Geometry Workbook For Dummies

Introduction 1

About This Book 1

Conventions Used in This Book 2

How to Use This Book 2

Foolish Assumptions 2

How This Book is Organized 3

Part I: Getting Started 3

Part II: Triangles 3

Part III: Polygons 3

Part IV: Circles 4

Part V: 3-D Geometry and Coordinate Geometry 4

Part VI: The Part of Tens 4

Icons Used in This Book 4

Where to Go from Here 5

Part I: Getting Started 7

Chapter 1: Introducing Geometry and Geometry Proofs! 9

Making the Right Assumptions: What is Geometry? 9

If-Then Logic: If You Bought This Book, Then You Must Love Geometry! 12

Discovering Geometry Proofs 14

Solutions for Introducting Geometry and Geometry Proofs! 17

Chapter 2: Points, Segments, Lines, Rays, and Angles 19

Basic Definitions 19

Union and Intersection Problems 20

Division in the Ranks: Bisection and Trisection 22

Perfect Hilarity for Perpendicularity 24

You Complete Me: Complementary and Supplementary Angles 26

Adding and Subtracting Segments and Angles 30

Multiplying and Dividing Angles and Segments 33

X Marks the Spot: Using Vertical Angles 37

Switching It Up with the Transitive and Substitution Properties 39

Solutions for Points, Segments, Lines, Rays, and Angles 42

Part II: Triangles 49

Chapter 3: Triangle Fundamentals and Other Cool Stuff 51

Triangle Types and Triangle Basics 51

Altitudes, Area, and the Super Hero Formula 55

Balancing Things Out with Medians and Centroids 58

Three More “Centers” of a Triangle 59

The Pythagorean Theorem 64

Pythagorean Triple Triangles 67

Unique Degrees: Two Special Right Triangles 70

Solutions for Triangle Fundamentals and Other Cool Stuff 74

Chapter 4: Congruent Triangles 83

Sizing Up Three Ways to Prove Triangles Congruent 83

Corresponding Parts of Congruent Triangles Are Congruent (CPCTC) 89

Isosceles Rules: If Sides, Then Angles; If Angles, Then Sides 93

Two More Ways to Prove Triangles Congruent 96

The Two Equidistance Theorems 99

Solutions for Congruent Triangles 104

Part III: Polygons 113

Chapter 5: Quadrilaterals: Your Fine, Four-Sided Friends 115

Double-Crossers: Transversals and Their Parallel Lines 115

Quadrilaterals: It’s a Family Affair 120

Properties of the Parallelogram and the Kite 123

Properties of Rhombuses, Rectangles, and Squares 127

Properties of Trapezoids and Isosceles Trapezoids 130

Proving That a Quadrilateral is a Parallelogram or a Kite 132

Proving That a Quadrilateral is a Rhombus, Rectangle, or Square 136

Solutions for Quadrilaterals: Your Fine, Four-Sided Friends 139

Chapter 6: Area, Angles, and the Many Sides of Polygon Geometry 151

Square Units: Finding the Area of Quadrilaterals 151

A Standard Formula for the Area of Regular Polygons 155

More Fantastically Fun Polygon Formulas 157

Solutions for Area, Angles, and the Many Sides of Polygon Geometry 160

Chapter 7: Similarity: Size Doesn’t Matter 165

Defining Similarity: The Sincerest Flattery 165

Proving Triangles Similar 168

Corresponding Sides and CSSTP — Cats Stalk Silently Then Pounce 172

Similar Rights: The Altitude-on-Hypotenuse Theorem 175

Three More Theorems Involving Proportions 178

Solutions for Similarity: Size Doesn’t Matter 183

Part IV: Circles 193

Chapter 8: Circular Reasoning 195

The Segments Within: Radii and Chords 195

Introducing Arcs, Chords, and Central Angles 199

Touching on Radii and Tangents 202

Solutions for Circular Reasoning 206

Chapter 9: Scintillating Circle Formulas 211

Not Quite Coming Full Circle: Area and “Perimeter” of Circles, Sectors, and Segments 211

Angles, Circles, and Their Connections: The Angle-Arc Theorems and Formulas 214

The Power Theorems That Be 217

Solutions for Scintillating Circle Formulas 221

Part V: 3-D Geometry and Coordinate Geometry 227

Chapter 10: 2-D Stuff Standing Up 229

Lines Perpendicular to Planes: They’re All Right 229

Parallel, Perpendicular, and Intersecting Lines and Planes 233

Solutions for 2-D Stuff Standing Up 238

Chapter 11: Solid Geometry: Digging into Volume and Surface Area 243

Starting with Flat-Top Figures 243

Sharpening Your Skills with Pointy-Top Figures 246

Rounding Out Your Understanding with Spheres 249

Solutions for Solid Geometry 251

Chapter 12: Coordinate Geometry, Courtesy of Descartes 257

Formulas, Schmormulas: Slope, Distance, and Midpoint 257

Mastering Coordinate Proofs with Algebra 260

Using the Equations of Lines and Circles 261

Solutions for Coordinate Geometry, Courtesy of Descartes 264

Chapter 13: Transforming the (Geometric) World: Reflections, Rotations, and Translations 271

Reflections on Mirror Images 271

Les Transformations Géométriques 274

So You Say You Want a    Rotation? 276

Working with Glide Reflections 278

Solutions for Transforming the World 281

Part VI: The Part of Tens 285

Chapter 14: Ten (Plus) Incredibly Fantastic Strategies for Doing Proofs 287

Look for Congruent Triangles 287

Try to Find Isosceles Triangles 287

Look for Radii, and Draw More Radii 288

Look for Parallel Lines 288

Make a Game Plan 288

Use All the Givens 288

Check Your If-Then Logic 288

Work Backwards 289

Make Up Numbers for Segments and Angles 289

Think Like a Computer 289

Bonus! Number 11 (Like the Amp in Spinal Tap That Goes Up to 11): Do Something! 290

Chapter 15: Ten Things You Better Know (for Geometry), or Your Name is Mudd 291

The Pythagorean Theorem (the Queen of All Geometry Theorems) 291

Special Right Triangles 291

Area Formulas 292

Sum of Angles 292

Circle Formulas 292

Angle-Arc Theorems 292

Power Theorems 293

Coordinate Geometry Formulas 293

Volume Formulas 293

Surface Area Formulas 294

Index 295