basics of geometry - edl · chapter1 basics of geometry chapter study guide 2 1.1 patterns and...

CHAPTER

1Basics of Geometry

CHAPTER STUDY GUIDE 2

1.1 Patterns and Inductive Reasoning 3

1.2 Points, Lines, and Planes 10

1.3 Segments and Their Measures 17QUIZ 1, 25

1.4 Angles and Their Measures 26

1.5 Segment and Angle Bisectors 34

� CONCEPT ACTIVITY: Folding Bisectors, 33QUIZ 2, 42

1.6 Angle Pair Relationships 44

� TECHNOLOGY ACTIVITY: Angles and Intersecting Lines, 43

1.7 Introduction to Perimeter, Circumference, and Area 51QUIZ 3, 58

ASSESSMENT

Skill Review, 2Quizzes, 25, 42, 58Test Preparation Questions, 9, 16, 24, 31,41, 50, 57

Chapter Summary, Review, and Test, 59Chapter Standardized Test, 64Project, Chapter 1, 66

APPLICATIONHIGHLIGHTS

Airport Runways 1, 32Bacteria Growth 8Two-Point Perspective 15Long-Distance Rates 23Playing Darts 31Kite Design 37Strike Zone 40Paper Airplanes 41Bridges 49Millennium Dome 56

MATH & HISTORY

Geometric Constructions 25

INTERNETCONNECTIONS

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Application Links1, 25, 49, 56

Student Help7, 12, 22, 28, 39, 40, 41, 45,56

Career Links8, 30

Extra Challenge9, 16, 24, 32, 41

vi Contents

Basics of Geometry

�READING, WRITING, ANDNOTETAKING

Reading 2, 10, 11, 17, 18, 19, 26, 27, 28,34, 36, 44, 46Writing 41, 50Notetaking 2, 17–20, 27, 35, 51, 53, 59Vocabulary 2, 6, 13, 21, 29, 38, 47, 55, 60

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Contents vii

CHAPTER

2Reasoning and Proof


2.1 Conditional Statements 71

2.2 Definitions and Biconditional Statements 79

2.3 Deductive Reasoning 87

� CONCEPT ACTIVITY: Logic Puzzle, 86QUIZ 1, 95

2.4 Reasoning with Properties from Algebra 96

2.5 Proving Statements about Segments 102

2.6 Proving Statements about Angles 109

� CONCEPT ACTIVITY: Investigating Complementary Angles, 108QUIZ 2, 116

ASSESSMENT

Skill Review, 70Quizzes, 95, 116Test Preparation Questions, 78, 85, 94,101, 107, 115Chapter Summary, Review, and Test, 117Chapter Standardized Test, 122Algebra Review, 124


Robotics 69, 93Advertising 77Research Buggy 77Winds at Sea 84Zoology 90Fitness 97Auto Racing 98Wind-Chill Factor 99Pay Raises 100Optical Illusion 106Wall Trim 115

MATH & HISTORY

Recreational Logic Puzzles 95

INTERNETCONNECTIONS


Application Links69, 84, 95

Student Help72, 76, 88, 93, 97, 114

Career Links77, 90, 106

Extra Challenge78, 85, 94, 101, 107

Reasoning and Proof


Reading 71, 72, 74, 79, 80, 87, 88, 103Writing 76, 77, 84, 91, 93, 94, 101, 115Notetaking 70, 73, 96, 98, 102, 109–112,117Vocabulary 70, 75, 82, 91, 99, 104,112, 118


CHAPTER

3Perpendicular andParallel Lines


3.1 Lines and Angles 129

3.2 Proof and Perpendicular Lines 136

� CONCEPT ACTIVITY: Forming a Flow Proof, 135

3.3 Parallel Lines and Transversals 143

� TECHNOLOGY ACTIVITY: Parallel Lines and Angles, 142QUIZ 1, 149

3.4 Proving Lines are Parallel 150

3.5 Using Properties of Parallel Lines 157QUIZ 2, 164

3.6 Parallel Lines in the Coordinate Plane 165

3.7 Perpendicular Lines in the Coordinate Plane 172QUIZ 3, 178

ASSESSMENT

Skill Review, 128Quizzes, 149, 164, 178Test Preparation Questions, 134, 141,148,156, 163, 171, 177Chapter Summary, Review, and Test, 179Chapter Standardized Test, 184Cumulative Practice, Chapters 1–3, 186Project, Chapters 2 and 3, 188


