Index [] 30 °-60°-90° right ... angle bisectors of triangle, 313 circumcenter of triangle, ... Ceva’s Theorem, 456 Chord of a sphere, 680 Chord(s) of circles, 549 ...

Index A81

IndexIn

de

x

30°-60°-90° right triangles fi nding sine and cosine of 30°, 500

fi nding tangent with, 493

side lengths of, 475, 477

30°-60°-90° Triangle Theorem (Thm. 9.5), 477

45°-45°-90° (isosceles) right triangles fi nding sine and cosine of 45°, 500

side lengths, 475, 476

in standard position, 466

45°-45°-90° Triangle Theorem (Thm. 9.4), 476

AAA, See Angle-Angle (AA) Similarity

Theorem (Thm. 8.3)

AAS, See Angle-Angle-Side (AAS)

Absolute value, fi nding, 1

Acute angle, 39

Acute triangle in circumscribed circle, 315

classifying by angles, 236

classifying by Pythagorean

inequalities, 471

orthocenter of, 326

Addition Property of Equality, 92

Adjacent angles, 48–49

Adjacent arcs, 543

Ailles rectangle, 480

Algebraic Properties of Equality, 92

Algebraic reasoning, 91–95, 117

Distributive Property, 93

other properties of equality, 94

properties of equality, 92

Alternate exterior angles, 128

Alternate Exterior Angles Converse (Thm. 3.7), 139

Alternate Exterior Angles Theorem (Thm. 3.3), 132

exploring converses, 137

Alternate interior angles, 128

Alternate Interior Angles Converse (Thm. 3.6), 139

proving theorems about parallel

lines, 140

Alternate Interior Angles Theorem (Thm. 3.2), 132


proof of, 134

Altitude of triangle defi ned, 325

examples of segments and points in

triangles, 304, 327

using, 323, 325–327, 355

Ambiguous case of Law of Sines, 519

Analyzing Mathematical Relationships, Throughout. See for example:

area of sector of circle, 608

in corresponding parts of similar

polygons, 422

dimensions and surface area of right

cone, 657

dimensions and volume of

rectangular prism, 666

effects of changing dimensions, 624,

625

height and volume of cone, 675

isosceles triangles, 617

similar polygons with scale factor

of k, 424, 425

“and” (intersection), 732–733

Angle(s) and arc measures in circles, 565–567

circumscribed, 568

classifying, and types of, 39

complementary (See

Complementary angles)

congruent, 40

construction, copying an angle, 40

corresponding (See Corresponding

angles)

defi ned, 38

diagram interpretation, 51

fi nding angle measures, 47, 49, 50

(See also Angle measures)

inscribed, 557–559, 588

measuring and constructing, 37–42,

58

naming, 38

obtuse, 39, 512

pairs of, describing, 47–51, 58

adjacent angles, 48–49

complementary angles, 48–49

linear pair, 50

supplementary angles, 48–49

vertical angles, 50

pairs of, formed by transversals, 128

alternate exterior angles, 128

alternate interior angles, 128

consecutive interior angles, 128

corresponding angles, 128

proof of Symmetric Property of

Angle Congruence, 102, 110

Properties of Angle Congruence

(Thm. 2.2), 101

of triangles, 235–239, 294

angle measures of triangles,

237–239

classifying triangles by sides and

angles, 236–237

relating to sides, 339, 341–342

Angle Addition Postulate (Post. 1.4), 41

Angle-Angle-Side (AAS) congruence, 275, 277

identifying congruent triangles,

275

using Law of Sines to solve triangle,

514

Angle-Angle-Side (AAS) Congruence Theorem (Thm. 5.11), 275

Angle-Angle (AA) Similarity Theorem (Thm. 8.3), 432

proof of, 432

triangle similarity theorems

compared, 443

using, 433–434

Angle bisector(s) construction, bisecting an angle, 42

defi ned, 42

examples of segments and points

in triangles, 304, 327

fi nding angle measures, 42

points on, 305

proportionality in triangle, 453

using, 308–309

Angle Bisector Theorem (Thm. 6.3), 308

converse of, 308

Angle of depression, 501

Angle of elevation, 494

Angle measures fi nding with dynamic geometry

software, 176

in kite, 405

in polygons

exterior, 363, 366–367

interior, 363–366

in regular polygons, 617

in rhombus, 394

in triangles, 237–239

types of angles, 39

using properties of equality with, 94

Angle of rotation, 194

Angle-Side-Angle (ASA) congruence, 274, 276, 277

copying a triangle using ASA,

276

A82 Index

using Law of Sines to solve triangle,

514

Angle-Side-Angle (ASA) Congruence Theorem (Thm. 5.10), 274

Angles Inside the Circle Theorem (Thm. 10.15), 567

Angles Outside the Circle Theorem (Thm. 10.16), 567

Another Way corresponding angles, 132

probability, sample space and

outcomes, 706

segments of secants and tangents,

576

sketching a diagram, 86

solving right triangle, 507

Table of Trigonometric Ratios, 506

triangles and Laws of Cosines or

Sines, 515

Antipodal points, 687, 688

Apothem of regular polygon, 615,

617

Arc Addition Postulate (Post. 10.1), 543

Arc length, 599–600

Arc measures, 541–545, 587

fi nding, 542–543

fi nding from angle relationships in

circles, 566–567

fi nding with congruent chords, 550

identifying congruent arcs, 544

of intercepted arc, 559

of minor and major arcs, 542

proving circles are similar, 545

Area, See also Circumference and area;

Surface area and volume

of circle, 605–606, 637

of composite fi gures, 596, 619, 637

in coordinate plane, 29–33, 57

fi nding, 31

effects of changing dimensions,

623–626, 632

fi nding after dilation, 420

of hexagon, 619

of kite, 616, 619

of nonagon, 618

of octagon, 618

of parallelogram, 596, 619

of polygons and composite fi gures,

615–619, 631

of rectangle, 596, 637

of regular polygons, 615, 617–618

of rhombus, 616

of sector of circle, 605, 608–609,

619

of semicircle, 637

of similar polygons, 425

of spherical triangles, 690

of trapezoid, 619

of triangle, 637

of triangle, square, and rectangle, 31

of triangle, using trigonometric

ratios, 512

of two-dimensional fi gures, 595

using to fi nd probability, 708

Areas of Similar Polygons (Thm. 8.2), 425

Arithmetic mean, compared to

geometric mean, 481, 488

Arithmetic sequence, nth term of, 63

ASA, See Angle-Side-Angle (ASA)

Auxiliary line, 238

Axiom(s), 12

Axis of revolution, 642

BBase of isosceles triangle, 256

of solid, 640

of trapezoid, 402

Base angles of isosceles triangle, 256

of trapezoid, 402

Base Angles Theorem (Thm. 5.6), 256

converse of, 256

corollaries to, 257

Base edge of pyramid, 654

Basics of geometry angles, describing pairs of, 47–51,

58

angles, measuring and constructing,

37–42, 58

midpoint formulas, 19–23, 57

perimeter and area in coordinate

plane, 29–33, 57

points, lines, and planes, 3–7, 56

segments, measuring and

constructing, 11–15, 56

Bayes’ Theorem, 728

Between, 13

Biconditional statement(s) defi ned, and writing, 69

and defi nitions, 234

reading theorems, and rewriting, 394

Binomial distribution(s), 745–748,

754

constructing, 748

defi ned, 747

interpreting, 748

Binomial experiments, 747

Binomials, multiplying, 531

Birthday problem, 744

Bisecting angles, 42, See also Angle

bisector(s)

Bisecting segments, See Segment

bisector(s)

Bisector, perpendicular, See

Perpendicular bisector(s)

Bisectors of triangles, 313–318, 354

angle bisectors of triangle, 313

circumcenter of triangle, 314–316

circumscribing circle about triangle,

315–316

incenter of triangle, 317–318

inscribing circle within triangle, 318

perpendicular bisectors of triangle,

313

CCavalieri, Bonaventura, 664

Cavalieri’s Principle, 664–665, 672

Center of arc, 40

Center of circle, 532, 534

Center of dilation, 212

Center of regular polygon, 615, 617

Center of rotation, 194

Center of sphere, 680

Center of symmetry, 197

Central angle of circle defi ned, 541, 542

and inscribed angles, 557

Central angle of regular polygon, 617

Centroid Theorem (Thm. 6.7), 324

Centroid of triangle defi ned, 324

examples of segments and points

in triangles, 327

fi nding, 325

Ceva’s Theorem, 456

Chord of a sphere, 680

Chord(s) of circles, 549–552, 587

defi ned, 534

intersection with tangent on circle,

566

perpendicular to diameter, 549

using congruent chords

to fi nd arc measure, 550

to fi nd circle’s radius, 552

using diameter, 551

using perpendicular bisectors, 551

Circle(s) angle relationships in circles,

565–569, 588–589

fi nding angle and arc measures,

566–567

using circumscribed angles, 568

arc measures, 541–545, 587

identifying congruent arcs, 544

proving circles are similar, 545

area of, 605–606, 630, 637

chords, 549–552, 587

Index A83

Ind

ex

circumference of, 598

circumscribed about triangle,

315–316

congruent, 544

in coordinate plane, 579–582, 590

equations of circles, 580–581

writing coordinate proofs

involving circles, 582

defi ned, 532, 534

diameter of

chord perpendicular to, 549, 551

defi ned, 534

drawing by using string, 533

inscribed angles, 557–559, 588

inscribed polygons, 557, 560–561,

588

lines and segments that intersect

circles, 533–537, 586

radius of

defi ned, 534

fi nding with congruent chords,

552

fi nding with segments, 576

fi nding with tangent, 536

relationships with tangent circles,

532

segment relationships in circles,

573–576, 589

Circular arc, 541

Circumcenter Theorem (Thm. 6.5), 314

Circumcenter of triangle circumscribing circle about triangle,

316

defi ned, 314


triangles, 327

fi nding, 316

types of triangles with

circumscribed circles, 315

Circumference and area arc length, 597, 599–600, 630

areas (See also Area)

