| Introduction |
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1 | (1) |
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1 | (1) |
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Conventions Used in This Book |
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2 | (1) |
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2 | (1) |
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3 | (1) |
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How This Book is Organized |
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3 | (3) |
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Getting Started with Geometry Basics |
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4 | (1) |
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4 | (1) |
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Triangles: Polygons of the Three-Sided Variety |
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4 | (1) |
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Polygons of the Four-or-More-Sided Variety |
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5 | (1) |
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Working with Not-So-Vicious Circles |
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5 | (1) |
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Going Deep with 3-D Geometry |
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5 | (1) |
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Placement, Points, and Pictures: Alternative Geometry Topics |
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6 | (1) |
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6 | (1) |
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6 | (1) |
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6 | (1) |
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7 | (2) |
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Part I: Getting Started With Geometry Basics |
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9 | (38) |
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11 | (10) |
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Studying the Geometry of Shapes |
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12 | (2) |
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12 | (1) |
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12 | (1) |
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13 | (1) |
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Getting Acquainted with Geometry Proofs |
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14 | (4) |
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Easing into proofs with an everyday example |
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15 | (1) |
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Turning everyday logic into a proof |
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15 | (1) |
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Sampling a simple geometrical proof |
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16 | (2) |
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When Am I Ever Going to Use This? |
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18 | (1) |
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When you'll use your knowledge of shapes |
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18 | (1) |
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When you'll use your knowledge of proofs |
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19 | (1) |
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Why You Won't Have Any Trouble with Geometry |
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19 | (2) |
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Building Your Geometric Foundation |
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21 | (14) |
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Getting Down with Definitions |
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21 | (4) |
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25 | (1) |
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Lines, Segments, and Rays Pointing Every Which Way |
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26 | (3) |
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Singling out horizontal and vertical lines |
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26 | (1) |
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Doubling up with pairs of lines |
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27 | (2) |
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Investigating the Plane Facts |
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29 | (1) |
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30 | (5) |
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Goldilocks and the three angles: Small, large, and just ``right'' |
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30 | (1) |
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Angle pairs: Often joined at the hip |
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31 | (4) |
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Sizing Up Segments and Analyzing Angles |
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35 | (12) |
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Measuring Segments and Angles |
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35 | (4) |
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35 | (1) |
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36 | (3) |
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Adding and Subtracting Segments and Angles |
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39 | (1) |
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Cutting in Two or Three: Bisection and Trisection |
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40 | (3) |
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Bisecting and trisecting segments |
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40 | (2) |
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Bisecting and trisecting angles |
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42 | (1) |
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Proving (Not Jumping to) Conclusions about Figures |
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43 | (4) |
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Part II: Introducing Proofs |
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47 | (42) |
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49 | (10) |
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Getting the Lay of the Land: The Components of a Formal Geometry Proof |
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49 | (2) |
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Reasoning with If-Then Logic |
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51 | (5) |
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52 | (1) |
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You've got your reasons: Definitions, theorems, and postulates |
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53 | (2) |
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Bubble logic for two-column proofs |
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55 | (1) |
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Horsing Around with a Two-Column Proof |
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56 | (3) |
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Your Starter Kit of Easy Theorems and Short Proofs |
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59 | (20) |
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Doing Right and Going Straight: Complementary and Supplementary Angles |
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59 | (4) |
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Addition and Subtraction: Eight No-Big-Deal Theorems |
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63 | (7) |
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63 | (4) |
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67 | (3) |
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Like Multiples and Like Divisions? Then These Theorems Are for You! |
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70 | (3) |
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The X-Files: Congruent Vertical Angles Are Out There |
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73 | (2) |
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Pulling the Switch with the Transitive and Substitution Properties |
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75 | (4) |
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The Ultimate Guide to Tackling a Longer Proof |
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79 | (10) |
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80 | (1) |
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81 | (1) |
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Making Sure You Use If-Then Logic |
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81 | (2) |
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Chipping Away at the Problem |
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83 | (2) |
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Jumping Ahead and Working Backward |