Sailing 127, 152Escalators 133Circuit Boards 140Botany 144Earth’s Circumference 145Rainbows 148Snow Making 156Zip Line 170Ray Tracing 174Needlepoint 176Sculpture 177

MATH & HISTORY

Measuring Earth’sCircumference 164

INTERNETCONNECTIONS


Application Links127, 145, 154, 163, 164

Student Help131, 140, 147, 152, 161, 167,176


Extra Challenge134, 141, 148, 156, 177

viii Contents

Perpendicular and Parallel Lines


Reading 128, 129, 131, 165, 167, 174Writing 134, 148, 155, 160, 163, 177, 178Notetaking 128, 130, 137–138, 143, 150,157, 166, 172, 179Vocabulary 128, 132, 138, 146, 153, 168,175, 180


Contents ix

CHAPTER

4Congruent Triangles


4.1 Triangles and Angles 194

� CONCEPT ACTIVITY: Investigating Angles of Triangles, 193

4.2 Congruence and Triangles 202QUIZ 1, 210

4.3 Proving Triangles are Congruent: SSS and SAS 212

� CONCEPT ACTIVITY: Investigating Congruent Triangles, 211

4.4 Proving Triangles are Congruent: ASA and AAS 220QUIZ 2, 227

� TECHNOLOGY ACTIVITY: Investigating Triangles and Congruence, 228

4.5 Using Congruent Triangles 229

4.6 Isosceles, Equilateral, and Right Triangles 236

4.7 Triangles and Coordinate Proof 243QUIZ 3, 250

ASSESSMENT

Skill Review, 192Quizzes, 210, 227, 250Test Preparation Questions, 201, 209,219, 226, 235, 242, 249Chapter Summary, Review, and Test, 251Chapter Standardized Test, 256Algebra Review, 258


Bridges 191, 234Weaving 195Wing Deflectors 200Stamps 204Crop Circles 207Origami 208Structural Support 215Astronomy 222Orienteering 225Color Wheel 241

MATH & HISTORY

Triangles in Architecture210

INTERNETCONNECTIONS



Student Help199, 203, 217, 230, 237, 243,245

Career Links200, 234

Extra Challenge201, 209, 219, 226, 235, 242,249

Congruent Triangles


Reading 192, 195, 196, 202, 229, 236Writing 209, 218, 226, 250Notetaking 192, 194–197, 203, 205,212–213, 220, 236–238, 251Vocabulary 192, 198, 205, 216, 223, 239,246, 252


CHAPTER

5Properties of Triangles


5.1 Perpendiculars and Bisectors 264

� CONCEPT ACTIVITY: Investigating Perpendicular Bisectors, 263

5.2 Bisectors of a Triangle 272

5.3 Medians and Altitudes of a Triangle 279QUIZ 1, 285

� TECHNOLOGY ACTIVITY: Investigating Concurrent Lines, 286

5.4 Midsegment Theorem 287

5.5 Inequalities in One Triangle 295

� TECHNOLOGY ACTIVITY: Side Lengths and Angle Measures, 294

5.6 Indirect Proof and Inequalities in Two Triangles 302QUIZ 2, 308

ASSESSMENT

Skill Review, 262Quizzes, 285, 308Test Preparation Questions, 271, 278,284,292, 301, 307Chapter Summary, Review, and Test, 309Chapter Standardized Test, 314Project, Chapters 4 and 5, 316


Goalkeeping 261, 270Roof Truss 267Early Aircraft 269Mushroom Rings 277Center of Population 280Electrocardiograph 283Fractals 291Surveying Land 298Kitchen Triangle 299Hiking 308