of circles and sectors, 605–609,

630

of composite fi gures, 619

of polygons, 615–618

circumference, 597–601, 630


623–626, 632

Circumference of circle, 598

Circumference of Earth, 603

Circumscribed angle, 568

Circumscribed Angle Theorem (Thm. 10.17), 568

Circumscribed circle, 560–561

Classifying angles, 39

lines, pairs of, 124, 125

polygons, 30, 365

quadrilaterals, 362, 393, 406

solids, 640

triangles by sides and angles,

236–237

Clockwise rotation, 194

Coin fl ip, 706, 737, 745

Coincident lines, example of, 124, 125

Collinear points, 4Combination(s), 740–741, 754

counting, 740

defi ned, 740

fi nding probability using, 741

formula, 740–741

Common Errors angles

adjacent, 48

approximation, 569

linear pair of, 50

names and angle measures, 49

naming, 38

symbol compared to less than

symbol, 341

and vertex, 258

area of semicircle, 609

calculator, inverse sine feature, 515

conditional statement and

contrapositive, 67

diameter of sphere, 681

indirect proofs, 341

pay attention to units, 600

probability

and binomial distribution, 748

overlapping events, 733

protractor scales, 39

rays, 5

transformation order, 196

triangles

congruence, 275

fi nding angles in, 516

geometric mean of right triangle,

485

proportional, 483

redrawing, 259

Common external tangent, 535

Common internal tangent, 535

Common tangent, 535

Compass, 15

Complement of event, 707–708

Complementary angles defi ned, 48–49

sine and cosine of, 498

Completing the square solving quadratic equations by, 531

in standard equation of circle, 581

Component form of vector, 178

Composite fi gure(s) area of, 596, 619, 631, 637

defi ned, 596

Composite solids surface area of, 649, 656

volumes of, 667, 675, 683

Composition of dilations, 221

Composition of rigid motions, 243–244

Composition Theorem (Thm. 4.1), 180

Composition of transformations, 180

Compositions performing, 180

performing with rotations, 196

Compound event(s), 732–733

Compound inequalities, writing, 303

Concave polygons, 30

Concentric circles, 535

Concept Summary Interpreting a Diagram, 51

Segments, Lines, Rays, and Points

in Triangles, 327

Triangle Congruence Theorems, 277

Triangle Similarity Theorems, 443

Types of Proofs, Symmetric

Property of Angle

Congruence, 110

Ways to Prove a Quadrilateral is a

Parallelogram, 383

Writing a Two-Column Proof, 102

Conclusion, in conditional statement,

66

Concurrent, lines, rays, or segments,

314

Conditional probability comparing, 725

defi ned, 715

fi nding with a table, 717

fi nding with conditional relative

frequencies, 724

Conditional relative frequency, 723–724

Conditional statement(s), 65–70, 116

biconditional statements, 69, 234,

394

defi ned, 66

in if-then form, 66

negation, 66

related conditionals, 67

true or false determination, 65

truth tables, 70

using defi nitions, 68

writing, 66–67

A84 Index

Cones frustum of, 678 lateral surface of, 655 oblique, 655 surface areas of, 653, 655–657, 695 volumes of, 671, 673, 675, 696–697 changing dimensions in, 675 formula for, 673Congruence, properties of, 101–102Congruence transformation, 205Congruence and transformations,

203–207, 227 congruence transformations, 205 identifying congruent fi gures, 204 refl ections in intersecting lines, 203,

207 refl ections in parallel lines, 203, 206 using theorems about, 206–207Congruent angles, 40Congruent arcs, identifying, 544Congruent Central Angles Theorem

(Thm. 10.4), 544Congruent circles, 544Congruent Circles Theorem

(Thm. 10.3), 544Congruent Complements Theorem

(Thm. 2.5), 107Congruent Corresponding Chords

Theorem (Thm. 10.6), 550Congruent fi gures defi ned, 204 using properties of, 245Congruent Parts of Parallel Lines

Corollary, 378Congruent polygons, 243–246, 294 using corresponding parts, 244–245 using Third Angles Theorem

(Thm. 5.4), 246Congruent segments, 15Congruent Supplements Theorem

(Thm. 2.4), 107Congruent triangles angles of triangles, 235–239, 294 congruent polygons, 243–246, 294 coordinate proofs, 287–290, 298 equilateral and isosceles triangles,

255–259, 295 proving triangle congruence by ASA and AAS, 273–277,

296–297 by SAS, 249–252, 295 by SSS, 265–269, 296 using, 281–284, 297Conjecture defi ned, 75, 76, 234 making and testing, 77

reasoning with, 75

writing, about angles of triangle, 235

writing, about isosceles triangles,

255

Consecutive integers, 77

Consecutive interior angles, 128

Consecutive Interior Angles Converse (Thm. 3.8), 139

Consecutive Interior Angles Theorem (Thm. 3.4), 132


Consecutive vertices, 364

Constant of proportionality, 601

Construction(s) bisecting a segment, 21

bisecting an angle, 42

centroid of triangle, 324

circumscribing circle about triangle,

316

copying a segment, 15

copying a triangle

using ASA, 276

using SAS, 252

using SSS, 268

copying an angle, 40

defi ned, 15

of a dilation, 214

of equilateral triangle, 258


parallel lines, 139

perpendicular bisector, 149

perpendicular line, 149

point along directed line segment,

451

proving, 284

square inscribed in circle, 561

tangent to circle, 537

Contingency table, 722

Contradiction, Proof by, 340

Contrapositive defi ned, of conditional statement, 67

truth table for, 70

Contrapositive of Triangle Proportionality Theorem, 451

Converse defi ned, of conditional statement, 67

truth table for, 70

Converses of theorems Alternate Exterior Angles Converse

Theorem (Thm. 3.7), 139

Alternate Interior Angles Converse


Converse of Angle Bisector


Converse of Base Angles Theorem

(Thm. 5.7), 256

Converse of Hinge Theorem

(Thm. 6.13), 348

Converse of Perpendicular Bisector Theorem (Thm. 6.2), 306

Converse of Pythagorean Theorem (Thm. 9.2), 470

Converse of Triangle Proportionality Theorem (Thm. 8.7), 450

Corresponding Angles Converse Theorem (Thm. 3.5), 138

Isosceles Trapezoid Base Angles Converse (Thm. 7.15), 403

Parallelogram Diagonals Converse (Thm. 7.10), 382

Parallelogram Opposite Angles Converse (Thm. 7.8), 380

Parallelogram Opposite Sides Converse (Thm. 7.7), 380

Perpendicular Chord Bisector Converse (Thm. 10.8), 550

Convex polygons, 30Coordinate (of point), 12Coordinate plane circles in, 579–582, 590 classifying triangle in, 237 dilating fi gures in, 211, 213 midpoint of segment in, 23 midsegments in, 334 parallelograms in, 375, 384, 396 perimeter and area in, 29–33, 57 placing fi gures in, 288 refl ecting fi gures in, 185 rotating fi gures in, 193, 195 slopes of lines in, 156 translating a fi gure in, 177, 179 trapezoid in, 402 trapezoid midsegment, 404Coordinate proof(s), 287–290, 298 applying variable coordinates, 289 defi ned, 288 placing fi gure in coordinate plane,

288 writing, 287, 288, 290Coordinate Rule for Dilations, 213Coordinate Rules for Refl ections, 187Coordinate Rules for Rotations

about the Origin, 195Coplanar circles, 532, 535Coplanar points, 4Corollaries, See also Theorems Congruent Parts of Parallel Lines

