Contents Getting Started About This Book Eight Standards for Mathematical Practice Test-Taking Strategies

Chapter R: Algebra Review R.1

R.2

R.3

R.4

R.5

R.6

1 1 3

4

Solving Equations A-CED.4; A-REI.3

5

Solving One-Variable Equations Solving Literal Equations Ratios and Proportions

5 6 7

Solving Inequalities A-REI.3

9

Addition Property of Inequality Multiplication Property of Inequality

9 10

The Slope-Intercept Form of a Line

11

Slope of a Line Horizontal and Vertical Lines Graphing Equations Using Slope and Intercept Determining an Equation from Two Points

11 12 13 14

Solving a System of Equations A-REI.5; A-REI.6

16

Solving by Substitution Solving by Elimination

16 17

Multiplying Polynomials A-APR.1

19

Multiplying a Binomial by a Monomial Multiplying Binomials Using FOIL

19 20

Factoring Polynomials A-SSE.2

21

Factoring x 1 bx 1 c Difference of Squares

21 22

Simplifying Square Roots Completing the Square A-REI.4a Graphing Parabolas A-APR.3; A-REI.10; F-BF.3

23

Graphing Parabolas of the Form y 5 ax2 Graphing Parabolas in Vertex Form

26 28

2

R.7 R.8 R.9

1

24 26

Contents

iii

R.10 Area and Perimeter Fundamentals

30

Squares, Rectangles, and Triangles Circles

30 31

R.11 Pythagorean Theorem

33

Chapter 1: Geometry Fundamentals

36

1.1 Geometry Essentials G-CO.1; G-CO.12

37

Points, Lines, and Planes Angles Construction: Copying Segments and Angles

37 39 40

1.2 Measuring Distances G-GPE.6

43

Segment Addition The Distance and Midpoint Formulas Ratios and Line Segments

1.3 Transformations and Congruent Figures

43 45 49 G-CO.5; G-CO.6; G-CO.7

Rigid Motions Rigid Motion and Congruent Figures

50 51

1.4 Translation in a Coordinate Plane G-CO.2; G-CO.5; G-CO.6 1.5 Rotation G-CO.2; G-CO.3; G-CO.4; G-CO.5; G-CO.6 Rotation Fundamentals Multi-Part Problem Practice G-CO.2; G-CO.3; G-CO.4; G-CO.5; G-CO.6

Reflecting an Image About an Axis Lines of Symmetry Rotational Symmetry Three-Dimensional Symmetry

1.7 Composition of Transformations G-CO.2; G-CO.4; G-CO.5; G-CO.6 Order of Composite Transformations Glide Reflections Parallel Reflections Reflections and Rotations Multi-Part Problem Practice

Chapter 1 Review

The globe icon

iv   Contents 

62

70 70 75 77 78 79 79 80 81 82 84 85

Key to the icons: The computer icon

56 62 64 70

Mathematics of 180° and 90° Rotations

1.6 Reflection, Rotation, and Symmetry

50

indicates Digital Activities that can be found at www.amscomath.com. indicates where Real-World Model Problems are found in the text.

Chapter 2: Similar Figures and Dilation 2.1 Similar Figures G-SRT.5 2.2 Dilation and Similar Figures

88 89

G-CO.2; G-SRT.1a; G-SRT.1b; G-SRT.2; G-SRT.5

Dilation and Proportions Dilation and Angles Centers of Dilation

93 93 94 95 101

Multi-Part Problem Practice

2.3 Similarity, Polygons, and Circles

G-SRT.2; G-SRT.5

Similarity and Polygons Similarity and Circles Similar Polygons and Area

102 102 102 107

2.4 Similarity and Transformations G-CO.2; G-CO.5

109 112

Multi-Part Problem Practice

Chapter 2 Review Cumulative Review for Chapters 1–2

112 115

Chapter 3: Reasoning

118

3.1 Inductive Reasoning

119

Finding Patterns in Numbers Finding Patterns in Figures Counterexamples

119 120 122 124

Multi-Part Problem Practice

3.2 Conditional Statements

124

Applying Conditional Statements

3.3 Deductive Reasoning

126 G-CO.9

Reasoning and Proofs

3.4 Reasoning in Geometry

130 130

G-CO.9

Angles and Their Relationships Vertical Angles Properties of Equality and Congruence Proving Relationships About Angles Reasoning and Graphs Multi-Part Problem Practice

