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Previous edition: 9781118853269
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Allen Ma and Amber Kuang are math teachers at John F. Kennedy High School in Bellmore, NY. Allen has taught geometry for more than 25 years, has coached the math team, and is a former honors math research coordinator. Amber has taught all levels of math, from algebra to calculus, for 20 years.
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
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