Download Index A81 30°-60°-90° right triangles finding sine and cosine of 30

Index
30°-60°-90° right triangles
finding sine and cosine of 30°, 500
finding tangent with, 493
side lengths of, 475, 477
30°-60°-90° Triangle Theorem
(Thm. 9.5), 477
45°-45°-90° (isosceles) right triangles
finding sine and cosine of 45°, 500
side lengths, 475, 476
in standard position, 466
45°-45°-90° Triangle Theorem
(Thm. 9.4), 476
A
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
defined, 42
examples of segments and points
in triangles, 304, 327
finding 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
finding 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
Index
A81
Index
AA, See Angle-Angle (AA) Similarity
Theorem (Thm. 8.3)
AAS, See Angle-Angle-Side (AAS)
Absolute value, finding, 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
exploring converses, 137
proof of, 134
Altitude of triangle
defined, 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)
defined, 38
diagram interpretation, 51
finding 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
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
finding, 542–543
finding from angle relationships in
circles, 566–567
finding 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 figures, 596, 619, 637
in coordinate plane, 29–33, 57
finding, 31
effects of changing dimensions,
623–626, 632
finding after dilation, 420
of hexagon, 619
of kite, 616, 619
of nonagon, 618
of octagon, 618
of parallelogram, 596, 619
of polygons and composite figures,
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
A82
Index
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 figures, 595
using to find 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
B
Base
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)
defined, and writing, 69
and definitions, 234
reading theorems, and rewriting, 394
Binomial distribution(s), 745–748,
754
constructing, 748
defined, 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
C
Cavalieri, 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
defined, 541, 542
and inscribed angles, 557
Central angle of regular polygon, 617
Centroid Theorem (Thm. 6.7), 324
Centroid of triangle
defined, 324
examples of segments and points
in triangles, 327
finding, 325
Ceva’s Theorem, 456
Chord of a sphere, 680
Chord(s) of circles, 549–552, 587
defined, 534
intersection with tangent on circle,
566
perpendicular to diameter, 549
using congruent chords
to find arc measure, 550
to find circle’s radius, 552
using diameter, 551
using perpendicular bisectors, 551
Circle(s)
angle relationships in circles,
565–569, 588–589
finding 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
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 flip, 706, 737, 745
Coincident lines, example of, 124, 125
Collinear points, 4
Combination(s), 740–741, 754
counting, 740
defined, 740
finding 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
finding 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
defined, 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 figure(s)
area of, 596, 619, 631, 637
defined, 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
defined, 715
finding with a table, 717
finding with conditional relative
frequencies, 724
Conditional relative frequency,
723–724
Conditional statement(s), 65–70, 116
biconditional statements, 69, 234,
394
defined, 66
in if-then form, 66
negation, 66
related conditionals, 67
true or false determination, 65
truth tables, 70
using definitions, 68
writing, 66–67
Index
A83
Index
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
defined, 532, 534
diameter of
chord perpendicular to, 549, 551
defined, 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
defined, 534
finding with congruent chords,
552
finding with segments, 576
finding 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
defined, 314
examples of segments and points in
triangles, 327
finding, 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 figures, 619
of polygons, 615–618
circumference, 597–601, 630
effects of changing dimensions,
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
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, 673
Congruence, properties of, 101–102
Congruence transformation, 205
Congruence and transformations,
203–207, 227
congruence transformations, 205
identifying congruent figures, 204
reflections in intersecting lines, 203,
207
reflections in parallel lines, 203, 206
using theorems about, 206–207
Congruent angles, 40
Congruent arcs, identifying, 544
Congruent Central Angles Theorem
(Thm. 10.4), 544
Congruent circles, 544
Congruent Circles Theorem
(Thm. 10.3), 544
Congruent Complements Theorem
(Thm. 2.5), 107
Congruent Corresponding Chords
Theorem (Thm. 10.6), 550
Congruent figures
defined, 204
using properties of, 245
