**Pure and Applied Undergraduate Texts**

Volume: 51;
2021;
258 pp;
Softcover

MSC: Primary 51; 97; 20;

**Print ISBN: 978-1-4704-6307-6
Product Code: AMSTEXT/51**

List Price: $99.00

AMS Member Price: $79.20

MAA Member Price: $89.10

**Electronic ISBN: 978-1-4704-6443-1
Product Code: AMSTEXT/51.E**

List Price: $99.00

AMS Member Price: $79.20

MAA Member Price: $89.10

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#### Supplemental Materials

# Geometry Transformed: Euclidean Plane Geometry Based on Rigid Motions

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*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.

This book is published in cooperation with IAS/Park City Mathematics Institute

#### Readership

Undergraduate and graduate students interested in geometry aligned with Common Core standards (CCSSM).

# Table of Contents

## Geometry Transformed: Euclidean Plane Geometry Based on Rigid Motions

- Cover Cover11
- Title page i5
- Copyright ii6
- Contents vii9
- Introduction xi13
- Advice for Students and Less Experienced Geometers xiii15
- Information for More Experienced Geometers xv17
- A Chapter Guide for Instructors and Others xvii19
- Acknowledgments xxi23
- Chapter 1. Congruence and Rigid Motions 125
- Chapter 2. Axioms for the Plane 1135
- Chapter 3. Existence and Properties of Reflections 2347
- Chapter 4. Congruence of Triangles 3559
- Chapter 5. Rotation and Orientation 4569
- Chapter 6. Half-turns and Inequalities in Triangles 6791
- Chapter 7. Parallel Lines and Translations 83107
- The Euclidean Parallel Postulate 83107
- Transversals and Parallel Lines 85109
- Parallelograms 88112
- Rectangles 91115
- Midpoint Figures 93117
- Generalizing Parallelograms 96120
- Translations as Half-turn Products 98122
- Products of Translations 101125
- Direction from Translation 102126
- Direction and Rotation from Polar Angle 104128
- Vectors 107131
- Exercises and Explorations 108132

- Chapter 8. Dilations and Similarity 113137
- Similarity Theorems for Triangles 115139
- Right Triangles 117141
- The Regular Pentagon and Its Ratios 121145
- Ratios, Signed Ratios, and Scale Factors 123147
- Transversals of Parallels and Ratios 125149
- Parallel Segments and Centers of Dilation 128152
- Construction by Scaling Models 132156
- Harmonic Division 134158
- Composition of Dilations 136160
- Circles, Angles, and Ratios 139163
- Radical Axis, Intersections, and Triangle Existence 148172
- Centers of Dilation and the Midpoint Triangle 152176
- Exercises and Explorations 155179

- Chapter 9. Area and Its Applications 161185
- Chapter 10. Products and Patterns 177201
- Products of Rotations 178202
- Symmetry and 90-Degree Rotations 181205
- Triangles and 60 Degrees of Rotation 188212
- Translations and Symmetry 192216
- Tessellations and Symmetric Wallpaper Designs 194218
- Translations and Frieze Symmetry 200224
- Triple Line Reflection Products 208232
- Exercises and Explorations 211235

- Chapter 11. Coordinate Geometry 217241
- Axes and Coordinates 217241
- Midpoints, Half-turns, and Translations 218242
- Lines, Dilations, and Equations 221245
- Euclidean Geometry and Cartesian Coordinates 223247
- Perpendicular Lines in the Coordinate Plane 225249
- Graphs and Transformations 228252
- Unit Circle and Rotation Formula 229253
- Complex Numbers and Transformations of the Plane 231255
- Barycentric Coordinates 233257
- Vectors and Affine Transformations 240264
- Axioms and Models 245269
- Conclusion 249273
- Exercises and Explorations 249273

- Bibliography 253277
- Index 255279
- Back Cover Back Cover1284