Introduction 1
About This Book 1
Conventions Used in This Book 2
How to Use This Book 2
Foolish Assumptions 3
Icons Used in This Book 3
Beyond the Book 4
Where to Go from Here 4
Part 1: Getting Started with Geometry 5
Chapter 1: Introducing Geometry and Geometry Proofs 7
What Is Geometry? 7
Making the Right Assumptions 8
If-Then Logic: If You Bought This Book, Then You Must Love Geometry! 11
What’s a Geometry Proof? 12
Solutions 15
Chapter 2: Points, Segments, Lines, Rays, and Angles 17
Hammering Out Basic Definitions 17
Looking at Union and Intersection Problems 18
Uncovering More Definitions 20
Division in the Ranks: Bisection and Trisection 20
Perfect Hilarity for Perpendicularity 23
You Complete Me: Complementary and Supplementary Angles 25
X Marks the Spot: Vertical Angles 26
Solutions 28
Chapter 3: Your First Geometry Proofs 33
Ready to Try Some Proofs? 33
Proofs Involving Complementary and Supplementary Angles 34
Proofs Involving Adding and Subtracting Segments and Angles 37
Proofs Involving Multiplying and Dividing Angles and Segments 42
Proofs Involving the Transitive and Substitution Properties 46
Solutions 50
Part 2: Triangles, Proof and Non-proof Problems 55
Chapter 4: Triangle Fundamentals and Other Cool Stuff (No Proofs) 57
Triangle Types and Triangle Basics 58
Altitudes, Area, and the Super Hero Formula 61
Balancing Things Out with Medians and Centroids 65
Locating Three More “Centers” of a Triangle 66
The Pythagorean Theorem 71
Solving Pythagorean Triple Triangles 74
Unique Degrees: Two Special Right Triangles 78
Solutions 80
Chapter 5: Proofs Involving Congruent Triangles 91
Sizing Up Three Ways to Prove Triangles Congruent 91
Corresponding Parts of Congruent Triangles Are Congruent 97
Using Isosceles Triangle Rules: If Sides, Then Angles; If Angles, Then Sides 102
Exploring Two More Ways to Prove Triangles Congruent 105
Explaining the Two Equidistance Theorems 108
Solutions 113
Part 3: Polygons, Proof and Non-proof Problems 121
Chapter 6: Quadrilaterals: Your Fine, Four-Sided Friends (Including Proofs) 123
Double-Crossers: Transversals and Their Parallel Lines 124
Quadrilaterals: It’s a Family Affair 128
Discovering the Properties of the Parallelogram and the Kite 132
Properties of Rhombuses, Rectangles, and Squares 137
Unearthing the Properties of Trapezoids and Isosceles Trapezoids 141
Proving That a Quadrilateral Is a Parallelogram or a Kite 143
Proving That a Quadrilateral Is a Rhombus, Rectangle, or Square 147
Solutions 149
Chapter 7: Area, Angles, and the Many Sides of Polygon Geometry (No Proofs) 159
Square Units: Finding the Area of Quadrilaterals 159
The Standard Formula for the Area of Regular Polygons 163
More Fantastically Fun Polygon Formulas 165
Solutions 168
Chapter 8: Similarity: Size Doesn’t Matter (Including Proofs) 175
Defining Similarity 176
Proving Triangles Similar 179
Corresponding Sides and CSSTP — Cats Stalk Silently Then Pounce 183
Similar Rights: The Altitude-on-Hypotenuse Theorem 186
Discovering Three More Theorems Involving Proportions 190
Solutions 195
Part 4: Circles, Proof and Non-proof Problems 205
Chapter 9: Circular Reasoning (Including Proofs) 207
The Segments Within: Radii and Chords 207
Introducing Arcs and Central Angles 211
Touching on Radii and Tangents 215
Solutions 218
Chapter 10: Scintillating Circle Formulas (No Proofs) 223
Pizzas, Slices, and Crusts: Finding Area and “Perimeter” of Circles, Sectors, and Segments 223
Angles, Circles, and Their Connections: The Angle-Arc Theorems and Formulas 226
The Power Theorems That Be 230
Solutions 233
Part 5: 3-d Geometry: Proof and Non-proof Problems 239
Chapter 11: 2-D Stuff Standing Up (Including Proofs) 241
Lines Perpendicular to Planes: They’re All Right 241
Parallel, Perpendicular, and Intersecting Lines and Planes 245
Solutions 249
Chapter 12: Solid Geometry: Digging into Volume and Surface Area (No Proofs) 253
Starting with Flat-Top Figures 253
Sharpening Your Skills with Pointy-Top Figures 256
Rounding Out Your Understanding with Spheres 259
Solutions 261
Part 6: Coordinate Geometry, Loci, and Constructions: Proof and Non-proof Problems 269
Chapter 13: Coordinate Geometry, Courtesy of Descartes (Including Proofs) 271
Formulas, Schmormulas: Slope, Distance, and Midpoint 272
Mastering Coordinate Proofs with Algebra 275
Using the Equations of Lines and Circles 276
Solutions 279
Chapter 14: Transforming the (Geometric) World: Reflections, Rotations, and Translations (No Proofs) 285
Reflections on Mirror Images 286
Lost in Translation 289
So You Say You Want a Rotation? 292
Working with Glide Reflections 294
Solutions 297
Chapter 15: Laboring Over Loci and Constructions (No Proofs) 301
Tackling Locus Problems 301
Compass and Straightedge Constructions 306
Solutions 311
Chapter 16: Ten Things You Better Know (for Geometry), or Your Name Is Mudd 319
The Pythagorean Theorem (the Queen of All Geometry Theorems) 319
Special Right Triangles 320
Area Formulas 320
Sum of Angles 320
Circle Formulas 321
Angle-Arc Theorems 321
Power Theorems 321
Coordinate Geometry Formulas 322
Volume Formulas 322
Surface Area Formulas 322
Index 323