Congruent Parts of Parallel Lines
Corollary, 378
Congruent polygons, 243–246, 294
using corresponding parts, 244–245
using Third Angles Theorem
(Thm. 5.4), 246
Congruent segments, 15
Congruent Supplements Theorem
(Thm. 2.4), 107
Congruent 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, 297
Conjecture
defined, 75, 76, 234
making and testing, 77
reasoning with, 75
writing, about angles of triangle, 235
A84
Index
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
exploring converses, 137
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
defined, 15
of a dilation, 214
of equilateral triangle, 258
inscribing circle within triangle, 318
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
defined, of conditional statement, 67
truth table for, 70
Contrapositive of Triangle
Proportionality Theorem,
451
Converse
defined, of conditional statement, 67
truth table for, 70
Converses of theorems
Alternate Exterior Angles Converse
Theorem (Thm. 3.7), 139
Alternate Interior Angles Converse
Theorem (Thm. 3.6), 139
Converse of Angle Bisector
Theorem (Thm. 6.4), 308
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, 30
Coordinate (of point), 12
Coordinate plane
circles in, 579–582, 590
classifying triangle in, 237
dilating figures in, 211, 213
midpoint of segment in, 23
midsegments in, 334
parallelograms in, 375, 384, 396
perimeter and area in, 29–33, 57
placing figures in, 288
reflecting figures in, 185
rotating figures in, 193, 195
slopes of lines in, 156
translating a figure in, 177, 179
trapezoid in, 402
trapezoid midsegment, 404
Coordinate proof(s), 287–290, 298
applying variable coordinates, 289
defined, 288
placing figure in coordinate plane,
288
writing, 287, 288, 290
Coordinate Rule for Dilations, 213
Coordinate Rules for Reflections, 187
Coordinate Rules for Rotations
about the Origin, 195
Coplanar circles, 532, 535
Coplanar points, 4
Corollaries, 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
D
Deductive reasoning, 75–79, 116
compared to inductive reasoning, 78,
79
defined, 78
using correct logic, 64
using with laws of logic, 78
Defined terms of geometry, 5
Definitions
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
defined, 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
defined, 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
defined, 212
finding 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 dimensions and volume, 674
sphere, radius and volume, 683
Dimensions of two-dimensional
figures, 623–626
changing non-proportionally,
624–625
changing proportionally, 623, 626
Directed line segment
constructing point along, 451
defined, 157
partitioning, 157
Disjoint events, 732, 753
Distance between points
defined, 12
finding minimum distance, 189
on a sphere, 687, 689
using circumference and arc length
to find, 600
Distance Formula, 14
using, 233
Distance from a point to a line
defined, 148
finding, 164
Distributive Property, 93
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
E
Earth’s circumference, 603
Edge of polyhedron, 639, 640
Endpoints, 5
Enlargement, 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
Index
A85
Index
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, defined, 239
Corresponding angles
in congruent polygons, 244–245
defined, 128
Corresponding Angles Converse
(Thm. 3.5), 138
constructing parallel lines, 139
Corresponding Angles Theorem
(Thm. 3.1), 132
exploring converses, 137
Corresponding lengths, in similar
polygons, 423
Corresponding part(s)
defined, in congruent polygons,
244–245
of similar polygons, 422
Corresponding sides, in congruent
polygons, 244–245
Cosine ratio, 497–501, 524–525
of 45° and 30° angles, 500
of complementary angles, 498
defined, 498
finding 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, 2
Cylinders
defined, 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
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
defined, 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
F
Faces of polyhedron, 639, 640
Factorials, 738
Favorable outcomes, 707
Flawed reasoning, 64
Flow proof, 106
Flowchart proof
concept summary of, 110
defined, 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
A86
Index
permutations, 739
population density, 607
prism
lateral area, 646
surface area, 646
volume, 664
pyramid
lateral area of, 654
surface area of, 654
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
surface area of, 680
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
G
Geometric mean
compared to arithmetic mean, 481,
488
defined, 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 reflection(s), 188
Golden ratio, 430
Graph theory, 280
Graphing calculator
combinations, 740
permutations, 739
Graphing a circle, 581
Great circle, 680
H
Heads 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) Congruence