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85 | (1) |
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86 | (2) |
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Writing Out the Finished Proof |
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88 | (1) |
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Part III: Triangles: Polygons of the Three-Sided Variety |
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89 | (60) |
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Grasping Triangle Fundamentals |
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91 | (16) |
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Taking In a Triangle's Sides |
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91 | (3) |
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Scalene triangles: Akilter, awry and askew |
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92 | (1) |
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Isosceles triangles: Nice pair o' legs |
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93 | (1) |
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Equilateral triangles: All parts are created equal |
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93 | (1) |
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Introducing the Triangle Inequality Principle |
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94 | (2) |
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Getting to Know Triangles by Their Angles |
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96 | (1) |
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96 | (6) |
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96 | (2) |
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Determining a triangle's area |
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98 | (4) |
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Locating the ``Centers'' of a Triangle |
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102 | (5) |
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Balancing on the centroid |
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102 | (2) |
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Finding three more ``centers'' of a triangle |
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104 | (3) |
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Regarding Right Triangles |
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107 | (16) |
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Applying the Pythagorean Theorem |
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107 | (6) |
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Perusing Pythagorean Triple Triangles |
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113 | (5) |
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The Fab Four Pythagorean triple triangles |
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113 | (2) |
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Families of Pythagorean triple triangles |
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115 | (3) |
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Getting to Know Two Special Right Triangles |
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118 | (5) |
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The 45° - 45° - 90° triangle --- half a square |
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118 | (2) |
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The 30° - 60° - 90° triangle --- half of an equilateral triangle |
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120 | (3) |
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Completing Congruent Triangle Proofs |
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123 | (26) |
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Introducing Three Ways to Prove Triangles Congruent |
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123 | (8) |
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SSS: Using the side-side-side method |
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124 | (2) |
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SAS: Taking the side-angle-side approach |
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126 | (2) |
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ASA: Taking the angle-side-angle tack |
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128 | (3) |
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CPCTC: Taking Congruent Triangle Proofs a Step Further |
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131 | (4) |
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131 | (1) |
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132 | (3) |
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Eying the Isosceles Triangle Theorems |
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135 | (2) |
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Trying Out Two More Ways to Prove Triangles Congruent |
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137 | (4) |
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AAS: Using the angle-angle-side theorem |
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137 | (3) |
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HLR: The right approach for right triangles |
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140 | (1) |
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Going the Distance with the Two Equidistance Theorems |
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141 | (4) |
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Determining a perpendicular bisector |
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142 | (1) |
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Using a perpendicular bisector |
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143 | (2) |
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Making a Game Plan for a Longer Proof |
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145 | (2) |
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Running a Reverse with Indirect Proofs |
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147 | (2) |
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Part IV: Polygons of the Four-or-More Sided Variety |
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149 | (76) |
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The Seven Wonders of the Quadrilateral World |
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151 | (22) |
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Getting Started with Parallel-Line Properties |
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151 | (6) |
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Crossing the line with transversals: Definitions and theorems |
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152 | (1) |
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Applying the transversal theorems |
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153 | (2) |
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Working with more than one transversal |
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155 | (2) |
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Meeting the Seven Members of the Quadrilateral Family |
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157 | (5) |
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Looking at quadrilateral relationships |
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158 | (1) |
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Working with auxiliary lines |
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159 | (3) |
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Giving Props to Quads: The Properties of Quadrilaterals |
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162 | (11) |
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Properties of the parallelogram |
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162 | (4) |
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Properties of the three special parallelograms |
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166 | (3) |
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169 | (2) |
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Properties of the trapezoid and the isosceles trapezoid |
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171 | (2) |
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Proving That You've Got a Particular Quadrilateral |
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173 | (14) |
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Putting Properties and Proof Methods Together |
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173 | (2) |
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Proving That a Quadrilateral Is a Parallelogram |
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175 | (5) |
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Surefire ways of ID-ing a parallelogram |
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176 | (1) |
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Trying some parallelogram proofs |
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177 | (3) |
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Proving That a Quadrilateral Is a Rectangle, Rhombus, or Square |