Corollary, 378 Corollary to the Base Angles

Theorem (Cor. 5.2), 257 Corollary to the Converse of Base

Angles Theorem (Cor. 5.3), 257

Corollary to the Polygon Interior Angles Theorem (Cor. 7.1),

365

Index A85

Ind

ex

Corollary to the Triangle Sum

Theorem (Cor. 5.1), 239

Rectangle Corollary (Cor 7.3), 392

Rhombus Corollary (Cor 7.2), 392

Square Corollary (Cor 7.4), 392

Corollary to a theorem, defi ned, 239

Corresponding angles in congruent polygons, 244–245

defi ned, 128

Corresponding Angles Converse (Thm. 3.5), 138

constructing parallel lines, 139

Corresponding Angles Theorem (Thm. 3.1), 132


Corresponding lengths, in similar

polygons, 423

Corresponding part(s) defi ned, in congruent polygons,

244–245


Corresponding sides, in congruent

polygons, 244–245

Cosine ratio, 497–501, 524–525

of 45° and 30° angles, 500

of complementary angles, 498

defi ned, 498

fi nding leg lengths, 499

inverse, 506

Counterclockwise rotation, 194

Counterexample, 77

Cross section, 641

Cube cross sections of, 641

diagonal of, 652

Platonic solid, 639

Cuboctahedron, 652

Customary units of length, 2Cylinders defi ned, 647

effect of changing linear dimensions

on surface area, 649

effect of changing radius on volume,

667

oblique, 647

right cylinders

lateral area of, 647–648

surface area of, 647–649, 695

volumes of, 663–667, 696

formula for, 665

DDeductive reasoning, 75–79, 116

compared to inductive reasoning, 78,

79

defi ned, 78

using correct logic, 64

using with laws of logic, 78

Defi ned terms of geometry, 5Defi nitions as biconditional statements, 234

as conditional statement, 68

Degrees converting between radians and, 601

measure of angle, 39

Dependent events, 713–716, 752

comparing to independent events,

716

defi ned, 713, 715

determination of, 713

probability of, 715, 716

Diagonal of cube, 652

Diagonal of polygon, 364

Diagrams identifying postulates from, 85

interpreting, 51, 83

sketching and interpreting, 86

Diameter of circle

chord perpendicular to, 549, 551

defi ned, 534

of sphere, 680, 681, 687, 688

Die roll, 706

Dilation(s), 211–215, 228

comparing triangles after, 421

constructing, 214

coordinate rule for, 213

defi ned, 212

fi nding perimeter and area after, 420

identifying, 212

negative scale factor, 214

performing a composition of, 221

performing, in coordinate plane,

211, 213

scale factor, 212, 419

Dimensions, changing in a solid

cone

dimensions and surface area, 657

height and volume, 675

cylinder

linear dimensions and surface

area, 649

radius and volume, 667

rectangular prism, linear dimensions

and volume, 666

rectangular pyramid, linear dimen-

sions and volume, 674

sphere, radius and volume, 683

Dimensions of two-dimensional fi gures, 623–626

changing non-proportionally,

624–625

changing proportionally, 623, 626

Directed line segment constructing point along, 451

defi ned, 157

partitioning, 157

Disjoint events, 732, 753

Distance between points defi ned, 12

fi nding minimum distance, 189

on a sphere, 687, 689

using circumference and arc length

to fi nd, 600

Distance Formula, 14

using, 233

Distance from a point to a line defi ned, 148

fi nding, 164


Division Property of Equality, 92

Dodecagon, 367

Dodecahedron, 639

Dynamic geometry software basic drawings of lines, segments,

and rays, 3

calculating sine and cosine ratios,

497

calculating tangent ratios, 491

constructing chords, 549

drawing perpendicular bisector, 304

drawing triangles, 249

side lengths and angle measures,

176

EEarth’s circumference, 603

Edge of polyhedron, 639, 640

Endpoints, 5Enlargement, 212

Equations of circles, writing and graphing,

580–581

of lines

writing in point-slope form, 123

writing in slope-intercept form,

123

of parallel and perpendicular lines,

161–164, 170

distance from point to line, 164

writing, 161, 162–163

of perpendicular line, 303

solving with variables on both sides,

233

writing for perpendicular bisectors,

309

Equiangular polygon, 365

Equiangular triangle, 236, 257

Equidistant (point), 306

A86 Index

Equidistant Chords Theorem (Thm. 10.9), 552

Equilateral polygon, 365

Equilateral triangle classifying, 236

constructing, 258

and equiangular triangle, 257

line symmetry, 197

using, 258

Equivalent statements, 67

Eratosthenes, 603

Euclidean geometry, compared with

spherical geometry, 688–689

Event(s) compound, 732–733

defi ned, 706

probability of complement of,

707–708

Experimental probability, 709, 713

Exterior of the angle, 38

Exterior Angle Inequality Theorem, 346

Exterior Angle Theorem (Thm. 5.2), 238

Exterior angles, 237, See also

Alternate exterior angles

measures of polygons, 363, 366–367

External segment, 575

External Tangent Congruence Theorem (Thm. 10.2), 536

FFaces of polyhedron, 639, 640

Factorials, 738

Favorable outcomes, 707

Flawed reasoning, 64

Flow proof, 106

Flowchart proof concept summary of, 110

defi ned, 106

matching reasons in, 105

Formulas arc length, 599

circle

area of, 605–606, 637

circumference of, 598

combinations, 740

cone

lateral area of, 655

surface area of, 655

volume of, 673

cylinder

lateral area, 647

surface area, 647

volume, 665

Distance Formula, 14

kite area, 616

permutations, 739

population density, 607

prism

lateral area, 646

surface area, 646

volume, 664

pyramid



volume, 672

Pythagorean Theorem (Thm. 9.1),

468

rectangle, area of, 637

regular polygon, area, 617

rhombus area, 616

sectors, area, 605, 608–609

semicircle, area of, 637

sphere


volume of, 682

spherical cap, volume, 686

spherical triangles, area, 690

triangle, area of, 637

Frequency(ies), probability and

two-way tables, 722–724

Fundamental Counting Principle, 738

GGeometric mean compared to arithmetic mean, 481,

488

defi ned, 481, 484

using, 484–485

Geometric Mean (Altitude) Theorem (Thm. 9.7), 484

Geometric Mean (Leg) Theorem (Thm. 9.8), 484

Geometric probability, 708

Geometric relationships, proving,

105–110, 118

Glide refl ection(s), 188

Golden ratio, 430

Graph theory, 280

Graphing calculator combinations, 740

permutations, 739

Graphing a circle, 581

Great circle, 680

HHeads and tails, 706

Hexagon, area of, 619

Hinge Theorem (Thm. 6.12), 348

converse of, 348

using, 349

Histograms analyzing, 745

making, 703

Horizontal component, 178

Horizontal lines, 158

Horizontal stretch, and nonrigid

transformation, 215

Hypotenuse-Leg (HL) CongruenceTheorem (Thm. 5.9), 268–269, 277

Hypotenuse of right triangle, 268

Hypothesis, in conditional statement,

66

IIcosahedron, 639

If-then form, of conditional statement,

66

Image, 178

Incenter Theorem (Thm. 6.6), 317

Incenter of triangle defi ned, 317


triangles, 327


using, 317

Independent events, 713–716, 752

comparing to dependent events, 716

defi ned, 713, 714

determination of, 713, 714

probability of, 714, 715

Indirect measurement of river, 281, 283

using geometric mean of right

triangle, 485

