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ContentsContentsContentsContentsContentsContentsContentsContentsContentsContentsContentsContentsContents

UNIT 1 PROOF, PARALLEL AND PERPENDICULAR LINES Planning the Unit 1a Unit 1 Overview 1 Getting Ready 2

Activity 1 Geometric Figures—What's My Name? 3 Lesson 1-1 Basic Geometric Figures 3 Lesson 1-2 More Geometric Figures 7 Activity 1 Practice 11

Activity 2 Logical Reasoning—Riddle Me This 13 Lesson 2-1 Inductive Reasoning 13 Lesson 2-2 Deductive Reasoning 18 Activity 2 Practice 23

Activity 3 The Axiomatic System of Geometry—Back to the Beginning 25 Lesson 3-1 Geometric Definitions and Two-Column Proofs 25 Lesson 3-2 Conditional Statements 29 Lesson 3-3 Converse, Inverse, and Contrapositive 32 Activity 3 Practice 35

Embedded Assessment 1 Geometric Figures and Basic Reasoning— The Art and Math of Folding Paper 37

Activity 4 Segment and Angle Measurement—It All Adds Up 39 Lesson 4-1 Segments and Midpoints 39 Lesson 4-2 Angles and Angle Bisectors 45 Activity 4 Practice 49

Activity 5 The Distance and Midpoint Formulas—We ♥ Descartes 51 Lesson 5-1 Distance on the Coordinate Plane 51 Lesson 5-2 Midpoint on the Coordinate Plane 56 Activity 5 Practice 59

Embedded Assessment 2 Distance, Midpoint, and Angle Measurement— A Walk in the Park 61

Activity 6 Proofs about Line Segments and Angles—Now I’m Convinced 63 Lesson 6-1 Justifying Statements 63 Lesson 6-2 Two-Column Geometric Proofs 66 Activity 6 Practice 71

Activity 7 Parallel and Perpendicular Lines—Patios by Madeline 73 Lesson 7-1 Parallel Lines and Angle Relationships 73 Lesson 7-2 Proving Lines are Parallel 79 Lesson 7-3 Perpendicular Lines 84 Activity 7 Practice 87

To the Teacher xi

Instructional Units

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Activity 8 Equations of Parallel and Perpendicular Lines— Skateboard Geometry 89 Lesson 8-1 Slopes of Parallel and Perpendicular Lines 89 Lesson 8-2 Writing Equations 92 Activity 8 Practice 97

Embedded Assessment 3 Angles, Parallel Lines, and Perpendicular Lines— Graph of Steel 99

UNIT 2 TRANSFORMATIONS, TRIANGLES, AND QUADRILATERALS Planning the Unit 101a Unit 2 Overview 101 Getting Ready 102

Activity 9 Translations, Reflections, and Rotations—The Marching Cougars 103 Lesson 9-1 Transformations 103 Lesson 9-2 Translations 108 Lesson 9-3 Reflection 112 Lesson 9-4 Rotations 120 Activity 9 Practice 127

Activity 10 Compositions and Congruence—More Rigid Motions 129 Lesson 10-1 Compositions of Transformations 129 Lesson 10-2 Congruence 134 Activity 10 Practice 139

Embedded Assessment 1 Transformations—Designing the Plaza 141

Activity 11 Congruence Transformations and Triangle Congruence— Truss Your Judgment 143 Lesson 11-1 Congruent Triangles 143 Lesson 11-2 Congruence Criteria 147 Lesson 11-3 Proving and Applying the Congruence Criteria 154 Lesson 11-4 Extending the Congruence Criteria 160 Activity 11 Practice 165

Activity 12 Flowchart Proofs—Go with the Flow 167 Lesson 12-1 Flowchart Proofs 167 Lesson 12-2 Three Types of Proofs 175 Activity 12 Practice 177

Embedded Assessment 2 Congruence, Triangles, and Proof—Building a Fitness Center 179

Activity 13 Properties of Triangles—Best Two Out of Three 181 Lesson 13-1 Angle Relationships in Triangles 181 Lesson 13-2 Isosceles Triangles 186 Activity 13 Practice 189

vi SpringBoard® Mathematics Geometry

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Activity 14 Concurrent Segments in Triangles—What’s the Point? 191 Lesson 14-1 Altitudes of a Triangle 191 Lesson 14-2 Medians of a Triangle 195 Lesson 14-3 Perpendicular Bisectors and Angle Bisectors of a Triangle 198 Activity 14 Practice 201