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180 | (5) |
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Revving up for rectangle proofs |
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181 | (1) |
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Waxing rhapsodic about rhombus proofs |
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182 | (2) |
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Squaring off with square proofs |
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184 | (1) |
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Proving That a Quadrilateral Is a Kite |
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185 | (2) |
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Polygon Formulas: Area, Angles, and Diagonals |
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187 | (16) |
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Calculating the Area of Quadrilaterals |
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187 | (8) |
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Setting forth the quadrilateral area formulas |
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188 | (1) |
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Getting behind the scenes of the formulas |
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188 | (2) |
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Trying a few area problems |
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190 | (5) |
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Finding the Area of Regular Polygons |
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195 | (4) |
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Presenting polygon area formulas |
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195 | (1) |
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Tackling more area problems |
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196 | (3) |
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Using Polygon Angle and Diagonal Formulas |
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199 | (4) |
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Interior and exterior design: Exploring polygon angles |
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199 | (1) |
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Handling the ins and outs of a polygon angle problem |
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200 | (1) |
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Criss-crossing with diagonals |
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201 | (2) |
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Similarity: Same Shape, Different Size |
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203 | (22) |
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Getting Started with Similar Figures |
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204 | (5) |
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Defining and naming similar polygons |
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204 | (1) |
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How similar figures line up |
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205 | (2) |
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Solving a similarity problem |
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207 | (2) |
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Proving Triangles Similar |
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209 | (4) |
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210 | (1) |
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Using SSS- to prove triangles similar |
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211 | (1) |
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Working through an SAS~ proof |
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212 | (1) |
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CASTC and CSSTP, the Cousins of CPCTC |
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213 | (3) |
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Working through a CASTC proof |
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214 | (1) |
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215 | (1) |
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Splitting Right Triangles with the Altitude-on-Hypotenuse Theorem |
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216 | (3) |
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Getting Proportional with Three More Theorems |
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219 | (6) |
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The Side-Splitter Theorem: It'll make you split your sides |
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219 | (2) |
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Crossroads: The Side-Splitter Theorem extended |
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221 | (2) |
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The Angle-Bisector Theorem |
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223 | (2) |
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Part V: Working With Not-So-Vicious Circles |
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225 | (38) |
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Coming Around to Circle Basics |
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227 | (16) |
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The Straight Talk on Circles: Radii and Chords |
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228 | (4) |
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Defining radii, chords, and diameters |
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228 | (1) |
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Introducing five circle theorems |
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229 | (1) |
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229 | (2) |
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Using extra radii to solve a problem |
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231 | (1) |
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Pieces of the Pie: Arcs and Central Angles |
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232 | (5) |
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Three circle definitions for your mathematical pleasure |
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232 | (1) |
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Six scintillating circle theorems |
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233 | (2) |
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Trying your hand at some proofs |
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235 | (2) |
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Going Off on a Tangent about Tangents |
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237 | (6) |
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Introducing the tangent line |
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237 | (1) |
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The common-tangent problem |
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238 | (3) |
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Taking a walk on the wild side with a walk-around problem |
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241 | (2) |
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Circle Formulas and Theorems |
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243 | (20) |
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Chewing on the Pizza Slice Formulas |
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243 | (6) |
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244 | (3) |
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Finding sector and segment area |
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247 | (1) |
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Pulling it all together in a problem |
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248 | (1) |
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Digesting the Angle-Arc Theorems and Formulas |
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249 | (8) |
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250 | (2) |
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252 | (2) |
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254 | (2) |
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Keeping your angle-arc formulas straight |
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256 | (1) |
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Powering Up with the Power Theorems |
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257 | (6) |
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Striking a chord with the Chord-Chord Power Theorem |
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257 | (2) |
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Touching on the Tangent-Secant Power Theorem |
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259 | (1) |
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Seeking out the Secant-Secant Power Theorem |
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260 | (2) |
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Condensing the power theorems into a single idea |