Indirect proof defi ned, 340

used in Triangle Larger Angle


writing, 340, 356

Indirect reasoning, 340

Inductive reasoning, 75–79, 116

compared to deductive reasoning,

78, 79

defi ned, 76

using with conjecture, 76–77

Inferring the truth, 64

Initial point, of vector, 178

Inscribed angle(s) defi ned, 557, 558

fi nding measure of angle, 559

fi nding measure of intercepted arc,

559

Inscribed Angles of a Circle Theorem (Thm. 10.11), 559

Index A87

Ind

ex

Inscribed polygon(s) constructing square inscribed in

circle, 561

defi ned, 557, 560

Inscribed Quadrilateral Theorem (Thm. 10.13), 560

Inscribed Right Triangle Theorem (Thm. 10.12), 560

Intercepted arc, 557, 558

Interior of the angle, 38

Interior angle measures of polygons, 363–366

fi nding interior angle measure, 365

number of sides of polygon, 365

sum of angle measures, 364

Interior angles, 237, See also

Alternate interior angles

Intersecting lines and circles, 566

example of, 124, 125

refl ections, 207

Intersection defi ned, 6

of lines and planes, 3, 6

Intersection of events, 732–733

Inverse defi ned, of conditional statement, 67

truth table for, 70

Inverse cosine, 506

Inverse operations, 92

Inverse sine, 506

Inverse tangent, 506

Inverse of the Triangle Proportionality Theorem, 451

Inverse trigonometric ratios, 506

Isometry, 180

Isomorphic polygons, 280

Isosceles right triangle fi nding sine and cosine of 30°, 500

side lengths, 475, 476

in standard position, 466

Isosceles trapezoid defi ned, 402

using properties of, 403

Isosceles Trapezoid Base Angles Converse (Thm. 7.15), 403

Isosceles Trapezoid Base Angles Theorem (Thm. 7.14), 403

Isosceles Trapezoid Diagonals Theorem (Thm. 7.16), 403

Isosceles triangle(s), 255–259, 295,

See also Isosceles right

triangle

classifying, 236

median and altitude of, 327

using, 258–259

using Base Angles Theorem,

256–257

writing conjecture about, 255

JJoint frequency, 722

Joint relative frequency, 723–724

KKite(s) area of, 616, 619

defi ned, 405

fi nding angle measures in, 405, 414

Kite Diagonals Theorem (Thm. 7.18), 405

Kite Opposite Angles Theorem (Thm. 7.19), 405

LLateral area defi ned, of polyhedron, 645, 646

of regular pyramid, 654

of right cone, 655

of right cylinder, 647–648

of right prism, 646–647

Lateral edges of prisms, 646

of pyramid, 654

Lateral faces of prisms, 646

Lateral surface of a cone, 655

Law of Cosines, 511–512, 515–516,

526

defi ned, 515

solving triangles

with SAS case, 515

with SSS case, 516

Law of Cosines (Thm. 9.10), 515

Law of Detachment, 78

Law of Sines, 511–514, 526

ambiguous case of, 519

areas of triangles, 512

defi ned, 513

solving triangles

with AAS case, 514

with ASA case, 514

with SSA case, 513

Law of Sines (Thm. 9.9), 513

Law of Syllogism, 78

Laws of Logic, 78

Legs of isosceles triangle, 256

of right triangle

defi ned, 268

fi nding, with sine and cosine

ratios, 499

fi nding, with tangent ratio, 493

of trapezoid, 402

Likelihoods, and probabilities, 704,

706

Line(s) in coordinate plane, characteristics

of, 124

fi nding distance to a point, 148, 164

intersecting circles, 566

intersections with planes, 3

Line Intersection Postulate

(Post. 2.3), 84, 85

Line-Point Postulate (Post. 2.2), 84

pairs of

classifying, 124, 125

identifying parallel and

perpendicular lines, 126–127

Plane Intersection Postulate

(Post. 2.7), 84, 85

Plane-Line Postulate (Post. 2.6), 84,

85

that intersect circles, 533–537, 586

Two Point Postulate (Post. 2.1), 84

undefi ned term, and naming, 4

writing equations of lines

in point-slope form, 123

in slope-intercept form, 123

Line Intersection Postulate (Post. 2.3), 84, 85

Line perpendicular to plane, 86


Line of refl ection, 186

Line segment(s), See also Segment(s)

defi ned, 5

directed, partitioning, 157

Line symmetry, 189

Line of symmetry, 189

Linear pair (of angles), 50

Linear Pair Perpendicular Theorem (Thm. 3.10), 150

Linear Pair Postulate (Post. 2.8), 108,

133

Lines Perpendicular to a Transversal Theorem (Thm. 3.12), 150

Literal equations, rewriting, 63

Logic, deductive reasoning and fl awed

reasoning, 64

Logically equivalent statements, 70

MMajor arc, 542

Marginal frequency, 722

Marginal relative frequency, 723

Measure of an angle, 39

Measure of an Inscribed Angle Theorem (Thm. 10.10), 558

Measure of a major arc, 542

Measure of a minor arc, 542

A88 Index

Measurement, indirect, 281, 283

Median of trapezoid, 404

Median of triangle, 323–325, 327, 355

defi ned, 324


triangles, 304, 327

Metric units of length, 2Midpoint(s), 19–23, 57

defi ned, 20

of line segment, fi nding, 19, 22

and segment bisectors, 20–21

of a segment in a coordinate plane,

fi nding, 23

Midpoint Formula, using, 233

Midpoint of a segment in a coordinate plane, 23

Midsegment of a trapezoid, 404

Midsegment triangle, 334

Midsegment of a triangle, 333–336,

355

defi ned, 334

examples of segments in triangles,

304

using in coordinate plane, 334

using Triangle Midsegment

Theorem (Thm. 6.8), 335–336

Minor arc, 541, 542

Modeling with Mathematics, Throughout. See for example:

basics of geometry

area of shed fl oor, 33

planes in sulfur hexafl uoride, 7

circles, Northern Lights, 569

probabilities and likelihoods, 704

reasoning and proofs, city street, 95

right triangles, angle measures, 239

right triangles and trigonometry

angle of elevation and height of

tree, 494

equilateral triangle road sign, 478

similarity, swimming pool, 424

transformations, golf website, 181

triangles, neighborhood distances,

336

volumes of prisms, rectangular

chest, 666

Multiplication Property of Equality, 92

Multi-step equations, using structure

to solve, 361

Mutually exclusive events, 732

Nn factorial, 738

n-gon, 30

area of, 617

Negation, of conditional statement, 66

Negative scale factor, 214Negative slope, 156Nets for three-dimensional fi gures,

638 defi ned, for a polyhedron, 646 for a pyramid, 638Nonagon, area of, 618Nonrigid transformation, 215nth term, of arithmetic sequence, 63

Number line, partitioning a segment

on, 22

OOblique cone, 655Oblique cylinder, 647Oblique prism, 646Obtuse angle defi ned, 39 trigonometric ratios for, 512Obtuse triangle in circumscribed circle, 315 classifying by angles, 236 classifying by Pythagorean

inequalities, 471 orthocenter of, 326Octagon, area of, 618Octahedron, 639Opposite rays, 5Opposite Sides Parallel and

Congruent Theorem (Thm. 7.9), 382, 383

Opposite of statement, See Negation“or” (union), 732–733Orthocenter of triangle defi ned, 325 examples of segments and points

in triangles, 327 fi nding, 326 type of triangle, and location, 326Outcomes defi ned, 706 favorable, 707Overlapping events defi ned, 732

fi nding probability of, 733, 753

PPairs of angles, See Angle(s), pairs ofPairs of lines, See Lines, pairs ofParagraph proof concept summary of, 110 defi ned, 108Parallel lines constructing, 139 defi ned, and identifying, 126–127