MATH & HISTORY

Optimization 285

INTERNETCONNECTIONS


Application Links261, 291

Student Help273, 280, 291, 300, 306


Extra Challenge271, 278, 284, 293, 301, 307

x Contents

Properties of Triangles


Reading 262, 264, 266, 272, 274, 279,287, 303Writing 271, 277, 284, 292, 307Notetaking 262, 265–266, 273–274, 279,281, 288, 295–297, 302–303, 309Vocabulary 262, 267, 275, 282, 290, 298,305, 310


Contents xi

CHAPTER

6Quadrilaterals


6.1 Polygons 322

� CONCEPT ACTIVITY: Classifying Shapes, 321

6.2 Properties of Parallelograms 330

� TECHNOLOGY ACTIVITY: Investigating Parallelograms, 329

6.3 Proving Quadrilaterals are Parallelograms 338QUIZ 1, 346

6.4 Rhombuses, Rectangles, and Squares 347

6.5 Trapezoids and Kites 356QUIZ 2, 363

6.6 Special Quadrilaterals 364

6.7 Areas of Triangles and Quadrilaterals 372

� CONCEPT ACTIVITY: Areas of Quadrilaterals, 371QUIZ 3, 380

ASSESSMENT

Skill Review, 320Quizzes, 346, 363, 380Test Preparation Questions, 328, 337,345, 355, 362, 370, 379Chapter Summary, Review, and Test, 381Chapter Standardized Test, 386Cumulative Practice, Chapters 1–6, 388Algebra Review, 390


Scissors Lift 319, 336Traffic Signs 326Plant Shapes 327Furniture Design 333Hinged Box 339Changing Gears 343Carpentry 350Layer Cake 361Gem Cutting 369Energy Conservation 377

MATH & HISTORY

History of Finding Area 346

INTERNETCONNECTIONS



Student Help335, 341, 344, 349, 357, 369


Extra Challenge328, 337, 345, 355, 362, 379

Quadrilaterals


Reading 320, 322, 323, 324, 330, 347, 356,364, 372Writing 328, 335–337, 355, 370, 379Notetaking 320, 324, 330, 338, 340,348–349, 356–358, 365, 372, 374, 381Vocabulary 320, 325, 333, 351, 359,376, 382


CHAPTER

7Transformations


7.1 Rigid Motion in a Plane 396

� CONCEPT ACTIVITY: Motion in a Plane, 395

7.2 Reflections 404

� CONCEPT ACTIVITY: Reflections in the Plane, 403

7.3 Rotations 412

� TECHNOLOGY ACTIVITY: Investigating Double Reflections, 411QUIZ 1, 420

7.4 Translations and Vectors 421

7.5 Glide Reflections and Compositions 430

� CONCEPT ACTIVITY: Multiple Transformations, 429

7.6 Frieze Patterns 437QUIZ 2, 444

ASSESSMENT

Skill Review, 394Quizzes, 420, 444Test Preparation Questions, 402, 410,419, 428, 436, 443Chapter Summary, Review, and Test, 445Chapter Standardized Test, 450Project, Chapters 6 and 7, 452


Architecture 393, 435Construction 398Machine Embroidery 401Kaleidoscopes 406Logo Design 415Wheel Hubs 418Pentominoes 432Stenciling 435Pet Collars 441Celtic Knots 443

MATH & HISTORY

History of DecorativePatterns 420

INTERNETCONNECTIONS



Student Help400, 408, 414, 424, 426, 434,443


Extra Challenge402, 410, 419, 428, 436

xii Contents

Transformations


Reading 394, 396, 397, 404, 405, 412, 415,421, 423, 430, 438Writing 401, 419, 434, 442–443Notetaking 394, 404, 412, 414, 421, 431,438, 445Vocabulary 394, 399, 407, 416, 425, 433,440, 446


Contents xiii

CHAPTER

8Similarity


8.1 Ratio and Proportion 457

8.2 Problem Solving in Geometry with Proportions 465

8.3 Similar Polygons 473

� CONCEPT ACTIVITY: Making Conjectures about Similarity, 472QUIZ 1, 479

8.4 Similar Triangles 480

8.5 Proving Triangles are Similar 488QUIZ 2, 496

8.6 Proportions and Similar Triangles 498

� TECHNOLOGY ACTIVITY: Investigating Proportional Segments, 497

8.7 Dilations 506QUIZ 3, 513

� TECHNOLOGY ACTIVITY: Exploring Dilations, 514

ASSESSMENT

Skill Review, 456Quizzes, 479, 496, 513Test Preparation Questions, 464, 471,478,487, 495, 505, 512Chapter Summary, Review, and Test, 515Chapter Standardized Test, 520Algebra Review, 522


