| Introduction |
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1 | (4) |
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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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3 | (1) |
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4 | (1) |
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4 | (1) |
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PART 1 GETTING STARTED WITH GEOMETRY BASICS |
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5 | (38) |
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Chapter 1 Introducing Geometry |
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7 | (10) |
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Studying the Geometry of Shapes |
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8 | (2) |
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8 | (1) |
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8 | (2) |
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10 | (1) |
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Getting Acquainted with Geometry Proofs |
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10 | (4) |
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Easing into proofs with an everyday example |
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11 | (1) |
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Turning everyday logic into a proof |
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12 | (1) |
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Sampling a simple geometrical proof |
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13 | (1) |
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When Am I Ever Going to Use This? |
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14 | (2) |
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When you'll use your knowledge of shapes |
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14 | (1) |
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When you'll use your knowledge of proofs |
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15 | (1) |
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Why You Won't Have Any Trouble with Geometry |
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16 | (1) |
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Chapter 2 Building Your Geometric Foundation |
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17 | (14) |
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Getting Down with Definitions |
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17 | (4) |
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21 | (1) |
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Lines, Segments, and Rays Pointing Every Which Way |
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22 | (3) |
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Singling out horizontal and vertical lines |
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22 | (1) |
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Doubling up with pairs of lines |
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23 | (2) |
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Investigating the Plane Facts |
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25 | (1) |
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26 | (5) |
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Goldilocks and the three angles: Small, large, and just "right" |
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26 | (2) |
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Angle pairs: Often joined at the hip |
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28 | (3) |
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Chapter 3 Sizing Up Segments and Analyzing Angles |
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31 | (12) |
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Measuring Segments and Angles |
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31 | (5) |
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32 | (1) |
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33 | (3) |
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Adding and Subtracting Segments and Angles |
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36 | (1) |
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Cutting in Two or Three: Bisection and Trisection |
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37 | (3) |
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Bisecting and trisecting segments |
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37 | (1) |
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Bisecting and trisecting angles |
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38 | (2) |
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Proving (Not Jumping to) Conclusions about Figures |
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40 | (3) |
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PART 2 INTRODUCING PROOFS |
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43 | (44) |
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Chapter 4 Prelude to Proofs |
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45 | (10) |
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Getting the Lay of the Land: The Components of a Formal Geometry Proof |
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46 | (2) |
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Reasoning with If-Then Logic |
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48 | (4) |
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48 | (1) |
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You've got your reasons: Definitions, theorems, and postulates |
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49 | (2) |
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Bubble logic for two-column proofs |
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51 | (1) |
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Horsing Around with a Two-Column Proof |
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52 | (3) |
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Chapter 5 Your Starter Kit of Easy Theorems and Short Proofs |
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55 | (20) |
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Doing Right and Going Straight: Complementary and Supplementary Angles |
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56 | (3) |
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Addition and Subtraction: Eight No-Big-Deal Theorems |
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59 | (7) |
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59 | (4) |
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63 | (3) |
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Like Multiples and Like Divisions? Then These Theorems Are for You! |
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66 | (3) |
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The X-Files: Congruent Vertical Angles Are Out There |
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69 | (2) |
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Pulling the Switch with the Transitive and Substitution Properties |
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71 | (4) |
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Chapter 6 The Ultimate Guide to Tackling a Longer Proof |
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75 | (12) |
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76 | (1) |
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77 | (1) |
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Making Sure You Use If-Then Logic |
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78 | (1) |
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Chipping Away at the Problem |
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79 | (2) |
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Jumping Ahead and Working Backward |
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81 | (2) |
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83 | (1) |
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Writing Out the Finished Proof |
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84 | (3) |
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PART 3 TRIANGLES: POLYGONS OF THE THREE-SIDED VARIETY |
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87 | (66) |
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Chapter 7 Grasping Triangle Fundamentals |
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89 | (18) |
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Taking In a Triangle's Sides |
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89 | (3) |
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Scalene triangles: Akilter, awry, and askew |
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90 | (1) |
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Isosceles triangles: Nice pair o' legs |
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91 | (1) |