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262 | (1) |
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Part VI: Going Deep with 3-D Geometry |
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263 | (24) |
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3-D Space: Proofs in a Higher Plane of Existence |
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265 | (8) |
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Lines Perpendicular to Planes |
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265 | (4) |
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Parallel, Perpendicular, and Intersecting Lines and Planes |
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269 | (4) |
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The four ways to determine a plane |
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269 | (1) |
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Line and plane interactions |
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270 | (3) |
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Getting a Grip on Solid Geometry |
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273 | (14) |
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Flat-Top Figures: They're on the Level |
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273 | (6) |
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Getting to the Point of Pointy-Top Figures |
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279 | (6) |
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Rounding Things Out with Spheres |
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285 | (2) |
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Part VII: Placement, Points, and Pictures: Alternative Geometry Topics |
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287 | (56) |
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289 | (18) |
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Getting Coordinated with the Coordinate Plane |
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289 | (2) |
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The Slope, Distance, and Midpoint Formulas |
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291 | (7) |
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291 | (3) |
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Going the distance with the distance formula |
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294 | (1) |
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Meeting each other halfway with the midpoint formula |
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295 | (1) |
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The whole enchilada: Putting the formulas together in a problem |
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295 | (3) |
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Proving Properties Analytically |
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298 | (4) |
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298 | (2) |
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Solving the problem algebraically |
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300 | (2) |
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Deciphering Equations for Lines and Circles |
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302 | (5) |
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302 | (1) |
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The standard circle equation |
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303 | (4) |
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Changing the Scene with Geometric Transformations |
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307 | (18) |
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Some Reflections on Reflections |
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308 | (4) |
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Getting oriented with orientation |
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309 | (1) |
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Finding a reflecting line |
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310 | (2) |
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Not Getting Lost in Translations |
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312 | (5) |
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A translation equals two reflections |
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312 | (2) |
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Finding the elements of a translation |
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314 | (3) |
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Turning the Tables with Rotations |
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317 | (4) |
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A rotation equals two reflections |
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317 | (1) |
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Finding the center of rotation and the equations of two reflecting lines |
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318 | (3) |
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Third Time's the Charm: Stepping Out with Glide Reflections |
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321 | (4) |
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A glide reflection equals three reflections |
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322 | (1) |
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Finding the main reflecting line |
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322 | (3) |
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Locating Loci and Constructing Constructions |
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325 | (18) |
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Meeting the Conditions with Loci |
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326 | (8) |
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The four-step process for locus problems |
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326 | (1) |
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Two-dimensional locus problems |
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326 | (6) |
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Three-dimensional locus problems |
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332 | (2) |
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Drawing with the Bare Essentials: Constructions |
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334 | (9) |
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334 | (3) |
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Bisecting angles and segments |
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337 | (2) |
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Two perpendicular line constructions |
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339 | (2) |
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Constructing parallel lines and using them to divide segments |
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341 | (2) |
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Part VIII: The Part of Tens |
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343 | (16) |
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Ten Things to Use as Reasons in Geometry Proofs |
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345 | (6) |
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345 | (1) |
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Vertical Angles Are Congruent |
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346 | (1) |
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The Parallel-Line Theorems |
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346 | (1) |
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Two Points Determine a Line |
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347 | (1) |
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All Radii of a Circle Are Congruent |
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347 | (1) |
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348 | (1) |
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348 | (1) |
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The Triangle Congruence Postulates and Theorems |
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349 | (1) |
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349 | (1) |
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The Triangle Similarity Postulates and Theorems |
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350 | (1) |
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Ten Cool Geometry Problems |
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351 | (8) |
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Eureka! Archimedes's Bathtub Revelation |
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351 | (1) |
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352 | (1) |
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353 | (1) |
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The Circumference of the Earth |
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354 | (1) |