Scale Drawing 455, 490Baseball Sculpture 463Paper Sizes 466Model Building 467Ramp Design 470Television Screens 477Aerial Photography 482Human Vision 487Pantograph 490Indirect Distances 494Shadow Puppets 508

MATH & HISTORY

Golden Rectangle 496

INTERNETCONNECTIONS



Student Help463, 466, 477, 482, 485, 491,504, 511


Extra Challenge471, 478, 487, 495

Similarity


Reading 456, 457, 459, 473, 506Writing 470, 505, 512Notetaking 456, 459, 465, 475, 481, 488,498–499, 515Vocabulary 456, 461, 468, 475, 483, 492,502, 509, 516


CHAPTER

9Right Trianglesand Trigonometry


9.1 Similar Right Triangles 527

9.2 The Pythagorean Theorem 535

9.3 The Converse of the Pythagorean Theorem 543

� TECHNOLOGY ACTIVITY: Investigating Sides and Angles of Triangles, 542QUIZ 1, 549

9.4 Special Right Triangles 551

� CONCEPT ACTIVITY: Investigating Special Right Triangles, 550

9.5 Trigonometric Ratios 558QUIZ 2, 566

9.6 Solving Right Triangles 567

9.7 Vectors 573QUIZ 3, 580

ASSESSMENT

Skill Review, 526Quizzes, 549, 566, 580Test Preparation Questions, 534, 541,548, 556, 565, 572, 579Chapter Summary, Review, and Test, 581Chapter Standardized Test, 586Cumulative Practice, Chapters 1–9, 588Project, Chapters 8 and 9, 590


Support Beam 525, 537Monorail 530Rock Climbing 532Construction 545Moon Craters 564Parade Balloon 565Space Shuttle 569Tug of War 577Skydiving 578Bumper Cars 579

MATH & HISTORY

Pythagorean TheoremProofs 557

INTERNETCONNECTIONS


Application Links525, 557

Student Help532, 540, 547, 552, 559, 569


Extra Challenge534, 541, 548, 556, 565, 572,579

xiv Contents

Right Trianglesand Trigonometry


Reading 530, 558, 568, 573, 575Writing 534, 541, 564, 571–572, 579Notetaking 526, 527, 529, 535, 543–544,551, 558, 575, 581Vocabulary 526, 531, 538, 545, 554, 562,570, 576, 582


Contents xv

CHAPTER

10Circles


Tangents to Circles 595

Arcs and Chords 603

Inscribed Angles 613

� CONCEPT ACTIVITY: Investigating Inscribed Angles, 612QUIZ 1, 620

Other Angle Relationships in Circles 621

Segment Lengths in Circles 629

� TECHNOLOGY ACTIVITY: Investigating Segment Lengths, 628QUIZ 2, 635

Equations of Circles 636

Locus 642

� TECHNOLOGY ACTIVITY: Investigating Points Equidistantfrom a Point and a Line, 641QUIZ 3, 648

ASSESSMENT

Skill Review, 594Quizzes, 620, 635, 648Test Preparation Questions, 602, 611,619, 627, 634, 640, 647Chapter Summary, Review, and Test, 649Chapter Standardized Test, 654Algebra Review, 656

10.7

10.6

10.5

10.4

10.3

10.2

10.1


Fireworks 593, 625Pick Ax 606Time Zone Wheel 609Avalanche Rescue Beacon

609Theater Design 614Carpenter’s Square 619Aquarium Tank 631Global Positioning System

634Cellular Phones 639Earthquake Epicenter 644

MATH & HISTORY

History of Timekeeping 648

INTERNETCONNECTIONS


Application Links593, 625, 631, 634, 639, 648

Student Help596, 610, 622, 633


Extra Challenge602, 611, 619, 627, 634, 640

Circles


Reading 595, 596, 603, 622, 630, 636, 644Writing 602, 611, 634Notetaking 594, 597–598, 604–606,613–615, 621–622, 629–630, 642, 649Vocabulary 594, 599, 607, 616, 632, 638,645, 650


