Mark Ryan has more than three decades' experience as a geometry teacher and tutor. He has a gift for mathematics and a gift for explaining it in plain English. He tutors students in all junior high and high school math courses as well as math test prep, and he's the founder of The Math Center on Chicago's North Shore. Ryan is the author of Geometry For Dummies, Geometry Essentials For Dummies, Calculus For Dummies, and several other math books.

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