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The Great Pyramid of Khufu |
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354 | (1) |
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355 | (1) |
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355 | (1) |
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356 | (1) |
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357 | (1) |
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357 | (2) |
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359 | (16) |
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Appendix A: Formulas and Other Important Stuff You Should Know |
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361 | (6) |
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361 | (1) |
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362 | (1) |
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363 | (2) |
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365 | (1) |
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Coordinate Geometry Stuff |
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365 | (2) |
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367 | (8) |
| Index |
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375 | |
| Introduction |
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1 | (1) |
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1 | (1) |
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Conventions Used in This Book |
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2 | (1) |
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2 | (1) |
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2 | (1) |
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How This Book Is Organized |
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3 | (1) |
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3 | (1) |
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3 | (1) |
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3 | (1) |
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4 | (1) |
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3-D Geometry and Coordinate Geometry |
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4 | (1) |
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4 | (1) |
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4 | (1) |
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5 | (2) |
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7 | (42) |
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Introducing Geometry and Geometry Proofs! |
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9 | (10) |
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Making the Right Assumptions: What Is Geometry? |
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9 | (3) |
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If-Then Logic: If You Bought This Book, Then You Must Love Geometry! |
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12 | (2) |
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Discovering Geometry Proofs |
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14 | (3) |
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Solutions for Introducting Geometry and Geometry Proofs! |
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17 | (2) |
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Points, Segments, Lines, Rays, and Angles |
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19 | (30) |
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19 | (1) |
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Union and Intersection Problems |
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20 | (2) |
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Division in the Ranks: Bisection and Trisection |
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22 | (2) |
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Perfect Hilarity for Perpendicularity |
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24 | (2) |
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You Complete Me: Complementary and Supplementary Angles |
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26 | (4) |
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Adding and Subtracting Segments and Angles |
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30 | (3) |
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Multiplying and Dividing Angles and Segments |
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33 | (4) |
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X Marks the Spot: Using Vertical Angles |
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37 | (2) |
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Switching It Up with the Transitive and Substitution Properties |
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39 | (3) |
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Solutions for Points, Segments, Lines, Rays, and Angles |
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42 | (7) |
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49 | (64) |
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Triangle Fundamentals and Other Cool Stuff |
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51 | (32) |
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Triangle Types and Triangle Basics |
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51 | (4) |
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Altitudes, Area, and the Super Hero Formula |
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55 | (3) |
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Balancing Things Out with Medians and Centroids |
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58 | (1) |
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Three More ``Centers'' of a Triangle |
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59 | (5) |
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64 | (3) |
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Pythagorean Triple Triangles |
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67 | (3) |
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Unique Degrees: Two Special Right Triangles |
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70 | (4) |
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Solutions for Triangle Fundamentals and Other Cool Stuff |
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74 | (9) |
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83 | (30) |
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Sizing Up Three Ways to Prove Triangles Congruent |
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83 | (6) |
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Corresponding Parts of Congruent Triangles Are Congruent (CPCTC) |
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89 | (4) |
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Isosceles Rules: If Sides, Then Angles; If Angles, Then Sides |
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93 | (3) |
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Two More Ways to Prove Triangles Congruent |
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96 | (3) |
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The Two Equidistance Theorems |
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99 | (5) |
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Solutions for Congruent Triangles |
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104 | (9) |
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113 | (80) |
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Quadrilaterals: Your Fine, Four-Sided Friends |
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115 | (36) |
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Double-Crossers: Transversals and Their Parallel Lines |
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115 | (5) |
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Quadrilaterals: It's a Family Affair |
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120 | (3) |
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Properties of the Parallelogram and the Kite |
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123 | (4) |
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Properties of Rhombuses, Rectangles, and Squares |
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127 | (3) |
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Properties of Trapezoids and Isosceles Trapezoids |
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130 | (2) |
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Proving That a Quadrilateral Is a Parallelogram or a Kite |
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132 | (4) |
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Proving That a Quadrilateral Is a Rhombus, Rectangle, or Square |
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136 | (3) |
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Solutions for Quadrilaterals: Your Fine, Four-Sided Friends |
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139 | (12) |
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Area, Angles, and the Many Sides of Polygon Geometry |
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151 | (14) |
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Square Units: Finding the Area of Quadrilaterals |