CHAPTER

11Area of Polygonsand Circles


Angle Measures in Polygons 661

Areas of Regular Polygons 669

Perimeters and Areas of Similar Figures 677

� CONCEPT ACTIVITY: Area Relationships in Similar Figures, 676QUIZ 1, 682

Circumference and Arc Length 683

� TECHNOLOGY ACTIVITY: Perimeters of Regular Polygons, 690

Areas of Circles and Sectors 691

Geometric Probability 699QUIZ 2, 705

� TECHNOLOGY ACTIVITY: Investigating Experimental Probability, 706

ASSESSMENT

Skill Review, 660Quizzes, 682, 705Test Preparation Questions, 668, 675,681, 689, 698, 704Chapter Summary, Review, and Test, 707Chapter Standardized Test, 712Project, Chapters 10 and 11, 714

11.6

11.5

11.4

11.3

11.2

11.1


Basaltic Columns 659, 673Foucault Pendulums 671Tiling 674Fort Jefferson 680Track 685Boomerangs 694Viking Longships 697Ship Salvage 703Archery Target 703Balloon Race 704

MATH & HISTORY

Approximating Pi 682

INTERNETCONNECTIONS


Application Links659, 671, 678, 682, 697

Student Help662, 674, 680, 684, 692, 700


Extra Challenge675, 681, 689, 698, 704

xvi Contents

Area of Polygonsand Circles


Reading 661, 663, 669, 670, 677, 683,684, 691, 692Writing 666, 673, 681, 697, 704Notetaking 660, 662–663, 669–670,677, 683, 691–692, 699, 707Vocabulary 660, 665, 672, 679, 686,695, 701, 708


Contents xvii

CHAPTER

12Surface Areaand Volume


Exploring Solids 719

Surface Area of Prisms and Cylinders 728

� CONCEPT ACTIVITY: Investigating Surface Area, 727

Surface Area of Pyramids and Cones 735QUIZ 1, 742

Volume of Prisms and Cylinders 743

� TECHNOLOGY ACTIVITY: Minimizing Surface Area, 750

Volume of Pyramids and Cones 752

� CONCEPT ACTIVITY: Investigating Volume, 751QUIZ 2, 758

Surface Area and Volume of Spheres 759

Similar Solids 766QUIZ 3, 772

ASSESSMENT

Skill Review, 718Quizzes, 742, 758, 772Test Preparation Questions, 726, 734,741, 749, 757, 765, 771Chapter Summary, Review, and Test, 773Chapter Standardized Test, 778Cumulative Practice, Chapters 1–12, 780

12.7

12.6

12.5

12.4

12.3

12.2

12.1


Spherical Buildings 717,764

Soccer Ball 722Wax Cylinder Records 733Pyramids 740Candles 748Nautical Prism 754Automatic Pet Feeder 756Baseball 760Ball Bearings 761Model Car 770

MATH & HISTORY

History of Containers 742

INTERNETCONNECTIONS



Student Help725, 730, 739, 747, 756, 760,764


Extra Challenge726, 734, 741, 749, 765, 771

Surface Areaand Volume


Reading 719, 720, 721, 728, 730, 735, 737,753, 759, 760, 766Writing 734, 757Notetaking 718, 721, 729–730, 736–737,743–744, 752, 759, 761, 767, 773Vocabulary 718, 723, 731, 738, 746, 755,762, 769, 774


SKILLS REVIEW HANDBOOK 783

EXTRA PRACTICE FOR CHAPTERS 1–12 803

POSTULATES AND THEOREMS 827

ADDITIONAL PROOFS 833

SYMBOLS, FORMULAS, AND TABLES 841

MIXED PROBLEM SOLVING 846

APPENDICES 858

GLOSSARY 876

ENGLISH-TO-SPANISH GLOSSARY 888

INDEX 901

SELECTED ANSWERS SA1

xviii Contents

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