College Geometry with GeoGebra

From two authors who embrace technology in the classroom and value the role of collaborative learning comes College Geometry Using GeoGebra, a book that is ideal for geometry courses for both mathematics and math education majors. The book's discovery-based approach guides students to explore geometric worlds through computer-based activities, enabling students to make observations, develop conjectures, and write mathematical proofs. This unique textbook helps students understand the underlying concepts of geometry while learning to use GeoGebra software—constructing various geometric figures and investigating their properties, relationships, and interactions. The text allows students to gradually build upon their knowledge as they move from fundamental concepts of circle and triangle geometry to more advanced topics such as isometries and matrices, symmetry in the plane, and hyperbolic and projective geometry. Emphasizing active collaborative learning, the text contains numerous fully-integrated computer lab activities that visualize difficult geometric concepts and facilitate both small-group and whole-class discussions. Each chapter begins with engaging activities that draw students into the subject matter, followed by detailed discussions that solidify the student conjectures made in the activities and exercises that test comprehension of the material. Written to support students and instructors in active-learning classrooms that incorporate computer technology, College Geometry with GeoGebra is an ideal resource for geometry courses for both mathematics and math education majors.

Preface Especially for Students Notes for Instructors Our Motivation, Philosophy, and Pedagogy Prerequisites and Chapter Dependencies Acknowledgments ONEUsing GeoGebra 1.1 Activities: Getting Started with GeoGebra 1.2 Discussion: Exploring and Conjecturing Some GeoGebra Tips Constructing −→ Exploring −→ Conjecturing: Inductive Reasoning Language of Geometry Explorations, Observations, Questions The Family of Quadrilaterals Angles Inscribed in Circles Rules of Logic 1.3 Exercises 1.4 Chapter Overview   TWO  Constructing → Proving  2.1 Activities  2.2 Discussion: Euclid’s Postulates and Constructions  Euclid’s Postulates  Congruence and Similarity  Constructions  Geometric Language Revisited  Conditional Statements: Implication  Using Robust Constructions to Develop a Proof  Angles and Measuring Angles  Constructing Perpendicular and Parallel Lines  Properties of Triangles  Euclid’s Parallel Postulate  Euclid’s Constructions in the Elements  Ideas About Betweenness  2.3 Exercises  2.4 Chapter Overview THREE Mathematical Arguments and Triangle Geometry 3.1 Activities 3.2 Discussion Deductive Reasoning Universal and Existential Quantifiers Negating a Quantified Statement Direct Proof and Disproof by Counterexample Step-by-Step Proofs Congruence Criteria for Triangles The Converse and the Contrapositive Concurrence Properties for Triangles Ceva’s Theorem and Its Converse Brief Excursion into Circle Geometry The Circumcircle of ΔABC The Nine-Point Circle: A First Pass Menelaus’ Theorem and Its Converse 3.3 Exercises 3.4 Chapter Overview FOUR Circle Geometry and Proofs  4.1 Activities  4.2 Discussion  Axiom Systems: Ancient and Modern Approaches  Language of Circles  Inscribed Angles  Mathematical Arguments  Additional Methods of Proof  Cyclic Quadrilaterals  Incircles and Excircles  Some Interesting Families of Circles  The Arbelos and the Salinon  Power of a Point  The Radical Axis  The Nine-Point Circle: A Second Pass  4.3 Exercises  4.4 Chapter Overview FIVE Analytic Geometry  5.1 Activities  5.2 Discussion  Points  Lines  Distance  Using Coordinates in Proofs  Another Look at the Radical Axis  Polar Coordinates  The Nine-Point Circle, Revisited  5.3 Exercises  5.4 Chapter Overview SIX Taxicab Geometry   6.1 Activities   6.2 Discussion   An Axiom System for Metric Geometry   Circles   Ellipses   Measuring Distance from a Point to a Line   Parabolas   Hyperbolas   Axiom Systems   6.3 Exercises   6.4 Chapter Overview SEVEN  Finite Geometries 7.1 Activities 7.2 Discussion An Axiom System for an Affine Plane An Axiom System for a Projective Plane Duality Relating Affine Planes to Projective Planes Coordinates for Finite Geometries 7.3 Exercises 7.4 Chapter Overview EIGHTTransformational Geometry  8.1 Activities  8.2 Discussion  Transformations  Isometries  Other Transformations  Composition of Isometries  Inverse Isometries  Using Isometries in Proofs  Isometries in Space 8.3 Exercises 8.4 Chapter Overview NINE  Isometries and Matrices 9.1 Activities 9.2 Discussion  Using Vectors to Represent Translations  Using Matrices to Represent Rotations  Using Matrices to Represent Reflections  Composition of Isometries  The General Form of a Matrix Representation  Using Matrices in Proofs  Similarity Transformations  9.3 Exercises  9.4 Chapter Overview TENSymmetry in the Plane  10.1 Activities  10.2 Discussion  Symmetries  Groups of Symmetries  Classifying Figures by Their Symmetries  Friezes and Symmetry  Wallpaper Symmetry  Tilings  10.3 Exercises  10.4 Chapter Overview     ELEVEN  Hyperbolic Geometry  Part I: Exploring a New Universe  11.1 Activities Part I  11.2 Discussion Part I   Hyperbolic Lines and Segments   The Poincaré Disk Model of the Hyperbolic Plane   Measuring Distance in the Poincaré Disk Model   Hyperbolic Circles   Hyperbolic Triangles   Circumcircles and Incircles of Hyperbolic Triangles   Congruence of Triangles in the Hyperbolic Plane  Part II: The Parallel Postulate in Hyperbolic Geometry  11.3 Activities Part II  11.4 Discussion Part II   The Hyperbolic and Elliptic Parallel Postulates   The Angle of Parallelism   The Exterior Angle Theorem   Quadrilaterals in the Hyperbolic Plane   Another Look at Triangles in the Hyperbolic Plane  Area in the Hyperbolic Plane  11.5 Exercises   The Upper-Half-Plane Model  11.6 Chapter Overview TWELVE Projective Geometry 12.1 Activities 12.2 Discussion   An Axiom System   Models for the Projective Plane   Duality   Coordinates for Projective Geometry   Projective Transformations 12.3 Exercises 12.4 Chapter Overview APPENDIX A Trigonometry A.1 Activities A.2 Discussion   Right Triangle Trigonometry   Unit Circle Trigonometry   Solving Trigonometric Equations   Double Angle Formulas  Angle Sum Formulas  Half-Angle Formulas  The Law of Sines and the Law of Cosines A.3 Exercises APPENDIX B Calculating with Matrices B.1 Activities B.2 Discussion   Linear Combinations of Vectors   Dot Product of Vectors   Multiplying a Matrix Times a Vector   Multiplying Two Matrices  The Determinant of a Matrix B.3 Exercises BIBLIOGRAPHY INDEX