example of, 124, 125

identifying, slopes of, 158

proofs with, 137–141, 169


Corresponding Angles Converse


proving Alternate Interior Angles

Converse, 140

Transitive Property of Parallel

Lines (Thm. 3.9), 141

properties of, 132–134

Alternate Exterior Angles


Alternate Interior Angles


Consecutive Interior Angles


Corresponding Angles Theorem

(Thm. 3.1), 132

proportionality with three lines, 452

proving theorems about, 140

Refl ections in Parallel Lines


and transversals, 131–134, 168

writing equations of, 162, 170

Parallel and perpendicular lines equations of, 161–164, 170

identifying, 361

pairs of lines and angles, 125–128,

168

parallel lines and transversals,

131–134, 168

proofs with parallel lines, 137–141,

169

proofs with perpendicular lines,

147–151, 169

slopes of lines, 155–158, 170

Parallel planes, 126

Parallel Postulate (Post. 3.1), 127, 689

Parallelogram(s) area of, 596, 619

in coordinate plane, 375, 384, 396

defi ned, 372

diagonal lengths of, 382

effects of changing dimensions, 626

identifying and verifying, 380–383

properties of, 371–375, 412

properties of diagonals, 394–395

properties of special parallelograms,

391–396, 413–414

rotational symmetry, 197

side lengths of, 381

ways to prove quadrilateral is

parallelogram, 383

writing two-column proof, 374

Parallelogram Consecutive Angles Theorem (Thm. 7.5), 373

Parallelogram Diagonals Converse (Thm. 7.10), 382, 383

Index A89

Ind

ex

Parallelogram Diagonals Theorem (Thm. 7.6), 373

Parallelogram Opposite Angles Converse (Thm. 7.8), 380,

383

Parallelogram Opposite Angles Theorem (Thm. 7.4), 372

Parallelogram Opposite Sides Converse (Thm. 7.7), 380,

383

Parallelogram Opposite Sides Theorem (Thm. 7.3), 372

Partitioning a directed line segment, 157

Patterns in dilation, 420

using inductive reasoning to

describe a visual pattern, 76

Pentagonal prism, 640

Percent, fi nding, 703

Performance Tasks Bicycle Renting Stations, 353

Window Design, 629

Circular Motion, 585

Comfortable Horse Stalls, 55

Creating the Logo, 293

Induction and the Next Dimension,

115

Judging the Math Fair, 457

The Magic of Optics, 225

Navajo Rugs, 167

A New Dartboard, 751

Scissor Lifts, 411

Triathlon, 521

Water Park Renovation, 693

Perimeter in coordinate plane, 29–33, 57

fi nding, 31


623–626, 632

fi nding after dilation, 420


of triangle, square, and rectangle, 31

Perimeters of Similar Polygons (Thm. 8.1), 424

Permutation(s), 739, 754

counting, 738

defi ned, 738

fi nding probability using, 739

formulas, 739

Perpendicular bisector(s) constructing, 149

drawing, 304


triangles, 304, 327

points on, 305

using, 306–307

using chords of circles, 550–551

writing equations for, 309

Perpendicular Bisector Theorem (Thm. 6.1), 306

converse of, 306

Perpendicular Chord Bisector Converse (Thm. 10.8), 550

Perpendicular Chord Bisector Theorem (Thm. 10.7), 550

Perpendicular lines defi ned, 68

equation of, 303

example of, 124

identifying, 127, 361

slopes of, 158

proofs with, 147–151, 169

constructing perpendicular lines,

149

distance from point to line, 148,

164

proving theorems about

perpendicular lines, 150

writing equations of, 163, 170

Perpendicular Postulate (Post. 3.2), 127

Perpendicular Transversal Theorem (Thm. 3.11), 150

Plane(s) intersections with lines, 3

parallel, 126


(Post. 2.7), 84, 85

Plane-Line Postulate (Post. 2.6),

84, 85

Plane-Point Postulate (Post. 2.5),

84, 85

Three Point Postulate (Post. 2.4), 84


Plane Intersection Postulate (Post. 2.7), 84, 85

Plane-Line Postulate (Post. 2.6), 84, 85

Plane-Point Postulate (Post. 2.5), 84, 85

Platonic solids, 639

Point(s) fi nding distance to a line, 148, 164


(Post. 2.3), 84, 85


Plane-Line Postulate (Post. 2.6),

84, 85

Plane-Point Postulate (Post. 2.5),

84, 85




Point of concurrency defi ned, 314


triangles, 327

Point-slope form, writing equations

of lines in, 123

Point of tangency, 534

Polar coordinate system, 200

Polygon(s) angle measures in, 617

angles of, 363–367, 412

exterior angle measures of, 363,

366–367

interior angle measures of,

363–366

area of, 29, 615–619, 631

classifying types of, 30, 365

congruent, 243–246, 294

using corresponding parts,

244–245

using Third Angles Theorem

(Thm. 5.4), 246

convex compared to concave, 30

drawing regular, 37

inscribed, 557, 560–561, 588

similar, 421–426 (See also Similar

polygons)

Polygon Exterior Angles Theorem (Thm. 7.2), 366

Polygon Interior Angles Theorem (Thm. 7.1), 364

Polyhedron defi ned, 639, 640

lateral area, 645

surface area, 645

Population density, 607

Positive slope, 156

Postulate, defi ned, 12, 234

Postulates Angle Addition Postulate (Post. 1.4),

41

Arc Addition Postulate (Post. 10.1),

543


(Post. 2.3), 84


Linear Pair Postulate (Post. 2.8), 108

Parallel Postulate (Post. 3.1), 127

Perpendicular Postulate (Post. 3.2),

127


(Post. 2.7), 84

Plane-Line Postulate (Post. 2.6), 84

Plane-Point Postulate (Post. 2.5), 84

Protractor Postulate (Post. 1.3), 39

Refl ection Postulate (Post. 4.2), 188

Rotation Postulate (Post. 4.3), 196

A90 Index

Ruler Postulate (Post. 1.1), 12

Segment Addition Postulate

(Post. 1.2), 13


Translation Postulate (Post. 4.1),

180


Volume Addition Postulate, 669

Postulates and diagrams, 83–86, 117

diagrams, sketching and

interpreting, 83, 86

identifying postulates from a

diagram, 85

point, line, and plane postulates, 84

Precise Mathematical Language exactly two answers, 707

probabilities, 748

rounding trigonometric ratios and

lengths, 492

on statement of problem, 318

use π key on calculator, 598

Preimage, 178

Prime notation, 178

Prisms defi ned, 646

oblique, 646

pentagonal, 640, 647

right pentagonal prism, 647

right prism, 646

lateral areas of, 646–647

surface areas of, 646–647, 694

triangular prism, 672

volumes of, 663–667, 696

effect of changing linear

dimensions, 666

formula for, 664

Probability binomial distributions, 745–748, 754

of complements of events, 707–708

disjoint and overlapping events,

731–734, 753

experimental, 709, 713

frequencies, 722–724

geometric, 708

independent and dependent events,

713–717, 752

permutations and combinations, 754

sample spaces and, 705–709, 752

theoretical, 706–708

two-way tables, 721–725, 753

Probability distribution(s) constructing, 746

defi ned, 746

interpreting, 747

Probability experiment, 706

Probability of complement of event, 707–708

Probability of compound events, 732–733

Probability of dependent events, 715–716

Probability of an event defi ned, 704, 706

and likelihoods, 704, 706

Probability of independent events, 714–715

Problem-solving strategies fi nding probabilities of events, 715

inductive reasoning and deductive

reasoning, 79

Problems, solving simpler form of, 596

Proof(s), See also Reasoning and

proofs

with congruent triangles, that

triangles are congruent, 246

defi ned, 99, 100

with parallel lines, 137–141, 169


Corresponding Angles Converse

(Thm. 3.5), 138

proving Alternate Interior Angles

Converse (Thm. 3.6), 140

proving theorems about parallel

lines, 134

Transitive Property of Parallel

Lines (Thm. 3.9), 141

with perpendicular lines, 147–151,

169

constructing perpendicular lines,

149

distance from point to line, 148,

164

proving theorems about


proving a construction, 284

proving statements about segments

and angles, 99–102, 118

fl owchart proof, 106, 110

paragraph proof, 108, 110

two-column proofs, 100, 102,

110

using properties of congruence,

101

types of, 110

writing coordinate proofs involving

circles, 582

Proof by Contradiction, 340

Proofs of theorems Angle-Angle-Side (AAS)

Congruence Theorem

(Thm. 5.11), 275

Angle-Angle Similarity Theorem

(Thm. 8.3), 432

Angle-Side-Angle (ASA)

Congruence Theorem

(Thm. 5.10), 274

Base Angles Theorem (Thm. 5.6),

256

Circumcenter Theorem (Thm. 6.5),

314

Converse of the Hinge Theorem

(Thm. 6.13), 349

Kite Diagonals Theorem

(Thm. 7.18), 405

Parallelogram Diagonals Theorem

(Thm. 7.5), 374

Parallelogram Opposite Sides




Perpendicular Bisector Theorem

(Thm. 6.1), 306

Perpendicular Transversal Theorem

(Thm. 3.11), 150

Rhombus Diagonals Theorem

(Thm. 7.11), 394

Side-Angle-Side (SAS) Congruence


Side-Side-Side (SSS) Congruence


Side-Side-Side (SSS) Similarity


Similar Circles Theorem

(Thm. 10.5), 545

Slopes of Parallel Lines Theorem

(Thm. 3.13), 443

Slopes of Perpendicular Lines


Symmetric Property of Angle

Congruence (Thm. 2.2),

102, 110

Symmetric Property of Segment

Congruence (Thm. 2.1), 101

Triangle Larger Angle Theorem

(Thm. 6.10), 341

Triangle Midsegment Theorem

(Thm. 6.8), 335

Triangle Sum Theorem (Thm. 5.1),

238

Properties Addition Property of Equality, 92

Algebraic Properties of Equality, 92

of congruence, 101–102


Division Property of Equality, 92

Multiplication Property of Equality,

92

of parallel lines, 132–134

Refl exive Property, 94

Substitution Property of Equality, 92

Index A91

Ind

ex

Subtraction Property of Equality, 92

Symmetric Property, 94

Transitive Property, 94

Properties of Angle Congruence (Thm. 2.2), 101

Properties of Segment Congruence (Thm. 2.1), 101

Properties of Triangle Congruence (Thm. 5.3), 245

Proportionality, 449–453, 460

effects of changing dimensions of a

fi gure, 623, 626

fi nding relationships, 449

of three parallel lines, 452

with triangle angle bisector, 453

of triangles, 450–451

Proportions ratios forming, 419

solving, 465

Protractor Postulate (Post. 1.3), 39

Pyramids defi ned, regular pyramid, 653, 654

net for, 638

regular hexagonal pyramid, 654

surface areas of, 653–655, 657, 695

triangular, 640

triangular pyramid, 672

volumes of, 671–672, 674–675,

696–697


dimensions, 674

formula for, 672

Pythagorean Inequalities Theorem (Thm. 9.3), 471

Pythagorean Theorem, use of,

467–471, 522

classifying triangles as acute or

obtuse, 471

common triples and multiples, 468

converse of, 470

in Distance Formula, 14

proving without words, 467

right triangles, verifying, 470

using, 468–469

Pythagorean Theorem (Thm. 9.1), 468

Pythagorean triple, 468

QQuadratic equations, solving by

completing the square, 531

Quadrilateral area of, 29

classifi cations of, 362, 393, 406

identifying special, 406

with inscribed angles, 557

summary of ways to prove

parallelogram, 383

Quadrilaterals and other polygons angles of polygons, 363–367, 412

properties of parallelograms,

371–375, 412

properties of special parallelograms,

391–396, 413–414

properties of trapezoids and kites,

401–406, 414

proving quadrilateral is a

parallelogram, 379–384, 413

RRadians, measuring angles in, 601

Radicals, using properties of, 465

Radius of arc, 40

of circle

defi ned, 534

fi nding with congruent chords,

552

fi nding with segments, 576

fi nding with tangent, 536

of cone, 655

of cylinder, 647

of regular polygon, 617

of sphere, 680

Random variable, 746

Ratios, forming a proportion, 419

Rays, and naming, 5

Reading abbreviations: sin, cos, hyp., 498

abbreviations: tan, opp., adj., 492

approximately equal to, 14

biconditionals, 394

bisect, 20

bisector of circular arc, 550

center of circle circumscribed about

polygon, 617

circles, radius and diameter, 534

circum- prefi x, 315

compound inequality, 343

congruent angles, 40

congruent segments, 15

contradiction, 340

corresponding lengths, 423

dilation scale factor, 212

equilateral and equiangular

triangles, 257

inverse tangent, 506

negative reciprocals, 158

parallelogram notation, 205

raked stage, 508

rely on marked information, 406

right angle and right triangle, 14

scale factors, 215

statement of proportionality, 422

trapezoid midsegment, 404

triangle altitudes, 326

triangle area formula, 325

triangle classifi cations, 236

triangle notation, 31

two-way table, 722

Real-life problems, Throughout. See for example:

