Pure and Applied UNDERGRADUATE/TEXTS 51 Geometry Transformed Euclidean Plane Geometry Based on Rigid Motions James R.King AMS 我 te for IAS PCMI Park City Mathematics Institute
282 pages on 50 lb • Spine: 9/16 inch • Softcover • Trim Size 7 X 10 51 51 AMSTEXT This series was founded by the highly respected mathematician and educator, Paul J. Sally, Jr. Geometry Transformed Euclidean Plane Geometry Based on Rigid Motions James R. King For additional information and updates on this book, visit www.ams.org/bookpages/amstext-51 AMSTEXT/51 Geometry Transformed James R. King Many paths lead into Euclidean plane geometry. Geometry Transformed offers an expeditious yet rigorous route using axioms based on rigid motions and dilations. Since transformations are available at the outset, interesting theorems can be proved sooner; and proofs can be connected to visual and tactile intuition about symmetry and motion. The reader thus gains valuable experience thinking with transformations, a skill that may be useful in other math courses or applications. For students interested in teaching mathematics at the secondary school level, this approach is particularly useful since geometry in the Common Core State Standards is based on rigid motions. The only prerequisite for this book is a basic understanding of functions. Some previous experience with proofs may be helpful, but students can also learn about proofs by experiencing them in this book—in a context where they can draw and experiment. The eleven chapters are organized in a flexible way to suit a variety of curriculum goals. In addition to a geometrical core that includes finite symmetry groups, there are additional topics on circles and on crystallographic and frieze groups, and a final chapter on affine and Cartesian coordinates. The exercises are a mixture of routine problems, experiments, and proofs. 2-color cover: Pantone 432 C (Gray) and Pantone 1805 C (Red) Photo courtesy of James R. King IAS | PCMI American Mathematical Society Institute for Advanced Study Park City Mathematics Institute
Contents Introduction Advice for Students and Less Experienced Geometers Information for More Experienced Geometer w A Chapter Guide for Instructors and Others xv Acknowledgments Chapter 1.Congruence and Rigid Motions 1 Rigid Motions Informal Preview of a Problem Solution 34 Sameness Properties of Congruence Exercises and Explorations 59 Chapter 2.Axioms for the Plane Incidence Axiom 112 Distance and the Ruler Axiom Protractor Axiom and Angles Plane Separation 2479 Rigid Motions and Lines The Other Axioms Exercises and Explorations 204 Chapter 3.Existence and Properties of Reflections Deducing the Properties of Reflections IsoscelesTriangles and Kites 3379 Circles and Lines
Contents Introduction xi Advice for Students and Less Experienced Geometers xiii Information for More Experienced Geometers xv A Chapter Guide for Instructors and Others xvii Acknowledgments xxi Chapter 1. Congruence and Rigid Motions 1 Rigid Motions 3 Informal Preview of a Problem Solution 4 Sameness Properties of Congruence 5 Exercises and Explorations 9 Chapter 2. Axioms for the Plane 11 Incidence Axiom 12 Distance and the Ruler Axiom 12 Protractor Axiom and Angles 14 Plane Separation 17 Rigid Motions and Lines 19 The Other Axioms 20 Exercises and Explorations 21 Chapter 3. Existence and Properties of Reflections 23 Deducing the Properties of Reflections 23 Isosceles Triangles and Kites 27 Circles and Lines 29 vii
Contents Light,Angles,and Reflections 30 Paper Folding and Tools for Construction 31 Exercises and explorations 32 Chapter 4 Congruence of Triangles 35 Triangle Congruence Tests Applications of Triangle Congruence 3 Properties of Rigid Motions Midpoint Triangle and Angle Sum 04 Exercises and Explorations 42 Chapter 5.Rotation and Orientation Rotations and Double Reflections Rotation-Reflection Relations 50 Symmetry at a Point Orientation of a Plane 5 Orientation-Preserving and Orientation-Reversing Transformations Exercises and Explorations 62 Chapter 6.Half-turns and Inequalities in Triangles 67 Half-turn Properties Inequalities with Angles Circles and Distance to Lines Reflections and the Triangle Inequality 8759 Exercises and Explorations Chapter 7.Parallel Lines and Translations The Euclidean Parallel Postulate Transversals and Parallel Lines Parallelograms Rectangles Midpoint Figures Generalizing Parallelograms 335899%90 Translations as Half-turn Products Products of Translations Direction from Translation 2 Direction and Rotation from Polar Angle 1 Vectors 107 Exercises and Explorations 108