Embedded Assessment 3 Properties of Triangles—Where Does the Fountain Go? 203

Activity 15 Quadrilaterals and Their Properties—A 4-gon Hypothesis 205 Lesson 15-1 Kites and Triangle Midsegments 205 Lesson 15-2 Trapezoids 209 Lesson 15-3 Parallelograms 213 Lesson 15-4 Rectangles, Rhombuses, and Squares 216 Activity 15 Practice 221

Activity 16 More About Quadrilaterals—A 4-gon Conclusion 223 Lesson 16-1 Proving a Quadrilateral is a Parallelogram 223 Lesson 16-2 Proving a Quadrilateral is a Rectangle 227 Lesson 16-3 Proving a Quadrilateral is a Rhombus 230 Lesson 16-4 Proving a Quadrilateral is a Square 233 Activity 16 Practice 235

Embedded Assessment 4 Quadrilaterals—Lucy Latimer’s Logo 237

UNIT 3 SIMILARITY AND TRIGONOMETRY Planning the Unit 239a Unit 3 Overview 239 Getting Ready 240

Activity 17 Dilations and Similarity Transformations—Scaling Up/ Scaling Down 241 Lesson 17-1 Dilations 241 Lesson 17-2 Similarity Transformations 248 Lesson 17-3 Properties of Similar Figures 252 Activity 17 Practice 255

Activity 18 Similar Triangles—Measuring Up 257 Lesson 18-1 Similarity Criteria 257 Lesson 18-2 Using Similarity Criteria 262 Lesson 18-3 Triangle Proportionality Theorem 265 Activity 18 Practice 271

Embedded Assessment 1 Similarity in Polygons—Monitoring Progress 273

Activity 19 Geometric Mean—Do You Mean It? 275 Lesson 19-1 The Right Triangle Altitude Theorem 275 Lesson 19-2 The Geometric Mean 278 Activity 19 Practice 281

Activity 20 The Pythagorean Theorem and Its Converse—Is That Right? 283 Lesson 20-1 Pythagorean Theorem 283 Lesson 20-2 Converse of the Pythagorean Theorem 286 Activity 20 Practice 289

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Activity 21 Special Right Triangles—The Community Quilting Project 291 Lesson 21-1 45°-45°-90° Triangles 291 Lesson 21-2 30°-60°-90° Triangles 295 Activity 21 Practice 299

Embedded Assessment 2 Right Triangles—Powered by the Wind 301

Activity 22 Basic Trigonometric Relationships—The Sine of Things to Come 303 Lesson 22-1 Similar Right Triangles 303 Lesson 22-2 Trigonometric Ratios 308 Lesson 22-3 Using Trigonometric Ratios 312 Lesson 22-4 Solving Right Triangles 315 Activity 22 Practice 317

Activity 23 The Laws of Sines and of Cosines—There Ought to Be a Law 319 Lesson 23-1 The Law of Sines 319 Lesson 23-2 The Ambiguous Case 322 Lesson 23-3 The Law of Cosines 324 Lesson 23-4 Solving Triangles 327 Activity 23 Practice 329

Embedded Assessment 3 Trigonometry—Zipping Along 331

UNIT 4 CIRCLES, COORDINATES, AND CONSTRUCTIONS Planning the Unit 333a Unit 4 Overview 333 Getting Ready 334

Activity 24 Tangents and Chords—Off On a Tangent 335 Lesson 24-1 Circle Basics 335 Lesson 24-2 Theorems About Chords 339 Lesson 24-3 Tangent Segments 345 Activity 24 Practice 347

Activity 25 Arcs and Angles—Coming Full Circle 349 Lesson 25-1 Arcs and Central Angles 349 Lesson 25-2 Inscribed Angles 353 Lesson 25-3 Angles Formed by Chords 358 Lesson 25-4 Angles Formed by Tangents and Secants 363 Activity 25 Practice 369

Embedded Assessment 1 Circles—Vertigo Round 371

Activity 26 Coordinate Proofs—Prove It! 373 Lesson 26-1 Proving the Midpoint Formula 373 Lesson 26-2 Proofs about Slope 377 Lesson 26-3 Proving Concurrency of Medians 382 Lesson 26-4 Points Along a Line Segment 387 Activity 26 Practice 390

viii SpringBoard® Mathematics Geometry

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Activity 27 Equation of a Circle—Round and Round 391 Lesson 27-1 Circles on the Coordinate Plane 391 Lesson 27-2 Completing the Square to Find the Center and Radius 396 Activity 27 Practice 399