basics of geometry

angles in ball-return net, 49

planes in sulfur hexafl uoride, 7

circles, graphs of, earthquake and

seismograph, 582

circumference and distance traveled,

600

congruent triangles

bench with diagonal support, 267

sign on barn, 252

lateral area

of cylindrical soup can, 648

of traffi c cone, 656

parallel and perpendicular lines

in neighborhood layout, 151

sunlight angles, 134

probability

adults with pets, 709

diagnostic test for diabetes, 734

reasoning and proofs, percent raise,

93

relationships within triangles

biking, 350

bridge, 307

circumcenter or incenter for

lamppost placement, 318

distance in city, 315

soccer goal, 309

right triangles and trigonometry

angle of depression and skiing

on mountain, 501

angle of elevation and height of

tree, 494

equilateral triangle road sign, 478

roof height, 483

skyscrapers and support beams,

469

solving right triangles and raked

stage, 508

step angle of dinosaurs, 516

similarity

height of fl agpole, 434

triangles and shoe rack, 451

transformations

fi nding minimum distance, 189

golf website, 181

scale factor and length of image,

215

A92 Index

Reasoning, on similar triangles, 433Reasoning and proofs algebraic reasoning, 91–95 conditional statements, 65–70, 116 inductive and deductive reasoning,

75–79, 116 postulates and diagrams, 83–86, 117 proving geometric relationships,

105–110, 118 proving statements about segments

and angles, 99–102, 118Rectangle Ailles, 480 area of, 596, 637 defi ned, 392 diagonal lengths in, 395 effects of changing dimensions,

623–625 perimeter and area, 31Rectangle Corollary (Cor 7.3), 392Rectangle Diagonals Theorem

(Thm. 7.13), 395Rectangular pyramid, 674Reduction, 212Refl ection(s), 185–189, 226 coordinate rules for, 187 defi ned, 186 glide refl ections, 188 in horizontal and vertical lines, 186 in line y = x or y = −x, 187 performing, 186–187 triangle in coordinate plane, 185 triangle using refl ective device, 185Refl ection Postulate (Post. 4.2), 188Refl ections in Intersecting Lines

Theorem (Thm. 4.3), 207Refl ections in Parallel Lines Theorem

(Thm. 4.2), 206Refl exive Property, 94 triangle congruence, 245Regular hexagonal pyramid, 654Regular octagon, rotational symmetry,

197Regular polygon angle measures in, 617 area of, 615, 617–618, 631 defi ned, 365Regular pyramid defi ned, 653, 654 lateral areas of, 654 surface areas of, 653, 654Related conditional statements, 67Relationships among special

parallelograms, 393Relationships within triangles bisectors of triangles, 313–318, 354

indirect proof and inequalities in one

triangle, 339–343, 356

inequalities in two triangles,

347–350, 356

medians and altitudes of triangles,

323–327, 355

perpendicular and angle bisectors,

305–309, 354

triangle midsegments, 333–336, 355

Relative frequencies fi nding conditional, 723–724

fi nding joint and marginal, 723–724

Remember complete the square, 581

convex polygon, 364

distance between points, 148

dodecagon, 367

Fundamental Counting Principle,

738

inverse operations, 92

order of operations, 93

perimeter and area in coordinate

plane, 31


polygon in coordinate plane, 375

radical in simplest form, 476

slope-intercept form, 162

slopes, product of, 187

system of linear equations in two

variables, 164

triangle side lengths, 471, 477

Revolution, solids of, 642

Rhombus angle measures in, 394

area of, 616

defi ned, 392

Rhombus Corollary (Cor 7.2), 392

Rhombus Diagonals Theorem (Thm. 7.11), 394

Rhombus Opposite Angles Theorem (Thm. 7.12), 394

Right angle, 39

Right Angles Congruence Theorem (Thm. 2.3), 106

Right cone defi ned, 655


surface area of, 653, 655–657

Right cylinders defi ned, 647

lateral area of, 647–648

surface area of, 647–649


dimensions, 649

Right pentagonal prism, 647

Right prism defi ned, 646

lateral areas of, 646–647


Right Triangle Similarity Theorem (Thm. 9.6), 482

Right triangles in circumscribed circle, 315

classifying, 236

orthocenter of, 326

similar, 481–485, 523

identifying, 482–483

using geometric mean, 484–485

solving, 505–508, 525

using inverse trigonometric

ratios, 506

special, side lengths of

30°-60°-90° triangle, 475, 477

isosceles (45°-45°-90°), 475, 476

standard position for, 466

verifying with Pythagorean

Theorem, 470

Right triangles and trigonometry cosine ratio, 497–501, 524–525

Law of Cosines, 511–512, 515–516,

526

Law of Sines, 511–514, 526

Pythagorean Theorem, 467–471,

522

similar right triangles, 481–485, 523

sine ratio, 497–501, 524–525

solving right triangles, 505–508, 525

special right triangles, 475–478, 522

tangent ratio, 491–494, 524

Rigid motion defi ned, 180

using in congruent polygons,

243–244

Rotation(s), 193–197, 227

in coordinate plane, 193, 195

coordinate rules for rotations about

the origin, 195

defi ned, 194

direction, clockwise or

counterclockwise, 194

performing, 194–195

performing compositions with, 196

Rotation Postulate (Post. 4.3), 196

Rotational symmetry, 197

Ruler Postulate (Post. 1.1), 12

SSame-Side Interior Angles Theorem,

See Consecutive Interior

Angles Theorem (Thm. 3.4)

Sample space, 752

defi ned, 705, 706

fi nding, 705, 706

Sample space(s), 705–709

SAS, See Side-Angle-Side (SAS)

Index A93

Ind

ex

Scale factor defi ned, 212 of dilation, 419 negative, 214 units in, and fi nding, 215Scalene triangle, classifying, 236Secant, 534Secant segment, 575–576Sector of circle area of, 605, 608–609, 619, 630 defi ned, 605, 608Segment(s) construction, bisecting a segment,

21 defi ned, and naming, 5 fi nding length of, 20 fi nding midpoint of, 19, 22 length in proportional triangles, 450 measuring and constructing, 11–15,

56 congruent segments, 15 Distance Formula, 14 Ruler Postulate (Post. 1.1), 12 Segment Addition Postulate

(Post. 1.2), 13 partitioning a directed line segment,

157 partitioning on a number line, 22 proof of Symmetric Property of

Segment Congruence, 101 Properties of Segment Congruence

(Thm. 2.1), 101 relationships in circles, 573–576,

589 chords, secants, and tangents,

574–576 that intersect circles, 533–537, 586Segment Addition Postulate

(Post. 1.2), 13Segment bisector(s), 20 construction bisecting a segment, 21 of perpendicular bisector, 149 defi ned, 20 and midpoints, 20–21Segments of a chord, 574Segments of Chords Theorem

(Thm. 10.18), 574Segments of Secants and Tangents

Theorem (Thm. 10.20), 576Segments of Secants Theorem

(Thm. 10.19), 575Semicircle, 542 area of, 637Side-Angle-Side (SAS) congruence, 249–252, 277

construction, copying a triangle

using SAS, 252

and properties of shapes, 251 using Law of Cosines to solve

triangle, 515Side-Angle-Side (SAS) Congruence

Theorem (Thm. 5.5), 250Side-Angle-Side (SAS) Similarity

Theorem (Thm. 8.5), 442 triangle similarity theorems

compared, 443Side-Side-Angle (SSA), 268 special case for right triangles,

268–269 using Law of Sines to solve triangle,

513Side-Side-Side (SSS) congruence, 265–269, 277 construction, copying a triangle

using SSS, 268 using, 266–268 using Law of Cosines to solve

triangle, 516Side-Side-Side (SSS) Congruence

Theorem (Thm. 5.8), 266Side-Side-Side (SSS) Similarity

Theorem (Thm. 8.4), 440 proof of, 441 triangle similarity theorems

compared, 443 using, 440–441Sides classifying triangles by, 236–237 defi ned, of an angle, 38 fi nding side lengths in special right

triangles, 475–478, 522 fi nding side lengths with dynamic

geometry software, 176 lengths of, 342, 343 of polygons, 30 relating to angles of triangle, 339,