Chapter 3 Review Cumulative Review for Chapters 1–3

134 134 134 137 138 140 143 143 146

Contents    v

Chapter 4: Parallel and Perpendicular Lines 4.1 Parallel Lines and Angles

149

G-CO.1; G-CO.9

Parallel and Perpendicular Lines Corresponding Angles Alternate Interior and Exterior Angles Consecutive Interior Angles

4.2 More on Parallel Lines and Angles

148

149 150 151 153 G-CO.9

Angle Relationships Parallel Lines and Angles Multi-Part Problem Practice

4.3 Perpendicular Lines G-CO.9; G-CO.12 Pairs of Perpendicular Lines Construction: Perpendicular and Parallel Lines Distance and Perpendicular Lines

4.4 Parallel Lines, Perpendicular Lines, and Slope G-GPE.5 Determining When Lines Are Parallel or Perpendicular Rotations and Perpendicular Lines Distance Between a Point and a Line Multi-Part Problem Practice

4.5 Parallel Lines and Triangles G-CO.10

158 158 160 166 166 166 169 173 176 176 177 179 182 182

Interior and Exterior Angles Angles in a Right Triangle

182 185

Chapter 4 Review Cumulative Review for Chapters 1–4

187

Chapter 5: Congruent Triangles

194

5.1 Isosceles and Equilateral Triangles G-CO.10; G-CO.12 Determining Types of Triangles Isosceles Triangles and Theorems Equilateral Triangle Theorems Construction: Equilateral Triangle Multi-Part Problem Practice

5.2 Congruent Figures G-SRT.5 Proving Triangles Are Congruent Multi-Part Problem Practice

5.3 Proving Triangles Congruent with SSS and SAS G-CO.10; G-SRT.5 Side-Side-Side (SSS) Postulate Hypotenuse-Leg Theorem Side-Angle-Side (SAS) Postulate

vi   Contents 

191

195 195 198 200 200 205 205 205 209 209 209 211 213

5.4 Proving Triangles Congruent with ASA and AAS

G-CO.7; G-CO.8; G-CO.10;

G-SRT.5

218

Angle-Side-Angle (ASA) Postulate Rigid Motion and Congruent Triangles Introduction to Coordinate Proofs Angle-Angle-Side (AAS) Theorem

218 219 219 222

Chapter 5 Review Cumulative Review for Chapters 1–5

226 229

Chapter 6: Relationships Within Triangles 6.1 Midsegments

233

G-CO.10; G-GPE.4

240

Multi-Part Problem Practice

6.2 Perpendicular and Angle Bisectors G-CO.9; G-CO.12; G-SRT.5 Perpendicular Bisectors Angle Bisectors Construction: Angle Bisector

240 240 243 244 248

Multi-Part Problem Practice

6.3 Circumcenters

232

248

G-CO.12; G-C.3; G-GPE.6

Perpendicular Bisectors in a Triangle Circumcenters

248 250

6.4 Centroids and Orthocenters G-CO.10; G-GPE.4; G-GPE.6

254

Medians Altitudes

254 258

6.5 Incenters G-C.3

262



Summary of Circumcenters, Incenters, Centroids, and Orthocenters

6.6 Optional: Inequalities in One Triangle

265

The Hinge Theorem and Its Converse

268

6.7 Optional: Indirect Reasoning Chapter 6 Review Cumulative Review for Chapters 1–6

271 275 279

Chapter 7: Similarity and Trigonometry

282

7.1 Similarity: Angle-Angle and Side-Side-Side G-SRT.3; G-SRT.5

Angle-Angle (AA) Similarity Postulate Side-Side-Side (SSS) Similarity Theorem

7.2 Similar Triangles: Side-Angle-Side Theorem 7.3 Pythagorean Theorem G-SRT.8 Pythagorean Triples Optional: A Formula for Pythagorean Triples

263

283 283 287

G-SRT.5

291 298 302 303 Contents    vii

7.4 Similar Right Triangles

304

G-SRT.4; G-SRT.5

Converse of Pythagorean Theorem Classifying Triangles Using the Pythagorean Theorem

7.5 Special Right Triangles

312

G-SRT.8

317

Multi-Part Problem Practice

7.6 Trigonometric Ratios

G-SRT.6; G-SRT.7; G-SRT.8

Sine, Cosine, and Complementary Angles Sine, Cosine, and Tangent for Special Triangles