Activity 28 Equations of Parabolas—Throwing a Curve 401 Lesson 28-1 Parabolas on the Coordinate Plane 401 Lesson 28-2 Parabolas with Vertex (h, k) 405 Activity 28 Practice 409

Activity 29 Constructions—Constructive Thinking 411 Lesson 29-1 Constructions with Segments and Angles 411 Lesson 29-2 Constructions with Parallel and Perpendicular Lines 418 Lesson 29-3 Constructions with Circles 422 Activity 29 Practice 427

Embedded Assessment 2 Coordinates and Constructions—Location Matters 429

UNIT 5 EXTENDING TWO DIMENSIONS TO THREE DIMENSIONS Planning the Unit 431a Unit 5 Overview 431 Getting Ready 432

Activity 30 Deriving Area Formulas—Shape Up 433 Lesson 30-1 Areas of Rectangles and Parallelograms 433 Lesson 30-2 Areas of Triangles 437 Lesson 30-3 Areas of Rhombuses and Trapezoids 441 Activity 30 Practice 445

Activity 31 Regular Polygons—Plenty of Polygons 447 Lesson 31-1 Sum of the Measures of the Interior Angles of a Polygon 447 Lesson 31-2 Regular Polygons and Exterior Angles 453 Lesson 31-3 Area and Perimeter of Regular Polygons 457 Activity 31 Practice 461

Activity 32 Length and Area of Circles—Pi in the Sky 463 Lesson 32-1 Circumference and Area of a Circle 463 Lesson 32-2 Sectors and Arcs 467 Lesson 32-3 Circles and Similarity 470 Activity 32 Practice 475

Embedded Assessment 1 Area and Perimeter—Play Planning 477

Activity 33 Three-Dimensional Figures—What’s Your View? 479 Lesson 33-1 Prisms and Pyramids 479 Lesson 33-2 Cylinders and Cones 486 Lesson 33-3 Spheres and Solids of Rotation 489 Activity 33 Practice 493

Activity 34 Prisms and Cylinders—Exterior Experiences 495 Lesson 34-1 Surface Area of Prisms and Cylinders 495 Lesson 34-2 Volume of Prisms and Cylinders 500 Activity 34 Practice 505

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Activity 35 Pyramids and Cones—Perfect Packaging 507 Lesson 35-1 Surface Area of Pyramids and Cones 507 Lesson 35-2 Volume of Pyramids and Cones 513 Lesson 35-3 Density 518 Activity 35 Practice 521

Embedded Assessment 2 Surface Area and Volume—Action-Packed Measurements 523

Activity 36 Spheres—Isn’t that Spatial? 525 Lesson 36-1 Surface Area of Spheres 525 Lesson 36-2 Volume of Spheres 529 Lesson 36-3 Spherical Geometry 532 Activity 36 Practice 537

Activity 37 Changing Dimensions—Model Behavior 539 Lesson 37-1 Cubes and Spheres 539 Lesson 37-2 Pyramids and Cylinders 542 Activity 37 Practice 547

Embedded Assessment 3 Changing Dimensions of Spheres—Spherical Storage 549

UNIT 6 PROBABILITY Planning the Unit 551a Unit 6 Overview 551 Getting Ready 552

Activity 38 Sample Spaces—SpringBoard Superstar and More 553 Lesson 38-1 Probability of a Single Event 553 Lesson 38-2 Events Involving “And” and “Or” 559 Activi ty 38 Practice 565

Activity 39 Venn Diagrams and Probability Notation—Annabel HS 567 Lesson 39-1 Using a Venn Diagram to Represent a Sample Space 567 Lesson 39-2 Using Venn Diagrams to Represent “And,” “Or,” and “Not” 573 Activity 39 Practice 579

Activity 40 Addition Rule and Mutually Exclusive Events—Hector Street 581 Lesson 40-1 Applying the Addition Rule 581 Lesson 40-2 Adapting the Addition Rule for Mutually Exclusive Events 587 Activity 40 Practice 591

Embedded Assessment 1 Probability and the Addition Rule—Diane’s Books 593

Activity 41 Dependent Events—Coco Wildlife Conservation Trust 595 Lesson 41-1 Understanding Conditional Probability 595 Lesson 41-2 The Conditional Probability Formula 601 Lesson 41-3 Tree Diagrams 604 Activity 41 Practice 611

x SpringBoard® Mathematics Geometry

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