Theorem (Thm. 5.9),
268–269, 277
Hypotenuse of right triangle, 268
Hypothesis, in conditional statement,
66
I
Icosahedron, 639
If-then form, of conditional statement,
66
Image, 178
Incenter Theorem (Thm. 6.6), 317
Incenter of triangle
defined, 317
examples of segments and points in
triangles, 327
inscribing circle within triangle, 318
using, 317
Independent events, 713–716, 752
comparing to dependent events, 716
defined, 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
defined, 340
used in Triangle Larger Angle
Theorem (Thm. 6.10), 341
writing, 340, 356
Indirect reasoning, 340
Inductive reasoning, 75–79, 116
compared to deductive reasoning,
78, 79
defined, 76
using with conjecture, 76–77
Inferring the truth, 64
Initial point, of vector, 178
Inscribed angle(s)
defined, 557, 558
finding measure of angle, 559
finding measure of intercepted arc,
559
Inscribed Angles of a Circle Theorem
(Thm. 10.11), 559
using Base Angles Theorem,
256–257
writing conjecture about, 255
J
Joint frequency, 722
Joint relative frequency, 723–724
K
Kite(s)
area of, 616, 619
defined, 405
finding angle measures in, 405, 414
Kite Diagonals Theorem (Thm. 7.18),
405
Kite Opposite Angles Theorem
(Thm. 7.19), 405
L
Lateral area
defined, 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
defined, 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
defined, 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
defined, 268
finding, with sine and cosine
ratios, 499
finding, with tangent ratio, 493
of trapezoid, 402
Likelihoods, and probabilities, 704,
706
Line(s)
in coordinate plane, characteristics
of, 124
finding 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
undefined 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-Point Postulate (Post. 2.2), 84
Line of reflection, 186
Line segment(s), See also Segment(s)
defined, 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 flawed
reasoning, 64
Logically equivalent statements, 70
M
Major 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
Index
A87
Index
Inscribed polygon(s)
constructing square inscribed in
circle, 561
defined, 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
finding 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
reflections, 207
Intersection
defined, 6
of lines and planes, 3, 6
Intersection of events, 732–733
Inverse
defined, 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
finding sine and cosine of 30°, 500
side lengths, 475, 476
in standard position, 466
Isosceles trapezoid
defined, 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
Measurement, indirect, 281, 283
Median of trapezoid, 404
Median of triangle, 323–325, 327, 355
defined, 324
examples of segments and points in
triangles, 304, 327
Metric units of length, 2
Midpoint(s), 19–23, 57
defined, 20
of line segment, finding, 19, 22
and segment bisectors, 20–21
of a segment in a coordinate plane,
finding, 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
defined, 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 floor, 33
planes in sulfur hexafluoride, 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
N
n factorial, 738
n-gon, 30
area of, 617
Negation, of conditional statement, 66
A88
Index
Negative scale factor, 214
Negative slope, 156
Nets for three-dimensional figures,
638
defined, for a polyhedron, 646
for a pyramid, 638
Nonagon, area of, 618
Nonrigid transformation, 215
nth term, of arithmetic sequence, 63
Number line, partitioning a segment
on, 22
O
Oblique cone, 655
Oblique cylinder, 647
Oblique prism, 646
Obtuse angle
defined, 39
trigonometric ratios for, 512
Obtuse triangle
in circumscribed circle, 315
classifying by angles, 236
classifying by Pythagorean
inequalities, 471
orthocenter of, 326
Octagon, area of, 618
Octahedron, 639
Opposite rays, 5
Opposite Sides Parallel and
Congruent Theorem
(Thm. 7.9), 382, 383
Opposite of statement, See Negation
“or” (union), 732–733
Orthocenter of triangle
defined, 325
examples of segments and points
in triangles, 327
finding, 326
type of triangle, and location, 326
Outcomes
defined, 706
favorable, 707
Overlapping events
defined, 732
finding probability of, 733, 753
P
Pairs of angles, See Angle(s), pairs of
Pairs of lines, See Lines, pairs of
Paragraph proof
concept summary of, 110
defined, 108
Parallel lines
constructing, 139
defined, and identifying, 126–127
example of, 124, 125
identifying, slopes of, 158
proofs with, 137–141, 169
constructing parallel lines, 139
Corresponding Angles Converse
Theorem (Thm. 3.5), 138
proving Alternate Interior Angles
Converse, 140
Transitive Property of Parallel
Lines (Thm. 3.9), 141
properties of, 132–134
Alternate Exterior Angles
Theorem (Thm. 3.3), 132
Alternate Interior Angles
Theorem (Thm. 3.2), 132
Consecutive Interior Angles
Theorem (Thm. 3.4), 132
Corresponding Angles Theorem