7.8 Law of Cosines and Law of Sines

G-SRT.9; G-SRT.10; G-SRT.11

Trigonometry and Triangle Area The Law of Sines and Solving ASA and AAS Triangles

Chapter 7 Review Cumulative Review for Chapters 1–7

334 339 339 346 349

Chapter 8: C  ircles

352

8.1 Circles, Tangents, and Secants

G-CO.1; G-C.2; G-C.3; G-C.4; G-GMD.1

Circles, Circumferences, and Areas Circles, Segments, and Lines Construction: Circles Based on Triangles Tangents with a Common Endpoint

363

G-C.2

8.3 Inscribed Figures

363 364 G-CO.13; G-C.2; G-C.3

Construction: Regular Hexagon in a Circle Inscribed Polygons and Cyclic Quadrilaterals Cyclic Quadrilaterals and Their Diagonals Inscribed Right Triangles Multi-Part Problem Practice

8.4 More on Chords and Angles Tangents, Chords, and Circles Angles Inside and Outside a Circle Secants, Tangents, and Circles Chord Lengths in Circles Multi-Part Problem Practice

353 353 355 355 360

Arc Measure Properties of Chords

viii   Contents 

330 334

Multi-Part Problem Practice

8.2 Chords and Arcs

317 320 324

7.7 Optional: Inverses of Trigonometric Functions



307 308

G-C.2

370 372 373 375 376 380 380 380 381 383 385 388

8.5 Arc Lengths and Area

388

G-CO.1; G-C.5

Area of a Sector Radian Measure of Angles Area of a Sector with Angle in Radians

389 392 395

Chapter 8 Review Cumulative Review for Chapters 1-8

399 402

Chapter 9: Polygons 9.1 Parallelograms and Their Diagonals

406 G-CO.11; G-GPE.5

Sides and Angles in a Parallelogram Diagonals in a Parallelogram

407 408 412

9.2 Deciding if a Parallelogram Is Also a Rectangle, Square, or Rhombus G-CO.11

417

Identifying Rectangles, Rhombuses, and Squares

421

9.3 Deciding if a Quadrilateral Is a Parallelogram 9.4 Optional: Polygons and Their Angles

G-CO.11

Interior Angles in a Polygon Exterior Angles of a Convex Polygon Number of Diagonals in a Polygon

425 432 432 434 435

9.5 Trapezoids and Kites G-GPE.4

437

Trapezoids Midsegments and Trapezoids Kites

437 439 441

9.6 Areas and the Coordinate Plane G-CO.12; G-GPE.7; G-MG.1; G-MG.2; G-MG.3 Area of a Parallelogram and Rhombus Construction: Two Ways to Construct a Square Area of a Trapezoid Area of a Kite and Rhombus Using Diagonals Multi-Part Problem Practice

9.7 Area of Regular Polygons G-MG.1 9.8 Area and Trigonometry G-MG.1; G-MG.3; G-SRT.8 Multi-Part Problem Practice

Chapter 9 Review Cumulative Review for Chapters 1–9

445 445 447 450 452 456 457 461 464 465 468

Contents ix

Chapter 10: Solids

472

10.1 Three-Dimensional Figures, Cross-Sections, and Drawings G-GMD.4 Three-Dimensional Figures Cross-Sections The Pythagorean Theorem in Three Dimensions Isometric Drawing and Perspective

10.2 Surface Area

G-MG.1; G-MG.2; G-MG.3

Surface Area of Cubes and Boxes Surface Area of Prisms Surface Area of Cylinders Surface Area of Pyramids Surface Area of Cones Surface Area of Spheres Surface Area Summary

10.3 Volume

Volume of Cubes and Boxes Volume of Prisms and Cylinders Volume of Pyramids and Cones Volume of Spheres Volume Summary