viii Contents Light, Angles, and Reflections 30 Paper Folding and Tools for Construction 31 Exercises and Explorations 32 Chapter 4. Congruence of Triangles 35 Triangle Congruence Tests 36 Applications of Triangle Congruence 39 Properties of Rigid Motions 40 Midpoint Triangle and Angle Sum 41 Exercises and Explorations 42 Chapter 5. Rotation and Orientation 45 Rotations and Double Reflections 46 Rotation-Reflection Relations 50 Symmetry at a Point 51 Orientation of a Plane 55 Orientation-Preserving and Orientation-Reversing Transformations 60 Exercises and Explorations 62 Chapter 6. Half-turns and Inequalities in Triangles 67 Half-turn Properties 67 Inequalities with Angles 68 Circles and Distance to Lines 71 Reflections and the Triangle Inequality 75 Exercises and Explorations 79 Chapter 7. Parallel Lines and Translations 83 The Euclidean Parallel Postulate 83 Transversals and Parallel Lines 85 Parallelograms 88 Rectangles 91 Midpoint Figures 93 Generalizing Parallelograms 96 Translations as Half-turn Products 98 Products of Translations 101 Direction from Translation 102 Direction and Rotation from Polar Angle 104 Vectors 107 Exercises and Explorations 108
Contents Chapter 8.Dilations and Similarity 11 Similarity Theorems for Triangles Right Triangles The Regular Pentagon and Its Ratios Ratios,Signed Ratios,and Scale Factors Transversals of Parallels and Ratios Parallel Segments and Centers of Dilation Construction by Scaling Models Harmonic Division Composition of Dilation Circles.Angles.and Ratios Radical Axis,Intersections,and Triangle Existence Centers of Dilation and the Midpoint Triangle Exercises and Explorations Chapter 9.Area and Its Applications Areas of Triangles and Parallelograms Area Proofs of the Pythagorean Theorem Area and Scaling Area and the circle Affine Relationships and Area Exercises and Explorations Chapter 10.Products and Patterns Products of Rotations Symmetry and 90-Degree Rotations Triangles and 60 Degrees of Rotation Translations and Symmetry Tessellations and Symmetric Wallpaper Designs Translations and Frieze Symmetry Triple Line Reflection Products Exercises and Explorations Chapter 11.Coordinate Geometry Axes and Coordinates Midpoints,Half-turns,and Translations Lines,Dilations,and Equations Euclidean Geometry and Cartesian Coordinates Perpendicular Lines in the Coordinate Plane 25
Contents ix Chapter 8. Dilations and Similarity 113 Similarity Theorems for Triangles 115 Right Triangles 117 The Regular Pentagon and Its Ratios 121 Ratios, Signed Ratios, and Scale Factors 123 Transversals of Parallels and Ratios 125 Parallel Segments and Centers of Dilation 128 Construction by Scaling Models 132 Harmonic Division 134 Composition of Dilations 136 Circles, Angles, and Ratios 139 Radical Axis, Intersections, and Triangle Existence 148 Centers of Dilation and the Midpoint Triangle 152 Exercises and Explorations 155 Chapter 9. Area and Its Applications 161 Areas of Triangles and Parallelograms 161 Area Proofs of the Pythagorean Theorem 165 Area and Scaling 167 Area and the Circle 169 Affine Relationships and Area 172 Exercises and Explorations 175 Chapter 10. Products and Patterns 177 Products of Rotations 178 Symmetry and 90-Degree Rotations 181 Triangles and 60 Degrees of Rotation 188 Translations and Symmetry 192 Tessellations and Symmetric Wallpaper Designs 194 Translations and Frieze Symmetry 200 Triple Line Reflection Products 208 Exercises and Explorations 211 Chapter 11. Coordinate Geometry 217 Axes and Coordinates 217 Midpoints, Half-turns, and Translations 218 Lines, Dilations, and Equations 221 Euclidean Geometry and Cartesian Coordinates 223 Perpendicular Lines in the Coordinate Plane 225
Contents Graphs and Transformations 228 Unit Circle and Rotation Formula 229 Complex Numbers and Transformations of the Plane Barycentric Coordinates Vectors and Affine Transformations Axioms and Models 28 Conclusion Exercises and Explorations Bibliography 253 Index 255
x Contents Graphs and Transformations 228 Unit Circle and Rotation Formula 229 Complex Numbers and Transformations of the Plane 231 Barycentric Coordinates 233 Vectors and Affine Transformations 240 Axioms and Models 245 Conclusion 249 Exercises and Explorations 249 Bibliography 253 Index 255