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Geometry Connections Table of Contents Student Edition Chapter 1 Shapes and Transformations 1 Section 1.1 Course Introduction Problems 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 Creating a Quilt Using Symmetry Making Predictions and Investigating Results Perimeter and Area of Enlarging Tile Patterns Logical Arguments Building a Kaleidoscope 3 7 12 16 21 Section 1.2 Transformations and Symmetry 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 Spatial Visualization and Reflections Rigid Transformations: Rotations and Translations Using Transformations Using Transformations to Create Shapes Symmetry 26 31 36 40 44 Section 1.3 Shapes and Probability 1.3.1 Attributes and Characteristics of Shapes 1.3.2 More Characteristics of Shapes 1.3.3 Introduction to Probability Chapter Closure Chapter 2 Angles and Measurement 49 53 57 62 71 Section 2.1 Angle Relationships 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 Complementary, Supplementary, and Vertical Angles Angles Formed by Transversals More Angles Formed by Transversals Angles in a Triangle Applying Angle Relationships 73 78 83 89 93 Section 2.2 Area 2.2.1 2.2.2 2.2.3 2.2.4 Units of Measure Areas of Triangles and Composite Shapes Areas of Parallelograms and Trapezoids Heights and Area 98 102 105 110 Section 2.3 The Pythagorean Theorem 2.3.1 Squares and Square Roots 2.3.2 Triangle Inequality 2.3.3 The Pythagorean Theorem 113 117 121 Chapter Closure 125 Chapter 3 Justification and Similarity 133 Section 3.1 Introduction to Similarity 3.1.1 3.1.2 3.1.3 3.1.4 Similarity Proportional Growth and Ratios Using Ratios of Similarity Applications and Notation 135 139 144 147 Section 3.2 Triangle Similarity 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 Conditions for Triangle Similarity Creating a Flowchart Triangle Similarity and Congruence More Conditions for Triangle Similarity Determining Similarity Applying Similarity Chapter Closure Chapter 4 Trigonometry and Probability 152 157 161 165 169 173 176 185 Section 4.1 The Tangent Ratio 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 Constant Ratios in Right Triangles Connecting Slope Ratios to Specific Angles Expanding the Trig Table The Tangent Ratio Applying the Tangent Ratio 187 192 195 198 202 Section 4.2 Probability Models 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 Introduction to Probability Models Theoretical and Experimental Probability Using an Area Model Choosing a Probability Model Optional: Applications of Probability Methods Chapter Closure 204 209 212 216 221 224 Chapter 5 Completing the Triangle Toolkit 233 Section 5.1 Trigonometry 5.1.1 5.1.2 5.1.3 5.1.4 Sine and Cosine Ratios Selecting a Trig Tool Inverse Trigonometry Trigonometric Applications 235 239 243 246 Section 5.2 Special Right Triangles 5.2.1 Special Right Triangles 5.2.2 Pythagorean Triples 250 254 Section 5.3 Completing the Triangle Toolkit 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 Finding Missing Parts of Triangles Law of Sines Law of Cosines Ambiguous Triangles Choosing a Tool Chapter Closure Chapter 6 Congruent Triangles 257 262 265 269 273 278 287 Section 6.1 Congruent Triangles 6.1.1 6.1.2 6.1.3 6.1.4 Congruent Triangles Conditions for Triangle Congruence Flowcharts for Congruence Converses 287 293 297 301 Section 6.2 Closure Activities 6.2.1 6.2.2 6.2.3 6.2.4 6.2.5 Angles on a Pool Table Investigating a Triangle Creating a Mathematical Model Analyzing a Game Using Transformations and Symmetry to Create a Snowflake Chapter Closure 306 309 311 315 318 324 Chapter 7 Proof and Quadrilaterals 333 Section 7.1 Introduction to Chapters 7 - 12 7.1.1 7.1.2 7.1.3 7.1.4 Properties of a Circle Building a Tetrahedron Shortest Distance Problems Using Symmetry to Study Polygons 335 339 343 349 Section 7.2 Proof and Quadrilaterals 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 7.2.6 Special Quadrilaterals and Proof Properties of Rhombi More Proof with Congruent Triangles More Properties of Quadrilaterals Two-Column Proofs Explore-Conjecture-Prove 352 356 359 363 365 369 Section 7.3 Coordinate Geometry 7.3.1 Studying Quadrilaterals on a Coordinate Grid 7.3.2 Coordinate Geometry and Midpoints 7.3.3 Quadrilaterals on a Coordinate Plane Chapter Closure Chapter 8 Polygons and Circles 373 376 380 383 391 Section 8.1 Angles and Area of a Polygon 8.1.1 8.1.2 8.1.3 8.1.4 8.1.5 Pinwheels and Polygons Interior Angles of a Polygon Angles of Regular Polygons Regular Polygon Angle Connections Finding the Area of Regular Polygons 393 398 402 405 408 Section 8.2 Ratio of the Area of Similar Figures 8.2.1 Area Ratios of Similar Figures 8.2.2 Ratios of Similarity 413 417 Section 8.3 Area and Circumference of a Circle 8.3.1 A Special Ratio 8.3.2 Area and Circumference of Circles 8.3.3 Circles in Context Chapter Closure 421 424 428 434 Chapter 9 Solids and Constructions 441 Section 9.1 Solids 9.1.1 9.1.2 9.1.3 9.1.4 9.1.5 Three-Dimensional Solids Volume and Surface Area of Prisms Prisms and Cylinders Volumes of Similar Solids Ratios of Similarity 443 446 450 453 456 Section 9.2 Construction 9.2.1 9.2.2 9.2.3 9.2.4 Introduction to Construction Constructing Bisectors More Exploration with Constructions Finding a Centroid Chapter Closure Chapter 10 Circles and Expected Value 459 463 466 470 473 481 Section 10.1 Relationships Within Circles 10.1.1 10.1.2 10.1.3 10.1.4 10.1.5 Introduction to Chords Angles and Arcs Chords and Angles Tangents and Chords Problem Solving with Circles 483 487 492 496 499 Section 10.2 Expected Value 10.2.1 Designing Spinners 10.2.2 Expected Value 10.2.3 More Expected Value 502 507 510 Section 10.3 Equation of a Circle 10.3.1 The Equation of a Circle Chapter Closure 514 518 Chapter 11 Solids and Circles 525 Section 11.1 Pyramids, Cones, Spheres 11.1.1 11.1.2 11.1.3 11.1.4 11.1.5 Platonic Solids Pyramids Volume of a Pyramid Surface Area and Volume of a Cone Surface Area and Volume of a Sphere 527 533 536 541 545 Section 11.2 More Circle Relationships 11.2.1 Coordinates on a Sphere 11.2.2 Tangents and Arcs 11.2.3 Secant and Tangent Relationships Chapter Closure Chapter 12 Conics and Closure 548 553 556 564 571 Section 12.1 Conic Sections 12.1.1 12.1.2 12.1.3 12.1.4 12.1.5 Introduction to Conic Sections Graphing Parabolas Using the Focus and Directrix Circles and Ellipses The Hyperbola Conic Equations and Graphs 573 576 580 583 587 Section 12.2 Closure Activities 12.2.1 12.2.2 12.2.3 12.2.4 Using Coordinate Geometry and Construction to Explore Shapes Euler’s Formula for Polyhedra The Golden Ratio Using Geometry to Find Probability Appendices A B C Points, Lines, and Planes Euclidean Geometry Non-Euclidean Geometry 591 594 597 601 607 607 612 616 Glossary 621 Index 647 Index to List of Symbols and Equations 658