495

508 513

Multi-Part Problem Practice

513

G-MG.2

Congruent Solids Similar Solids

513 514 518

Multi-Part Problem Practice

Chapter 10 Review Cumulative Review for Chapters 1–10

519 522

Chapter 11: Conics

526 527

G-GMD.4

Conic Sections Circles Centered at the Origin

11.2 Parabolas at the Origin

527 528 G-GPE.2; G-GMD.4

Geometric Definition of a Parabola Graphing a Parabola at the Origin

11.3 Circles Translated from the Origin G-CO.4; G-C.1; G-GPE.1 The Standard Form of the Equation for a Circle Circles: Completing the Square Circles Are Similar

x   Contents 

483

495 497 498 500 501

10.4 Cavalieri’s Principle G-GMD.2

11.1 Circles at the Origin

473 475 476 476 483 484 485 486 488 489 491

G-GMD.1; G-GMD.3; G-MG.1; G-MG.3

10.5 Similar Solids

473

531 531 533 538 538 540 542

11.4 Optional: Parabolas Translated from the Origin

544 549

Multi-Part Problem Practice

11.5 Optional: Ellipses at the Origin

550

Geometric Definition of an Ellipse Ellipses Centered at the Origin Graphing an Ellipse at the Origin The Vertices, Axes, and Center of an Ellipse Derivations: Ellipse Equations

550 551 552 552 554 558

Multi-Part Problem Practice

11.6 Optional: Hyperbolas at the Origin

558

Geometric Definition of a Hyperbola Hyperbolas Centered at the Origin Graphing a Hyperbola at the Origin Derivation: Hyperbola Equation

558 559 561 564

Chapter 11 Review Cumulative Review for Chapters 1–11

567 571

Chapter 12: Probability 12.1 Introduction to Probability

574 575

S-CP.1; S-MD.6; S-MD.7

Experimental Probability Theoretical Probability and Sample Spaces Expected Value

575 576 583 585

Multi-Part Problem Practice

12.2 Permutations and Combinations S-CP.9

586

Tree Diagrams and the Fundamental Counting Principle Factorials Permutations Permutations of n Objects Taken r at a Time Combinations Permutation or Combination? Complex Counting Problems

12.3 Independent Events and the Multiplication Rule

586 588 588 590 592 593 594 S-CP.1; S-CP.2;

S-CP.5; S-MD.6; S-MD.7

599

Compound Events

602

12.4 Addition and Subtraction Rules S-CP.1; S-CP.7; S-MD.7 The Addition Rule Mutually Exclusive Events and the Addition Rule The Subtraction Rule The Origin of Probability Studies Multi-Part Problem Practice

605 605 607 608 610 613

Contents xi

12.5 Conditional Probability

S-CP.3; S-CP.4; S-CP.5; S-CP.6; S-CP.8; S-MD.7

Conditional Probability and Two-Way Tables Optional: Bayes’ Theorem

Chapter 12 Review Cumulative Review for Chapters 1–12

613 616 616 623 626

Glossary

630

Digital Activities and Real-World Model Problems

639

Index

640

xii

Contents

Getting Started About This Book Geometry is a full-year course, written to help students understand and explore the concepts of geometry as well as prepare them for the new end-of-course assessments. All instruction, model problems, and practice items were developed to support the latest college and career readiness standards, as well as core standards. Each chapter opens with lessonby-lesson alignment with the standards. The eight Mathematical Practice Standards are embedded throughout in selected Model Problems, extensive practice problem sets, and the comprehensive Chapter and Cumulative Reviews. In Geometry, students will use transformations to explore congruence and similarity and apply their knowledge to contextual problems. Students will also identify patterns and use inductive reasoning to make conjectures and learn the meaning and nature of mathematical proofs. By learning how to formally describe and analyze relationships among lines, parts of lines, planes, triangles, and circles, students will be able to apply geometric concepts in modeling situations. Students will use the rules of probability to compute probabilities and evaluate outcomes of decisions. Finally, throughout the text, algebraic skills learned in Algebra 1 are maintained, strengthened, and expanded as a bridge to Algebra 2 and Trigonometry. Each chapter incorporates multiple performance tasks that measure the ability of students to think critically and apply their knowledge in real-world situations. In addition, students and teachers have access to a companion Web site (www.amscomath.com) with activities and simulations linked directly to lessons in Geometry. Teachers also have the option to include a full range of digital simulations, electronic whiteboard lessons, videos, and interactive problems to stimulate conceptual understanding through the Digital Teacher’s Edition. Careful and consistent use of this text and the supporting materials will give students a firm grasp of geometry, prepare them for the end-of-course exams, and give them the tools they need to be college and career ready.