341–342 using side similarity to prove

triangle similarity, 439–442, 459

Similar arcs, 545Similar Circles Theorem (Thm. 10.5),

545Similar fi gures defi ned, 220 identifying, 175 right triangles (See Triangle

similarity, right triangles) triangles (See Triangle similarity)Similar polygons, 421–426, 458 areas of, 425 comparing triangles after dilation,

421

corresponding lengths, 423

corresponding parts of, 422

determining whether polygons are

similar, 426

perimeters of, 424

Similar right triangles, 481–485, 523



Similarity proportionality theorems, 449–453,

460

proving slope criteria using similar

triangles, 443–444

proving triangle similarity

by AA, 431–434, 458

by SAS, 442, 459

by SSS, 439–441, 459

similar polygons, 421–426, 458

and transformations, 219–222, 228,

422

and dilations, 219

performing composition of

dilations, 221

and rigid motions, 219

Similarity statements, 422

Similarity transformations, 220–222

describing, 222

performing, 220

Sine ratio, 497–501, 524–525

of 45° and 30° angles, 500

of complementary angles, 498

defi ned, 498

fi nding leg lengths, 499

inverse, 506

Sketching diagram, 86

intersections of lines and planes, 6

solids of revolution, 642

Skew lines, 126

Slant height of a regular pyramid, 654

Slant height of a right cone, 655

Slope, defi ned, 155, 156

Slope-intercept form, 162

writing equations of lines in, 123

Slopes of lines in coordinate plane, 156

defi ned, 155, 156

fi nding, 155, 156, 170

identifying parallel and


negative slope, 156

partitioning a directed line segment,

157

positive slope, 156

proving criteria using similar


undefi ned slope, 156

zero slope (slope of 0), 156

A94 Index

Slopes of Parallel Lines Theorem

(Thm. 3.13), 158, 162

proof of, 443

Slopes of Perpendicular Lines Theorem (Thm. 3.14), 158, 163

proof of, 444

Solid of revolution, 642

Solids, See also Composite solids;

Three-dimensional fi gures

changing dimensions in

cone, dimensions and surface

area, 657

cone, height and volume, 675

cylinder, linear dimensions and

surface area, 649

cylinder, radius and volume, 667

rectangular prism, linear

dimensions and volume, 666

rectangular pyramid, linear

dimensions and volume, 674

sphere, radius and volume, 683

cross sections of, 641

Platonic, 639

types of, 640

Solve a right triangle, 507

Spheres defi ned, 680

diameter of, 680, 681

fi nding distances on, 689

lines on, in spherical geometry, 687


formula for, 680

volumes of, 679, 682–683, 697


683

formula for, 682

Spherical cap, 686

Spherical geometry compared with Euclidean geometry,

688–689

fi nding areas of spherical triangles,

690

fi nding distances on a sphere, 689

Spherical triangles, area of, 690

Square defi ned, 392

perimeter and area, 31

Square Corollary (Cor 7.4), 392

SSA, See Side-Side-Angle (SSA)

SSS, See Side-Side-Side (SSS)

Standard equation of a circle, 580–581

Standard position for right triangle, 466

Straight angle, 39

Straightedge, 15

Structure in dilation, 420 to solve multi-step equation, 361Study Skills Analyzing Your Errors: Misleading

Directions, 145 Form a Final Exam Study Group,

661 Form a Weekly Study Group, Set Up

Rules, 489 Keeping a Positive Attitude, 201 Keeping Your Mind Focused, 27 Keeping Your Mind Focused During

Class, 389 Keeping Your Mind Focused While

Completing Homework, 555 Kinesthetic Learners, 613 Making a Mental Cheat Sheet, 729 Rework Your Notes, 331 Take Control of Your Class Time,

437 Using the Features of Your Textbook

to Prepare for Quizzes and Tests, 89

Visual Learners, 263Substitution Property of Equality, 92Subtend, 558Subtraction Property of Equality, 92Success of trial, 709Supplementary angles defi ned, 48–49 proving cases, 107Surface area of composite solid, 649, 656 defi ned, of prisms, 646 of polyhedron, 645 of regular pyramid, 653, 654, 657 of right cone, 653, 655–657 of right cylinder, 647–649 of right prism, 646–647, 649 of spheres, 679–681Surface area and volume of spheres, 679–683, 697 surface areas of prisms and cylinders,

645–649, 694–695 of pyramids and cones, 653–657,

695 volumes of prisms and cylinders, 663–667,

696 of pyramids and cones, 671–675,

696–697Syllogism example of, 64 Law of Syllogism, 78

Symmetric Property, 94

proof of angle congruence, 102, 110

proof of segment congruence, 101

triangle congruence, 245

Symmetry distinguishing between types of, 197

rotational, 197

TTangent(s) constructing to a circle, 537

defi ned, 534

fi nding radius of circle, 536

using properties of, 536–537

Tangent circles defi ned, 532, 535

drawing and identifying common

tangents, 535

Tangent and Intersected Chord Theorem (Thm. 10.14), 566

Tangent Line to Circle Theorem (Thm. 10.1), 536

Tangent ratio, 491–494, 524

calculating, 491

defi ned, 492

fi nding, 492–493

inverse, 506

Tangent segment, 575–576

Terminal point, of vector, 178

Tessellations, 209, 210

Tetrahedron, 639

Theorem, defi ned, 101, 234, See also

Corollaries; Postulates

Theorems 30°-60°-90° Triangle Theorem

(Thm. 9.5), 477

45°-45°-90° Triangle Theorem

(Thm. 9.4), 476

Alternate Exterior Angles Converse

(Thm. 3.7), 139

Alternate Exterior Angles Theorem

(Thm. 3.3), 132

Alternate Interior Angles Converse

(Thm. 3.6), 139

Alternate Interior Angles Theorem

(Thm. 3.2), 132

Angle-Angle-Side (AAS)

Congruence Theorem

(Thm. 5.11), 275

Angle-Angle (AA) Similarity


Angle Bisector Theorem (Thm. 6.3),

308

Angle-Side-Angle (ASA)

Congruence Theorem

(Thm. 5.10), 274

Angles Inside the Circle Theorem

(Thm. 10.15), 567

Index A95

Ind

ex

Angles Outside the Circle Theorem (Thm. 10.16), 567

Areas of Similar Polygons (Thm. 8.2), 425

Base Angles Theorem (Thm. 5.6), 256

Centroid Theorem (Thm. 6.7), 324 Ceva’s Theorem, 456 Circumcenter Theorem (Thm. 6.5),

