**Introduction 1**

What You’ll Find 1

Beyond the Book 1

Where to Go for Additional Help 2

**Part 1: The Questions 3**

**Chapter 1: Diving into Geometry 5**

The Problems You’ll Work On 5What to Watch Out For 5

Understanding Basic Geometric Definitions 6

Applying Algebra to Basic Geometric Definitions 7

Recognizing Geometric Terms 8

Properties and Postulates 8

Adjacent Angles, Vertical Angles, and Angles That Form Linear Pairs 10

Complementary and Supplementary Angles 10

Angles in a Triangle 11

**Chapter 2: Constructions 13**

The Problems You’ll Work On 13

What to Watch Out For 13

Creating Congruent Constructions 14

Constructions Involving Angles and Segments 14

Parallel and Perpendicular Lines 15

Creative Constructions 16

**Chapter 3: Geometric Proofs with Triangles 17**

The Problems You’ll Work On 17

What to Watch Out For 17

Triangle Congruence Theorems 18

Completing Geometric Proofs Using Triangle Congruence Theorems 21

Overlapping Triangle Proofs 24

Indirect Proofs 28

**Chapter 4: Classifying Triangles 31**

The Problems You’ll Work On 31

What to Watch Out For 31

Classifying Triangles by Their Sides 32

Properties of Isosceles, Equilateral, and Right Triangles 33

Classifying Triangles by Their Angles 34

Understanding the Classification of Triangles 35

Geometric Proofs Involving Isosceles Triangles 36

**Chapter 5: Investigating the Centers of a Triangle 41**

The Problems You’ll Work On 41

What to Watch Out For 41

The Incenter of a Triangle 42

Understanding the Orthocenter 42

Understanding Centroids 43

Finding the Centroid of a Triangle 44

The Circumcenter of a Triangle 45

Recognizing Triangle Centers 45

Constructing the Centers of a Triangle 47

The Euler Line 48

**Chapter 6: Similar Triangles 49**

The Problems You’ll Work On 49

What to Watch Out For 49

Understanding Similar Triangles 50

Midsegments 50

Creating Similar Triangles 51

Similar-Triangle Word Problems 52

Proving That Two Triangles Are Similar to Each Other 53

Proving That Corresponding Sides Are in Proportion 54

Proving with the Means and Extremes 55

**Chapter 7: The Right Triangle 59**

The Problems You’ll Work On 59

What to Watch Out For 59

Pythagorean Theorem 60

Right Triangle Proportions 60

Word Problems Involving Right-Triangle Proportions 62

Working with Special Right Triangles 63

Application of Special Right Triangles 63

Trigonometric Ratios 65

Applying the Trigonometric Ratios to Word Problems 66

**Chapter 8: Triangle Inequalities 67**

The Problems You’ll Work On 67

What to Watch Out For 67

Relationships between the Sides and Angles of a Triangle 68

Triangle Inequality Theorem 69

Finding the Missing Side Length 69

Isosceles Triangles 70

Using the Exterior Angle Theorem of a Triangle 70

Geometric Proofs Involving Triangle Inequality Theorems 72

**Chapter 9: Polygons 75**

The Problems You’ll Work On 75

What to Watch Out For 75

Naming Polygons 76

Understanding Angles of a Polygon 76

The Sum of the Interior and Exterior Angles of a Polygon 77

Finding the Area of Regular Polygons 79

**Chapter 10: Properties of Parallel Lines 81**

The Problems You’ll Work On 81

What to Watch Out For 81

Alternate Interior and Alternate Exterior Angles 82

Classifying Triangles by Their Angle Measurements 83

Finding Angle Measures Involving Parallel Lines 83

Reviewing Corresponding, Adjacent, and Vertical Angles 84

More Practice with Angles Involving Parallel Lines 85

Geometric Proof Incorporating Parallel Lines 86

Geometric Proof Incorporating Parallel Lines 87

**Chapter 11: Properties of Quadrilaterals 89**

The Problems You’ll Work On 89

What to Watch Out For 89

Properties of Parallelograms 90

Word Problems with Parallelograms 91

Properties of Rectangles 91

Finding the Diagonal of a Rectangle 92

Reviewing the Properties of a Rhombus 93

Diagonal Properties of a Rhombus 93

Properties of a Square 94

Properties of a Trapezoid 96

**Chapter 12: Coordinate Geometry 97**

The Problems You’ll Work On 97

What to Watch Out For 97

Determining Distance 98

Using the Midpoint Formula 98

Using the Slope Formula 99

Parallel and Perpendicular Lines 100

Writing the Equation of a Line in Slope-Intercept Form 101

Coordinate Geometry Proofs 102

**Chapter 13: Transformational Geometry 105**

The Problems You’ll Work On 105

What to Watch Out For 105

Rigid Motion 106

Reflecting Points over the x- and y-axes 109

Writing Equations for Lines of Reflection 110

Understanding Point Symmetry 110

Triangle Translations 111

Translating Points 112

Finding Translation Rules 113

Doing Dilations 113

Practicing with Rotations 114

Understanding the Rules for Rotations 115

Rigid Motion of Triangles 115

Compositions of Transformations 116

Glide Reflections and Direct and Indirect Isometries 117

Transformations of a Segment 118

Trying Rigid Motion Constructions 118

**Chapter 14: Exploring Circles 121**

The Problems You’ll Work On 121

What to Watch Out For 121

Working with the Circumference of a Circle 122

Understanding the Area of a Circle 122

Working with Sectors 123

Arc Length 124

The Equation of a Circle in Standard Form 124

**Chapter 15: Circle Theorems 127**

The Problems You’ll Work On 127

What to Watch Out For 127

Central Angles and Arcs 128

Inscribed Angles and Arcs 128

Angles Formed by Intersecting Chords of a Circle 129

Angles Formed by Secants and Tangents 131

The Intersecting Chord Theorem 132

Lengths of Tangents and Secants 135

Tangent and Radius 137

“BIG” Circle Problems 137

Circle Proofs 138

**Chapter 16: Three-Dimensional Geometry 141**

The Problems You’ll Work On 141

What to Watch Out For 141

Understanding Points, Lines, and Planes 142

Surface Area of Solid Figures 143

Calculating the Volume of Solid Figures 145

Rotations of Two-Dimensional Figures 146

**Chapter 17: Locus Problems 149**

The Problems You’ll Work On 149

What to Watch Out For 149

Basic Locus Theorems 150

Loci Using Coordinate Geometry 150

The Locus of Points Equidistant from One or Two Lines 151

The Locus of Points Equidistant from Two Points 151

Writing the Equation of a Circle 152

Compound Locus in Coordinate Geometry 152

Compound and Challenging Locus Problems 153

**Part 2: The Answers 155**

**Chapter 18: Answers and Explanations 157**

Index 439