Eight Standards for Mathematical Practice The mathematical practices are a common thread for students to think about and understand math as they progress from Kindergarten through high school. Students should use the mathematical practices as a method to break down concepts and solve problems, including representing problems logically, justifying conclusions, applying mathematics to practical situations, explaining the mathematics accurately to other students, or deviating from a known procedure to find a shortcut. MP 1 Make sense of problems and persevere in solving them. Attack new problems by analyzing what students already know. Students should understand that many different strategies can work. Ask leading questions to direct the discussion. Take time to think. • explain the meaning of the problem • analyze given information, constraints, and relationships • plan a solution route • try simpler forms of the initial problem • use concrete objects to help conceptualize • monitor progress and change course, if needed • continually ask, “Does this make sense?” Getting Started

1

MP 2 Reason abstractly and quantitatively. Represent problems with symbols and/or pictures. • make sense of quantities and their relationships • decontextualize—represent a situation symbolically and contextualize—consider what given symbols represent • create a clear representation of the problem • consider the units involved • attend to the meaning of numbers and variables, not just how to compute them • use properties of operations and objects MP 3 Construct viable arguments and critique the reasoning

of others.

Ask questions, defend answers, and/or make speculations using correct math vocabulary. • use assumptions, definitions, and previously established results • make conjectures and build a valid progression of statements • use counterexamples • justify conclusions and communicate them to others • determine whether the arguments of others seem right MP 4 Model with mathematics. Show the relevance of math by solving real-world problems. Look for opportunities to use math for current situations in and outside of school in all subject areas. • apply mathematics to solve everyday problems • analyze and chart relationships using diagrams, two-way tables, graphs, flowcharts, and formulas to draw conclusions • apply knowledge to simplify a complicated situation • interpret results and consider whether answers make sense MP 5 Use appropriate tools strategically. Provide an assortment of tools for students and let them decide which ones to use. • choose appropriately from existing tools (pencil and paper, concrete models, ruler, protractor, calculator, spreadsheet, dynamic geometry software, etc.) when solving mathematical problems • detect possible errors by using estimation or other mathematical knowledge • use technology to explore and compare predictions and deepen understanding of concepts MP 6 Attend to precision. Use precise and detailed language in math. Instead of saying “I don’t get it,” students should be able to elaborate on where they lost the connection. Students should specify units in their answers and correctly label diagrams. • speak and write precisely using correct mathematical language • state the meaning of symbols and use them properly • specify units of measure and label axes appropriately • calculate precisely and efficiently • express answers with the proper degree of accuracy MP 7 Look for and make use of structure. See patterns and the significance of given information and objects. Use these to solve more complex problems. • see the big picture • discern a pattern or structure

2    Getting Started 

• • • •

recognize the significance of given aspects apply strategies to similar problems step back for an overview and shift perspective see complicated things as being composed of several objects

MP 8 Look for and express regularity in repeated reasoning. Understand why a process works so students can apply it to new situations. • notice repeated calculations and look for both general methods and shortcuts • maintain oversight of the process while paying attention to the details • evaluate the reasonableness of intermediate results • create generalizations founded on observations

Test-Taking Strategies General Strategies • Become familiar with the directions and format of the test ahead of time. There will be both multiple-choice and extended response questions where you must show the steps you used to solve a problem, including formulas, diagrams, graphs, charts, and so on, where appropriate. • Pace yourself. Do not race to
answer every question immediately. On the other hand, do not linger over any
question too long. Keep in mind that you will need more time to complete the
extended response questions than to complete the multiple-choice questions. • Speed comes from practice. The more you practice, the faster you will become and the more comfortable you will be with the material. Practice as often as you
can.

Specific Strategies • Always scan the answer choices before beginning to work on a multiple-choice question. This will help you to focus on the kind of answer that is required. Are you looking for fractions, decimals, percents, integers, squares, cubes, and so
on? Eliminate choices that clearly do not answer the question asked. • Do not assume that your answer is correct just because it appears among the choices. The wrong choices are usually there because they represent common student errors. After you find an answer, always reread the problem to make sure you have chosen the answer to the question that is asked, not the question
 you have in your mind. • Sub-in. To sub-in means to substitute. You can sub-in friendly numbers for the variables to find a pattern and determine the solution to the problem. • Backfill. If a problem is simple enough and you want to avoid doing the more complex algebra, or if a problem presents a phrase such as x 5 ?, then just fill in
the answer choices that are given in the problem until you find the one that works. • Do the math. This is the ultimate strategy. Don’t go wild searching in your mind for tricks, gimmicks, or math magic to solve every problem. Most of the time the best way to get the right answer is to do the math and solve the problem.

Getting Started

3