314 Circumscribed Angle Theorem

(Thm. 10.17), 568 Composition Theorem (Thm. 4.1),

180 Congruent Central Angles Theorem

(Thm. 10.4), 544 Congruent Circles Theorem

(Thm. 10.3), 544 Congruent Complements Theorem

(Thm. 2.5), 107 Congruent Corresponding Chords

Theorem (Thm. 10.6), 550 Congruent Supplements Theorem

(Thm. 2.4), 107 Consecutive Interior Angles

Converse (Thm. 3.8), 139 Consecutive Interior Angles

Theorem (Thm. 3.4), 132 Contrapositive of the Triangle

Proportionality Theorem, 451 Converse of the Angle Bisector

Theorem (Thm. 6.4), 308 Converse of the Base Angles

Theorem (Thm. 5.7), 256 Converse of the Hinge Theorem

(Thm. 6.13), 348 Converse of the Perpendicular

Bisector Theorem (Thm. 6.2), 306

Converse of the Pythagorean Theorem (Thm. 9.2), 470

Converse of the Triangle Proportionality Theorem (Thm. 8.7), 450

Corresponding Angles Converse (Thm. 3.5), 138

Corresponding Angles Theorem (Thm. 3.1), 132

Equidistant Chords Theorem (Thm. 10.9), 552

Exterior Angle Inequality Theorem, 346

Exterior Angle Theorem (Thm. 5.2), 238

External Tangent Congruence Theorem (Thm. 10.2), 536

Geometric Mean (Altitude) Theorem

(Thm. 9.7), 484

Geometric Mean (Leg) Theorem

(Thm. 9.8), 484

Hinge Theorem (Thm. 6.12), 348

Hypotenuse-Leg (HL) Congruence


Incenter Theorem (Thm. 6.6), 317

Inscribed Angles of a Circle


Inscribed Quadrilateral Theorem

(Thm. 10.13), 560

Inscribed Right Triangle Theorem

(Thm. 10.12), 560

Inverse of the Triangle

Proportionality Theorem, 451

Isosceles Trapezoid Base Angles


Isosceles Trapezoid Base Angles


Isosceles Trapezoid Diagonals


Kite Diagonals Theorem

(Thm. 7.18), 405

Kite Opposite Angles Theorem

(Thm. 7.19), 405

Law of Cosines (Thm. 9.10), 515

Law of Sines (Thm. 9.9), 513

Linear Pair Perpendicular Theorem

(Thm. 3.10), 150

Lines Perpendicular to a Transversal


Measure of an Inscribed Angle


Opposite Sides Parallel and

Congruent Theorem

(Thm. 7.9), 382

Parallelogram Consecutive Angles


Parallelogram Diagonals Converse

(Thm. 7.10), 382

Parallelogram Diagonals Theorem

(Thm. 7.6), 373

Parallelogram Opposite Angles


Parallelogram Opposite Angles






Perimeters of Similar Polygons

(Thm. 8.1), 424

Perpendicular Bisector Theorem

(Thm. 6.1), 306

Perpendicular Chord Bisector


Perpendicular Chord Bisector


Perpendicular Transversal Theorem (Thm. 3.11), 150

Polygon Exterior Angles Theorem (Thm. 7.2), 366

Polygon Interior Angles Theorem (Thm. 7.1), 364

Properties of Angle Congruence (Thm. 2.2), 101

Properties of Segment Congruence (Thm. 2.1), 101

Properties of Triangle Congruence (Thm. 5.3), 245

Pythagorean Inequalities Theorem (Thm. 9.3), 471

Pythagorean Theorem (Thm. 9.1), 468

Rectangle Diagonals Theorem (Thm. 7.13), 395

Refl ections in Intersecting Lines Theorem (Thm. 4.3), 207

Refl ections in Parallel Lines Theorem (Thm. 4.2), 206

Rhombus Diagonals Theorem (Thm. 7.11), 394

Rhombus Opposite Angles Theorem (Thm. 7.12), 394

Right Angles Congruence Theorem (Thm. 2.3), 106

Right Triangle Similarity Theorem (Thm. 9.6), 482

Segments of Chords Theorem (Thm. 10.18), 574

Segments of Secants and Tangents Theorem (Thm. 10.20), 576

Segments of Secants Theorem (Thm. 10.19), 575

Side-Angle-Side (SAS) Congruence Theorem (Thm. 5.5), 250

Side-Angle-Side (SAS) Similarity Theorem (Thm. 8.5), 442

Side-Side-Side (SSS) Congruence Theorem (Thm. 5.8), 266

Side-Side-Side (SSS) Similarity Theorem (Thm. 8.4), 440

Similar Circles Theorem (Thm. 10.5), 545

Slopes of Parallel Lines Theorem (Thm. 3.13), 158

Slopes of Perpendicular Lines Theorem (Thm. 3.14), 158

Tangent and Intersected Chord Theorem (Thm. 10.14), 566

Tangent Line to Circle Theorem (Thm. 10.1), 536

Third Angles Theorem (Thm. 5.4), 246

Three Parallel Lines Theorem

(Thm. 8.8), 452

A96 Index

Transitive Property of Parallel Lines

(Thm. 3.9), 141

Trapezoid Midsegment Theorem

(Thm. 7.17), 404

Triangle Angle Bisector Theorem

(Thm. 8.9), 453

Triangle Inequality Theorem

(Thm. 6.11), 343

Triangle Larger Angle Theorem

(Thm. 6.10), 341

Triangle Longer Side Theorem

(Thm. 6.9), 341

Triangle Midsegment Theorem

(Thm. 6.8), 335

Triangle Proportionality Theorem

(Thm. 8.6), 450

Triangle Sum Theorem (Thm. 5.1),

237–238

Vertical Angles Congruence

Theorem (Thm. 2.6), 108–110

Theoretical probability, 706–708

defi ned, 707

fi nding, 707, 713

Third Angles Theorem (Thm. 5.4), 246

Three-dimensional fi gures, 639–642,

694, See also Solids

classifying solids, 640

cross sections, 641

Platonic solids, 639

solids of revolution, 642

Three-dimensional solids, nets for,

638

Three Parallel Lines Theorem (Thm. 8.8), 452

Three Point Postulate (Post 2.4), 84

Tools, See Dynamic geometry software

Transformation(s) congruence and, 203–207, 227

defi ned, 178

dilations, 211–215, 228

identifying, 175

refl ections, 185–189, 226

rotations, 193–197, 227

similarity and, 219–222, 228, 422

translations, 177–181, 226

Transitive Property, 94

triangle congruence, 245

Transitive Property of Parallel Lines (Thm. 3.9), 141

Translation(s), 177–181, 226

defi ned, 178

of fi gure in coordinate plane, 179

of fi gure using vector, 179

performing compositions, 180

performing translations, 178–179

of triangle in coordinate plane, 177

Translation Postulate (Post. 4.1), 180

Transversal(s) angles formed by, 128

defi ned, 128

and parallel lines, 131–134, 168

Trapezoid(s), 401–404, 414

area of, 619

in coordinate plane, 402

defi ned, 402

isosceles, 402–403

making conjecture about, 401

midsegment of, 404

properties of, 402–403

rotational symmetry, 197

Trapezoid Midsegment Theorem (Thm. 7.17), 404

Tree diagram, 737

Trials of probability experiment, 709

Triangle(s), See also Relationships

within triangles; Right triangle

altitude of, 323, 325–327, 355

angles of, 235–239, 294

angle measures of triangles,

237–239

classifying triangles by sides and

angles, 236–237

using angle-angle similarity,

432–434

area of, 1, 31, 637

spherical triangles, 690

using trigonometric ratios, 512

bisectors of (See Bisectors of

triangles)

centroid of, 324–325, 327

circumcenter of, 314–316, 327

classifying by Pythagorean

inequalities, 471

classifying by sides and angles,

236–237

comparing measures in, 348–349

congruent (See Congruent triangles)

construction, copying a triangle

using SAS, 252


624–625

equiangular, 236, 257

equilateral (See Equilateral triangle)

examples of segments, lines, rays,

and points in, 304, 327

incenter of, 317–318, 327

inequalities

in one triangle, 343

in two triangles, 347–350, 356

median of, 324–325, 355

midsegments, 333–336, 355

obtuse (See Obtuse triangle)

orthocenter, 325–327

perimeter of, 31

proportionality, 450–451, 453

proving congruence

by ASA and AAS, 273–277,

296–297

by SAS, 249–252, 295

by SSS, 265–269, 296

relating sides and angles, 339,

341–342, 356

similarity (See Triangle similarity)

Triangle Angle Bisector Theorem (Thm. 8.9), 453

Triangle Inequality Theorem (Thm. 6.11), 343

Triangle Larger Angle Theorem (Thm. 6.10), 341

Triangle Longer Side Theorem (Thm. 6.9), 341

Triangle Midsegment Theorem (Thm. 6.8), 335–336

Triangle Proportionality Theorem (Thm. 8.6), 450

contrapositive of, 451

converse of, 450

inverse of, 451

Triangle similarity deciding whether triangles are

similar, 439

proving by AA, 431–434, 458

proving by SAS, 442, 459

proving by SSS, 440–441, 459

proving slope criteria using similar


right triangles, 481–485, 523



Triangle Sum Theorem (Thm. 5.1), 237–238

Triangular prism, 672

Triangular pyramid, 640, 672

Trigonometric ratio(s), See also

Cosine ratio; Sine ratio;

Tangent ratio

defi ned, 492

fi nding areas of triangles, 512

Trigonometry, See Right triangles and

trigonometry

Truth table, 70

Truth value of statement, 70

Two-column proof concept summary of, 102, 110

defi ned, 100

writing, 100, 102

writing for parallelograms, 374

Two-dimensional fi gures area of, 595

effects of changing dimensions

Index A97

Ind

ex

non-proportionally, 624–625

proportionally, 623, 626

Two Point Postulate (Post. 2.1), 84, 688

Two-way frequency table, 722

Two-way table(s), 721–725, 753

defi ned, 722

making, 722

and Venn diagram, 721

UUndefi ned slope, 156

Undefi ned terms of geometry, 4Union of events, 732–733

Unit circle trigonometry, 466

Units of measure converting between customary and

metric units of length, 2

nonstandard units, to measure line

segments, 11

VVector(s) defi ned, 178

translating a fi gure using, 179

Venn diagram, 721

classifying parallelograms, 393

classifying quadrilaterals, 362

reasoning with, 75

Vertex of cone, defi ned, 655

defi ned, of angle, 38

in polygons, 30

of polyhedron, defi ned, 639, 640

of pyramid, 654

Vertex angle (of isosceles triangle), 256

Vertical angles, 50

Vertical Angles Congruence Theorem (Thm. 2.6), 108–110, 133

Vertical component, 178

Vertical lines, 158

Vertical stretch, and nonrigid

transformation, 215

Volume(s) of composite solid, 667, 675, 683

of cones, 671, 673, 675

of cylinders, 663–667, 696

defi ned, of solid, 664

of prisms, 663–667, 696

of pyramids, 671–672, 674–675

of spheres, 679, 682–683

of spherical cap, 686

Volume Addition Postulate, 669

WWheel of Theodorus, 480

Writing, Throughout. See for example: conjecture about angles of triangle,

235

conjecture about isosceles triangles,

255

a coordinate proof, 287, 288, 290

coordinate proofs involving circles,

582

an indirect proof, 340

XZero slope (slope of 0), 156

Index [] 30 °-60°-90° right ... angle bisectors of triangle, 313 circumcenter of triangle, ... Ceva’s Theorem, 456 Chord of a sphere, 680 Chord(s) of circles, 549 ...

Documents