(Thm. 3.1), 132
proportionality with three lines, 452
proving theorems about, 140
Reflections in Parallel Lines
Theorem (Thm. 4.2), 206
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
defined, 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
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
defined, 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
Plane Intersection Postulate
(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
undefined term, and naming, 4
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)
finding distance to a line, 148, 164
Line Intersection Postulate
(Post. 2.3), 84, 85
Line-Point Postulate (Post. 2.2), 84
Plane-Line Postulate (Post. 2.6),
84, 85
Plane-Point Postulate (Post. 2.5),
84, 85
Three Point Postulate (Post. 2.4), 84
Two Point Postulate (Post. 2.1), 84
undefined term, and naming, 4
Point of concurrency
defined, 314
examples of segments and points in
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
defined, 639, 640
lateral area, 645
surface area, 645
Population density, 607
Positive slope, 156
Postulate, defined, 12, 234
Postulates
Angle Addition Postulate (Post. 1.4),
41
Arc Addition Postulate (Post. 10.1),
543
Line Intersection Postulate
(Post. 2.3), 84
Line-Point Postulate (Post. 2.2), 84
Linear Pair Postulate (Post. 2.8), 108
Parallel Postulate (Post. 3.1), 127
Perpendicular Postulate (Post. 3.2),
127
Plane Intersection Postulate
(Post. 2.7), 84
Plane-Line Postulate (Post. 2.6), 84
Plane-Point Postulate (Post. 2.5), 84
Protractor Postulate (Post. 1.3), 39
Reflection Postulate (Post. 4.2), 188
Rotation Postulate (Post. 4.3), 196
Index
A89
Index
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, finding, 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
finding, 31
effects of changing dimensions,
623–626, 632
finding after dilation, 420
of similar polygons, 424
of triangle, square, and rectangle, 31
Perimeters of Similar Polygons
(Thm. 8.1), 424
Permutation(s), 739, 754
counting, 738
defined, 738
finding probability using, 739
formulas, 739
Perpendicular bisector(s)
constructing, 149
drawing, 304
examples of segments and points in
triangles, 304, 327
points on, 305
using, 306–307
Ruler Postulate (Post. 1.1), 12
Segment Addition Postulate
(Post. 1.2), 13
Three Point Postulate (Post. 2.4), 84
Translation Postulate (Post. 4.1),
180
Two Point Postulate (Post. 2.1), 84
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
defined, 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
defined, 746
interpreting, 747
Probability experiment, 706
Probability of complement of event,
707–708
A90
Index
Probability of compound events,
732–733
Probability of dependent events,
715–716
Probability of an event
defined, 704, 706
and likelihoods, 704, 706
Probability of independent events,
714–715
Problem-solving strategies
finding 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
defined, 99, 100
with parallel lines, 137–141, 169
constructing parallel lines, 139
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
perpendicular lines, 150
proving a construction, 284
proving statements about segments
and angles, 99–102, 118
flowchart 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
Converse (Thm. 7.7), 380
Parallelogram Opposite Sides
Theorem (Thm. 7.3), 372
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
Theorem (Thm. 5.5), 250
Side-Side-Side (SSS) Congruence
Theorem (Thm. 5.8), 266
Side-Side-Side (SSS) Similarity
Theorem (Thm. 8.4), 441
Similar Circles Theorem
(Thm. 10.5), 545
Slopes of Parallel Lines Theorem
(Thm. 3.13), 443
Slopes of Perpendicular Lines
Theorem (Thm. 3.14), 444
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
Distributive Property, 93
Division Property of Equality, 92
Multiplication Property of Equality,
92
of parallel lines, 132–134
Reflexive Property, 94
Substitution Property of Equality, 92
Q
Quadratic equations, solving by
completing the square, 531
Quadrilateral
area of, 29
classifications 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
R
Radians, measuring angles in, 601
Radicals, using properties of, 465
Radius
of arc, 40
of circle
defined, 534
finding with congruent chords,
552
finding with segments, 576
finding 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- prefix, 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 classifications, 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 hexafluoride, 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 traffic 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 flagpole, 434
triangles and shoe rack, 451
transformations
finding minimum distance, 189
golf website, 181
scale factor and length of image,
215
Index
A91
Index
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
figure, 623, 626
finding 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
