67) 


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) 


522  (10) 

Coordinates, Vectors, and Isometries in E3 


532  (13) 


545  (15) 


559  (1) 


560  
Appendixes 



A1  

Appendix B Review of Topics in Secondary School Geometry 


A2  

Appendix C The Geometer's Sketchpad: Brief Instructions 


A27  

Appendix D Unified Axiom System for the Three Classical Geometries 


A31  

Appendix E Answers to Selected Problems 


A35  

Appendix F Symbols, Definitions, Axioms, Theorems, and Corollaries 


A55  
Index 

I1  