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151 | (4) |
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A Standard Formula for the Area of Regular Polygons |
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155 | (2) |
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More Fantastically Fun Polygon Formulas |
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157 | (3) |
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Solutions for Area, Angles, and the Many Sides of Polygon Geometry |
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160 | (5) |
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Similarity: Size Doesn't Matter |
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165 | (28) |
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Defining Similarity: The Sincerest Flattery |
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165 | (3) |
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Proving Triangles Similar |
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168 | (4) |
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Corresponding Sides and CSSTP --- Cats Stalk Silently Then Pounce |
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172 | (3) |
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Similar Rights: The Altitude-on-Hypotenuse Theorem |
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175 | (3) |
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Three More Theorems Involving Proportions |
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178 | (5) |
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Solutions for Similarity: Size Doesn't Matter |
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183 | (10) |
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193 | (34) |
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195 | (16) |
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The Segments Within: Radii and Chords |
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195 | (4) |
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Introducing Arcs, Chords, and Central Angles |
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199 | (3) |
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Touching on Radii and Tangents |
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202 | (4) |
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Solutions for Circular Reasoning |
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206 | (5) |
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Scintillating Circle Formulas |
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211 | (16) |
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Not Quite Coming Full Circle: Area and ``Perimeter'' of Circles, Sectors, and Segments |
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211 | (3) |
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Angles, Circles, and Their Connections: The Angle-Arc Theorems and Formulas |
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214 | (3) |
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The Power Theorems That Be |
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217 | (4) |
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Solutions for Scintillating Circle Formulas |
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221 | (6) |
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Part V: 3-D Geometry and Coordinate Geometry |
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227 | (58) |
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229 | (14) |
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Lines Perpendicular to Planes: They're All Right |
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229 | (4) |
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Parallel, Perpendicular, and Intersecting Lines and Planes |
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233 | (5) |
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Solutions for 2-D Stuff Standing Up |
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238 | (5) |
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Solid Geometry: Digging into Volume and Surface Area |
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243 | (14) |
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Starting with Flat-Top Figures |
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243 | (3) |
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Sharpening Your Skills with Pointy-Top Figures |
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246 | (3) |
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Rounding Out Your Understanding with Spheres |
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249 | (2) |
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Solutions for Solid Geometry |
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251 | (6) |
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Coordinate Geometry, Courtesy of Descartes |
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257 | (14) |
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Formulas, Schmormulas: Slope, Distance, and Midpoint |
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257 | (3) |
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Mastering Coordinate Proofs with Algebra |
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260 | (1) |
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Using the Equations of Lines and Circles |
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261 | (3) |
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Solutions for Coordinate Geometry, Courtesy of Descartes |
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264 | (7) |
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Transforming the (Geometric) World: Reflections, Rotations, and Translations |
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271 | (14) |
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Reflections on Mirror Images |
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271 | (3) |
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Les Transformations Geometriques |
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274 | (2) |
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So You Say You Want a...Rotation? |
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276 | (2) |
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Working with Glide Reflections |
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278 | (3) |
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Solutions for Transforming the World |
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281 | (4) |
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Part VI: The Part of Tens |
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285 | (10) |
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Ten (Plus) Incredibly Fantastic Strategies for Doing Proofs |
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287 | (4) |
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Look for Congruent Triangles |
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287 | (1) |
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Try to Find Isosceles Triangles |
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287 | (1) |
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Look for Radii, and Draw More Radii |
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288 | (1) |
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288 | (1) |
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288 | (1) |
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288 | (1) |
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288 | (1) |
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289 | (1) |
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Make Up Numbers for Segments and Angles |
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289 | (1) |
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289 | (1) |
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Bonus! Number 11 (Like the Amp in Spinal Tap That Goes Up to 11): Do Something! |
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290 | (1) |
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Ten Things You Better Know (for Geometry), or Your Name Is Mudd |
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291 | (4) |
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The Pythagorean Theorem (the Queen of All Geometry Theorems) |
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291 | (1) |
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291 | (1) |
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292 | (1) |
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292 | (1) |
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292 | (1) |
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292 | (1) |
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293 | (1) |
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Coordinate Geometry Formulas |
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|
293 | (1) |
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293 | (1) |
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|
294 | (1) |
| Index |
|
295 | |