defined, 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
effect of changing linear
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
Reasoning, on similar triangles, 433
Reasoning 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, 118
Rectangle
Ailles, 480
area of, 596, 637
defined, 392
diagonal lengths in, 395
effects of changing dimensions,
623–625
perimeter and area, 31
Rectangle Corollary (Cor 7.3), 392
Rectangle Diagonals Theorem
(Thm. 7.13), 395
Rectangular pyramid, 674
Reduction, 212
Reflection(s), 185–189, 226
coordinate rules for, 187
defined, 186
glide reflections, 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 reflective device, 185
Reflection Postulate (Post. 4.2), 188
Reflections in Intersecting Lines
Theorem (Thm. 4.3), 207
Reflections in Parallel Lines Theorem
(Thm. 4.2), 206
Reflexive Property, 94
triangle congruence, 245
Regular hexagonal pyramid, 654
Regular octagon, rotational symmetry,
197
Regular polygon
angle measures in, 617
area of, 615, 617–618, 631
defined, 365
Regular pyramid
defined, 653, 654
lateral areas of, 654
surface areas of, 653, 654
Related conditional statements, 67
Relationships among special
parallelograms, 393
Relationships within triangles
bisectors of triangles, 313–318, 354
indirect proof and inequalities in one
triangle, 339–343, 356
A92
Index
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
finding conditional, 723–724
finding 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
perpendicular lines, 126
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
defined, 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
defined, 655
lateral area of, 655
surface area of, 653, 655–657
Right cylinders
defined, 647
lateral area of, 647–648
surface area of, 647–649
effect of changing linear
dimensions, 649
Right pentagonal prism, 647
Right prism
defined, 646
lateral areas of, 646–647
surface areas of, 646–647, 649
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
defined, 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
defined, 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
S
Same-Side Interior Angles Theorem,
See Consecutive Interior
Angles Theorem (Thm. 3.4)
Sample space, 752
defined, 705, 706
finding, 705, 706
Sample space(s), 705–709
SAS, See Side-Angle-Side (SAS)
and properties of shapes, 251
using Law of Cosines to solve
triangle, 515
Side-Angle-Side (SAS) Congruence
Theorem (Thm. 5.5), 250
Side-Angle-Side (SAS) Similarity
Theorem (Thm. 8.5), 442
triangle similarity theorems
compared, 443
Side-Side-Angle (SSA), 268
special case for right triangles,
268–269
using Law of Sines to solve triangle,
513
Side-Side-Side (SSS)
congruence, 265–269, 277
construction, copying a triangle
using SSS, 268
using, 266–268
using Law of Cosines to solve
triangle, 516
Side-Side-Side (SSS) Congruence
Theorem (Thm. 5.8), 266
Side-Side-Side (SSS) Similarity
Theorem (Thm. 8.4), 440
proof of, 441
triangle similarity theorems
compared, 443
using, 440–441
Sides
classifying triangles by, 236–237
defined, of an angle, 38
finding side lengths in special right
triangles, 475–478, 522
finding 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, 545
Similar Circles Theorem (Thm. 10.5),
545
Similar figures
defined, 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
identifying, 482–483
using geometric mean, 484–485
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
defined, 498
finding 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, defined, 155, 156
Slope-intercept form, 162
writing equations of lines in, 123
Slopes of lines
in coordinate plane, 156
defined, 155, 156
finding, 155, 156, 170
identifying parallel and
perpendicular lines, 158
negative slope, 156
partitioning a directed line segment,
157
positive slope, 156
proving criteria using similar
triangles, 443–444
undefined slope, 156
zero slope (slope of 0), 156
Index
A93
Index
Scale factor
defined, 212
of dilation, 419
negative, 214
units in, and finding, 215
Scalene triangle, classifying, 236
Secant, 534
Secant segment, 575–576
Sector of circle
area of, 605, 608–609, 619, 630
defined, 605, 608
Segment(s)
construction, bisecting a segment,
21
defined, and naming, 5
finding length of, 20
finding 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, 586
Segment Addition Postulate
(Post. 1.2), 13
Segment bisector(s), 20
construction
bisecting a segment, 21
of perpendicular bisector, 149
defined, 20
and midpoints, 20–21
Segments of a chord, 574
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
Semicircle, 542
area of, 637
Side-Angle-Side (SAS)
congruence, 249–252, 277
construction, copying a triangle
using SAS, 252
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 figures
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
defined, 680
diameter of, 680, 681
finding distances on, 689
