Introduction 1
About This Book 1
Conventions Used in This Book 2
Foolish Assumptions 2
Icons Used in This Book 3
Where to Go from Here 3
Chapter 1: An Overview of Geometry 5
The Geometry of Shapes 6
One-dimensional shapes 6
Two-dimensional shapes 6
Three-dimensional shapes 6
Geometry Proofs 6
Am I Ever Going to Use This? 7
When youâll use your knowledge of shapes 7
When youâll use your knowledge of proofs 8
Getting Down with Definitions 9
A Few Points on Points 11
Lines, Segments, and Rays 12
Horizontal and vertical lines 12
Doubling up with pairs of lines 13
Investigating the Plane Facts 14
Everybodyâs Got an Angle 14
Five types of angles 15
Angle pairs 16
Bisection and Trisection 18
Segments 18
Angles 18
Chapter 2: Geometry Proof Starter Kit 21
The Lay of the (Proof) Land 21
Reasoning with If-Then Logic 23
If-then chains of logic 24
Definitions, theorems, and postulates 25
Bubble logic 26
Complementary and Supplementary Angles 27
Addition and Subtraction 29
Addition theorems 29
Subtraction theorems 33
Like Multiples and Like Divisions 34
Congruent Vertical Angles 36
Transitivity and Substitution 37
Chapter 3: Tackling a Longer Proof 41
Making a Game Plan 42
Using All the Givens 42
Using If-Then Logic 43
Chipping Away at the Problem 45
Working Backward 47
Filling in the Gaps 49
Writing out the Finished Proof 49
Chapter 4: Triangle Fundamentals 51
Taking in a Triangleâs Sides 51
Scalene triangles 52
Isosceles triangles 52
Equilateral triangles 52
Triangle Classification by Angles 52
The Triangle Inequality Principle 53
Sizing up Triangle Area 54
A triangleâs altitude or height 54
Determining a triangleâs area 56
Regarding Right Triangles 57
The Pythagorean Theorem 58
Pythagorean Triple Triangles 60
The Fab Four triangles 61
Families of Pythagorean triple triangles 61
Two Special Right Triangles 64
The 45 - 45 - 90 triangle 64
The 30 - 60 - 90 triangle 66
Chapter 5: Congruent Triangle Proofs 69
Proving Triangles Congruent 69
SSS: The side-side-side method 70
SAS: Side-angle-side 72
ASA: The angle-side-angle tack 74
AAS: Angle-angle-side 74
Last but not least: HLR 75
Taking the Next Step with CPCTC 75
Defining CPCTC 76
Tackling a CPCTC proof 76
The Isosceles Triangle Theorems 79
The Two Equidistance Theorems 81
Determining a perpendicular bisector 81
Using a perpendicular bisector 83
Chapter 6: Quadrilaterals 85
Parallel Line Properties 85
Parallel lines with a transversal 85
The transversal theorems 87
The Seven Special Quadrilaterals 89
Working with Auxiliary Lines 90
The Properties of Quadrilaterals 93
Properties of the parallelogram 93
Properties of the three special parallelograms 95
Properties of the kite 98
Properties of the trapezoid and the isosceles trapezoid 99
Proving That Youâve Got a Particular Quadrilateral 100
Proving youâve got a parallelogram 100
Proving that youâve got a rectangle, rhombus, or square 103
Proving that youâve got a kite 104
Chapter 7: Polygon Formulas 107
The Area of Quadrilaterals 107
Quadrilateral area formulas 108
Why the formulas work 108
Trying a few area problems 110
The Area of Regular Polygons 113
The polygon area formulas 114
Tackling an area problem 114
Angle and Diagonal Formulas 115
Interior and exterior angles 116
A polygon angle problem 117
Criss-crossing with diagonals 118
Chapter 8: Similarity 119
Similar Figures 119
Defining similar polygons 119
How similar figures line up 121
Solving a similarity problem 122
Proving Triangles Similar 124
Tackling an AA proof 125
Using SSS~ 126
An SAS~ proof 127
Splitting Right Triangles with the Altitude-on-Hypotenuse Theorem 128
More Proportionality Theorems 130
The Side-Splitter Theorem 130
The Angle-Bisector Theorem 132
Chapter 9: Circle Basics 135
Radii, Chords, and Diameters 135
Five circle theorems 136
Using extra radii 136
Arcs and Central Angles 138
Tangents 138
The Pizza Slice Formulas 140
Determining arc length 140
Sector and segment area 141
The Angle-Arc Formulas 143
Angles on a circle 144
Angles inside a circle 144
Angles outside a circle 145
Keeping the formulas straight 146
The Power Theorems 147
The Chord-Chord Theorem 148
The Tangent-Secant Theorem 149
The Secant-Secant Theorem 149
Condensing the power theorems into a single idea 150
Chapter 10: 3-D Geometry 151
Flat-Top Figures 151
Pointy-Top Figures 154
Spheres 159
Chapter 11: Coordinate Geometry 161
The Coordinate Plane 161
Slope, Distance, and Midpoint 162
The slope dope 162
The distance formula 164
The midpoint formula 165
Trying out the formulas 166
Equations for Lines and Circles 167
Line equations 168
The circle equation 168
Chapter 12: Ten Big Reasons to Use in Proofs 171
The Reflexive Property 171
Vertical Angles are Congruent 171
The Parallel-Line Theorems 172
Two Points Determine a Line 172
All Radii are Congruent 173
If Sides, Then Angles 173
If Angles, Then Sides 173
Triangle Congruence 173
CPCTC 174
Triangle Similarity 174
Index 175