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Equilateral triangles: All parts are created equal |
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92 | (1) |
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Introducing the Triangle Inequality Principle |
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92 | (2) |
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Getting to Know Triangles by Their Angles |
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94 | (1) |
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94 | (6) |
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95 | (1) |
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Determining a triangle's area |
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96 | (4) |
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Locating the "Centers" of a Triangle |
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100 | (7) |
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Balancing on the centroid |
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100 | (3) |
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Finding three more "centers" of a triangle |
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103 | (4) |
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Chapter 8 Regarding Right Triangles |
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107 | (18) |
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Applying the Pythagorean Theorem |
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108 | (5) |
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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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114 | (2) |
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Families of Pythagorean triple triangles |
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116 | (2) |
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Getting to Know Two Special Right Triangles |
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118 | (7) |
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The 45°- 45°- 90° triangle --- half a square |
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119 | (1) |
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The 30°- 60°- 90° triangle --- half of an equilateral triangle |
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120 | (5) |
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Chapter 9 Completing Congruent Triangle Proofs |
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125 | (28) |
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Introducing Three Ways to Prove Triangles Congruent |
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126 | (7) |
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SSS: Using the side-side-side method |
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127 | (1) |
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SAS: Taking the side-angle-side approach |
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128 | (3) |
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ASA: Taking the angle-side-angle tack |
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131 | (2) |
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CPCTC: Taking Congruent Triangle Proofs a Step Further |
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133 | (4) |
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133 | (1) |
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134 | (3) |
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Eying the Isosceles Triangle Theorems |
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137 | (2) |
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Trying Out Two More Ways to Prove Triangles Congruent |
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139 | (4) |
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AAS: Using the angle-angle-side theorem |
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139 | (3) |
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HLR: The right approach for right triangles |
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142 | (1) |
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Going the Distance with the Two Equidistance Theorems |
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143 | (4) |
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Determining a perpendicular bisector |
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144 | (1) |
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Using a perpendicular bisector |
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145 | (2) |
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Making a Game Plan for a Longer Proof |
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147 | (2) |
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Running a Reverse with Indirect Proofs |
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149 | (4) |
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PART 4 POLYGONS OF THE FOUR-OR-MORE-SIDED VARIETY |
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153 | (82) |
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Chapter 10 The Seven Wonders of the Quadrilateral World |
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155 | (22) |
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Getting Started with Parallel-Line Properties |
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156 | (5) |
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Crossing the line with transversals: Definitions and theorems |
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156 | (1) |
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Applying the transversal theorems |
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157 | (3) |
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Working with more than one transversal |
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160 | (1) |
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Meeting the Seven Members of the Quadrilateral Family |
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161 | (5) |
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Looking at quadrilateral relationships |
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163 | (1) |
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Working with auxiliary lines |
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164 | (2) |
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Giving Props to Quads: The Properties of Quadrilaterals |
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166 | (11) |
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Properties of the parallelogram |
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166 | (4) |
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Properties of the three special parallelograms |
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170 | (3) |
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173 | (2) |
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Properties of the trapezoid and the isosceles trapezoid |
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175 | (2) |
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Chapter 11 Proving That You Have a Particular Quadrilateral |
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177 | (16) |
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Putting Properties and Proof Methods Together |
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178 | (2) |
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Proving That a Quadrilateral Is a Parallelogram |
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180 | (4) |
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Surefire ways of ID-ing a parallelogram |
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180 | (1) |
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Trying some parallelogram proofs |
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181 | (3) |
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Proving That a Quadrilateral Is a Rectangle, Rhombus, or Square |
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184 | (5) |
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Revving up for rectangle proofs |
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185 | (2) |
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Waxing rhapsodic about rhombus proofs |
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187 | (1) |
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Squaring off with square proofs |
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188 | (1) |
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Proving That a Quadrilateral Is a Kite |
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189 | (4) |
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Chapter 12 Polygon Formulas: Area, Angles, and Diagonals |
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193 | (18) |
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Calculating the Area of Quadrilaterals |
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193 | (8) |
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Setting forth the quadrilateral area formulas |