lines on, in spherical geometry, 687
surface areas of, 679–681, 697
formula for, 680
volumes of, 679, 682–683, 697
effects of changing dimensions,
683
formula for, 682
Spherical cap, 686
Spherical geometry
compared with Euclidean geometry,
688–689
finding areas of spherical triangles,
690
finding distances on a sphere, 689
Spherical triangles, area of, 690
Square
defined, 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
A94
Index
Structure
in dilation, 420
to solve multi-step equation, 361
Study 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, 263
Substitution Property of Equality, 92
Subtend, 558
Subtraction Property of Equality, 92
Success of trial, 709
Supplementary angles
defined, 48–49
proving cases, 107
Surface area
of composite solid, 649, 656
defined, 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, 679–681
Surface 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,
696–697
Syllogism
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
T
Tangent(s)
constructing to a circle, 537
defined, 534
finding radius of circle, 536
using properties of, 536–537
Tangent circles
defined, 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
defined, 492
finding, 492–493
inverse, 506
Tangent segment, 575–576
Terminal point, of vector, 178
Tessellations, 209, 210
Tetrahedron, 639
Theorem, defined, 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
Theorem (Thm. 8.3), 432
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
Geometric Mean (Leg) Theorem
(Thm. 9.8), 484
Hinge Theorem (Thm. 6.12), 348
Hypotenuse-Leg (HL) Congruence
Theorem (Thm. 5.9), 268
Incenter Theorem (Thm. 6.6), 317
Inscribed Angles of a Circle
Theorem (Thm. 10.11), 559
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
Converse (Thm. 7.15), 403
Isosceles Trapezoid Base Angles
Theorem (Thm. 7.14), 403
Isosceles Trapezoid Diagonals
Theorem (Thm. 7.16), 403
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
Theorem (Thm. 3.12), 150
Measure of an Inscribed Angle
Theorem (Thm. 10.10), 558
Opposite Sides Parallel and
Congruent Theorem
(Thm. 7.9), 382
Parallelogram Consecutive Angles
Theorem (Thm. 7.5), 373
Parallelogram Diagonals Converse
(Thm. 7.10), 382
Parallelogram Diagonals Theorem
(Thm. 7.6), 373
Parallelogram Opposite Angles
Converse (Thm. 7.8), 380
Parallelogram Opposite Angles
Theorem (Thm. 7.4), 372
Parallelogram Opposite Sides
Converse (Thm. 7.7), 380
Parallelogram Opposite Sides
Theorem (Thm. 7.3), 372
Perimeters of Similar Polygons
(Thm. 8.1), 424
Perpendicular Bisector Theorem
(Thm. 6.1), 306
Perpendicular Chord Bisector
Converse (Thm. 10.8), 550
Perpendicular Chord Bisector
Theorem (Thm. 10.7), 550
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
Reflections in Intersecting Lines
Theorem (Thm. 4.3), 207
Reflections 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
Index
A95
Index
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
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
defined, 707
finding, 707, 713
Third Angles Theorem (Thm. 5.4),
246
Three-dimensional figures, 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
defined, 178
dilations, 211–215, 228
identifying, 175
reflections, 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
defined, 178
of figure in coordinate plane, 179
of figure using vector, 179
performing compositions, 180
performing translations, 178–179
of triangle in coordinate plane, 177
A96
Index
Translation Postulate (Post. 4.1), 180
Transversal(s)
angles formed by, 128
defined, 128
and parallel lines, 131–134, 168
Trapezoid(s), 401– 404, 414
area of, 619
in coordinate plane, 402
defined, 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
effects of changing dimensions,
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
triangles, 443–444
right triangles, 481–485, 523
identifying, 482–483
using geometric mean, 484–485
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
defined, 492
finding 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
defined, 100
writing, 100, 102
writing for parallelograms, 374
Two-dimensional figures
area of, 595
effects of changing dimensions
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
defined, 722
making, 722
and Venn diagram, 721
U
Undefined slope, 156
Undefined terms of geometry, 4
Union 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
V
W
Wheel 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
X
Zero slope (slope of 0), 156
Index
Vector(s)
defined, 178
translating a figure using, 179
Venn diagram, 721
classifying parallelograms, 393
classifying quadrilaterals, 362
reasoning with, 75
Vertex
of cone, defined, 655
defined, of angle, 38
in polygons, 30
of polyhedron, defined, 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
defined, 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
Index
A97