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194 | (1) |
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Getting behind the scenes of the formulas |
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194 | (2) |
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Trying a few area problems |
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196 | (5) |
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Finding the Area of Regular Polygons |
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201 | (4) |
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Presenting polygon area formulas |
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201 | (1) |
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Tackling more area problems |
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202 | (3) |
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Using Polygon Angle and Diagonal Formulas |
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205 | (6) |
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Interior and exterior design: Exploring polygon angles |
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206 | (1) |
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Handling the ins and outs of a polygon angle problem |
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207 | (1) |
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Criss-crossing with diagonals |
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208 | (3) |
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Chapter 13 Similarity: Same Shape, Different Size |
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211 | (24) |
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Getting Started with Similar Figures |
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212 | (5) |
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Defining and naming similar polygons |
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212 | (1) |
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How similar figures line up |
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213 | (2) |
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Solving a similarity problem |
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215 | (2) |
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Proving Triangles Similar |
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217 | (5) |
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218 | (1) |
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Using SSS~ to prove triangles similar |
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219 | (2) |
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Working through an SAS~ proof |
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221 | (1) |
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CASTC and CSSTP, the Cousins of CPCTC |
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222 | (2) |
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Working through a CASTC proof |
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222 | (1) |
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223 | (1) |
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Splitting Right Triangles with the Altitude-on-Hypotenuse Theorem |
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224 | (3) |
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Getting Proportional with Three More Theorems |
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227 | (8) |
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The side-splitter theorem: It'll make you split your sides |
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227 | (2) |
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Crossroads: The side-splitter theorem extended |
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229 | (2) |
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The angle-bisector theorem |
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231 | (4) |
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PART 5 WORKING WITH NOT-SO-VICIOUS CIRCLES |
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235 | (42) |
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Chapter 14 Coming Around to Circle Basics |
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237 | (18) |
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The Straight Talk on Circles: Radii and Chords |
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238 | (5) |
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Defining radii, chords, and diameters |
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238 | (1) |
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Introducing five circle theorems |
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238 | (1) |
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239 | (1) |
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Using extra radii to solve a problem |
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240 | (3) |
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Pieces of the Pie: Arcs and Central Angles |
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243 | (4) |
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Three definitions for your mathematical pleasure |
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243 | (1) |
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Six scintillating circle theorems |
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244 | (1) |
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Trying your hand at some proofs |
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245 | (2) |
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Going Off on a Tangent about Tangents |
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247 | (8) |
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Introducing the tangent line |
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248 | (1) |
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The common-tangent problem |
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249 | (2) |
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Taking a walk on the wild side with a walk-around problem |
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251 | (4) |
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Chapter 15 Circle Formulas and Theorems |
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255 | (22) |
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Chewing on the Pizza Slice Formulas |
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256 | (6) |
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256 | (3) |
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Finding sector and segment area |
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259 | (2) |
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Pulling it all together in a problem |
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261 | (1) |
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Digesting the Angle-Arc Theorems and Formulas |
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262 | (8) |
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262 | (3) |
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265 | (1) |
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266 | (3) |
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Keeping your angle-arc formulas straight |
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269 | (1) |
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Powering Up with the Power Theorems |
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270 | (7) |
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Striking a chord with the chord-chord power theorem |
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270 | (2) |
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Touching on the tangent-secant power theorem |
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272 | (1) |
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Seeking out the secant-secant power theorem |
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272 | (3) |
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Condensing the power theorems into a single idea |
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275 | (2) |
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PART 6 GOING DEEP WITH 3-D GEOMETRY |
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277 | (26) |
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Chapter 16 3-D Space: Proofs in a Higher Plane of Existence |
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279 | (8) |
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Lines Perpendicular to Planes |
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279 | (4) |
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Parallel, Perpendicular, and Intersecting Lines and Planes |
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283 | (4) |
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The four ways to determine a plane |
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283 | (1) |
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Line and plane interactions |
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284 | (3) |
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Chapter 17 Getting a Grip on Solid Geometry |
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287 | (16) |
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Flat-Top Figures: They're on the Level |
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287 | (6) |
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Getting to the Point of Pointy-Top Figures |
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293 | (6) |
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Rounding Things Out with Spheres |
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299 | (4) |
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PART 7 PLACEMENT, POINTS, AND PICTURES: ALTERNATIVE GEOMETRY TOPICS |
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303 | (58) |
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Chapter 18 Coordinate Geometry |
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305 | (18) |
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Getting Coordinated with the Coordinate Plane |
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305 | (2) |
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The Slope, Distance, and Midpoint Formulas |
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307 | (7) |
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307 | (3) |
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Going the distance with the distance formula |
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310 | (1) |
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Meeting each other halfway with the midpoint formula |
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311 | (1) |
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The whole enchilada: Putting the formulas together in a problem |
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312 | (2) |
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Proving Properties Analytically |
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314 | (4) |
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Step 1 Drawing a general figure |
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314 | (2) |
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Step 2 Solving the problem algebraically |
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316 | (2) |
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Deciphering Equations for Lines and Circles |
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318 | (5) |
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318 | (1) |
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The standard circle equation |
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319 | (4) |
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Chapter 19 Changing the Scene with Geometric Transformations |
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323 | (20) |
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Some Reflections on Reflections |
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324 | (4) |
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Getting oriented with orientation |
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325 | (1) |
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Finding a reflecting line |
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326 | (2) |
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Not Getting Lost in Translations |
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328 | (5) |
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A translation equals two reflections |
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329 | (1) |
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Finding the elements of a translation |
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330 | (3) |
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Turning the Tables with Rotations |
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333 | (5) |
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A rotation equals two reflections |
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334 | (1) |
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Finding the center of rotation and the equations of two reflecting lines |
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334 | (4) |
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Third Time's the Charm: Stepping Out with Glide Reflections |
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338 | (5) |
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A glide reflection equals three reflections |
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338 | (1) |
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Finding the main reflecting line |
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339 | (4) |
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Chapter 20 Locating Loci and Constructing Constructions... |
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343 | (18) |
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Loci Problems: Getting in with the Right Set |
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344 | (7) |
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The four-step process for locus problems |
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344 | (1) |
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Two-dimensional locus problems |
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345 | (5) |
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Three-dimensional locus problems |
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350 | (1) |
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Drawing with the Bare Essentials: Constructions |
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351 | (10) |
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352 | (3) |
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Bisecting angles and segments |
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355 | (2) |
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Two perpendicular line constructions |
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357 | (1) |
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Constructing parallel lines and using them to divide segments |
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358 | (3) |
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361 | (16) |
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Chapter 21 Ten Things to Use as Reasons in Geometry Proofs |
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363 | (6) |
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363 | (1) |
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Vertical Angles Are Congruent |
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364 | (1) |
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The Parallel-Line Theorems |
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364 | (1) |
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Two Points Determine a Line |
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365 | (1) |
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All Radii of a Circle Are Congruent |
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365 | (1) |
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366 | (1) |
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366 | (1) |
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The Triangle Congruence Postulates and Theorems |
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366 | (1) |
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367 | (1) |
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The Triangle Similarity Postulates and Theorems |
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367 | (2) |
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Chapter 22 Ten Cool Geometry Problems |
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369 | (8) |
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Eureka! Archimedes's Bathtub Revelation |
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369 | (1) |
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370 | (1) |
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371 | (1) |
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The Circumference of the Earth |
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372 | (1) |
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The Great Pyramid of Khufu |
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373 | (1) |
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373 | (1) |
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373 | (1) |
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374 | (1) |
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375 | (1) |
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375 | (2) |
| Index |
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377 | |
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