Geometry

For questions 7-9, find the volume of each sphere or
hemisphere. Leave answers in terms of pi.

7. 8.

9.

10. Find the volume of a hemisphere if the circumference of
the great circle is 45 feet. Round to the nearest tenth.

11. Steel weighs .2904 pounds per inches3. How much does
a steel ball with a diameter of 6 inches weigh? Round to
the nearest tenth.

12. Nicole is exercising with a piece of equipment that is
hollow and in the shape of a hemisphere. The base of
the ball has an area of 169π in.2. What is the volume
of the air inside the ball to the nearest tenth?

answers 587

1. 4π(72); 196π in.2

2. 4π(92); 324π cm2

3. 1 (4)π(42) + π(42); 48π ft2
2

4. 4π(3.22); 127.3 m2

5. 1 (4)π(12) + π(12); 3π ft2
2

6. 4 π(2.83); 91.9 m3
3

7. 4 π(123); 2,304π mm3
3

8. ( 4 π(4.53)); 243 π = 121.5π in.3
3 2

9. 1 ( 4 π(23)); 16 π cm3
2 3 3

10. 1 ( 4 π(7.23)); 781.3 ft3
2 3

11. 0.2904( 4 π(33)); 32.8 lbs
3

12. A rea of the base = πr2; 169π = πr2; r2 = 169; r = 13
1 4 1 4
V= 2 × 3 πr3 = 2 × 3 π(133) = 4,599.1 in.3

588

Chapter 54

VOLUMES OF

COMPOSITE FIGURES

A 3-D COMPOSITE FIGURE is a shape made up of two or

more basic geometric solids.

We can split a composite figure P = perimeter of the base
into its basic geometric solids to B = area of the base
make calculations. r = radius of the base
h = height
Formulas used to calculate volume
l = slant height
in composite 3-D figures:

SOLID LATERAL SURFACE VOLUME

Cone AREA AREA

πrl B + πrl or 1 Bh or
πr2 + πrl 3

1 πr2h
3

589

SOLID LATERAL SURFACE VOLUME

Cylinder AREA AREA Bh or πr2h

2πrh 2B + 2πrh or
2πr2 + 2πrh

Hemisphere ( )1 1 4
2 3
2 (4πr2) + πr2 πr3

Prism Ph 2B + Ph Bh

Pyramid 1 Pl 1 Pl 1
2 2 3
B+ Bh

Sphere 4
3
4πr2 πr3

590

SURFACE AREA OF
COMPOSITE FIGURES

The surface area of a composite figure is the area that
covers the entire outside of the solid. To find the surface
area, add up the areas of the faces, including any curved
surfaces (only the parts on the outside).

EXAMPLE: Find the surface area of the composite figure.

The parts on the surface are the
lateral area of the pyramid, the lateral
area of the prism, and the bottom of
the composite figure, which is the base
of the prism.

Do not include the top base of
the prism (which is also the base
of the pyramid) because it is not
on the surface.

591

Total = Lateral area + Lateral area + Area of one of
surface area of the pyramid of the prism the prism’s bases

= 1 Pl + Ph + lw
2

= 1 (5 + 5 + 5 + 5)(3) + (5 + 5 + 5 + 5)(6) + 5 × 5
2

= 30 + 120 + 25

= 175

The surface area of the composite figure is 175 cm2.

EXAMPLE: Find the surface area of

the ice cream cone and the ice cream.

Separate the solid into the cone and
the hemisphere.

592

Total = Lateral area + 1 Surface area of a sphere
surface area of the cone 2

= πrl + 1 (4πr2)
2

= π(1.5)(4.5) + 1 [4π(1.5)2]
2

≈ 35.3

The surface area of the cone and
ice cream is approximately 35.3 in.2.

Note: For the hemisphere, use half not included

the surface area of a sphere because
the surface area of a hemisphere
adds the area of the great circle,
which is not on the surface, and so
not part of the surface area.

VOLUME OF COMPOSITE
FIGURES

To find the volume of a composite figure, separate the
shape into its basic solids. Then we find the volume of each
solid, using the volume formulas. Finally, add all the volumes
together.

593

EXAMPLE: Find the volume of

the composite figure.

Separate the solid into three
prisms.

Since the volume of each prism is
V = lwh, find the missing length,
width, and height of each solid.

Use the horizontal lengths to find the length of the red prism:

l = 50 - 14 - 15 = 21

Use the total height to find

the height of the green prism:

h = 42 - 9 = 33

The width is the same throughout
the solid, so the width of every

prism is:

w = 13

594

Now, we have all the information we need to find the volume.

Total = Volume of + Volume of + Volume of
Volume green prism blue prism red prism

= lwh + lwh + lwh

= (50)(13)(33) + (14)(13)(9) + (21)(13)(20)

= 28,548

The volume is 28,548 cm3.

595

EXAMPLE: Find the volume of

the solid.
If we subtract the volume of the
cylinder from the volume of the
rectangular prism, we end up
with the volume of the remaining solid.
The length of the prism is the diameter of the cylinder,
l=2m+2m=4m

Total volume = Volume of the prism - Volume of the cylinder
= lwh - πr2h
= (4)(13)(7) - π(2)2(7)
= 364 - 28π
≈ 276.0

The volume is approximately 276.0 m3.

596

w

For questions 1 and 2, find the surface area of each
composite figure. Round to the nearest tenth if necessary.

1. 2.

For questions 3-6, find the volume of the composite figures.
Round to the nearest tenth if necessary.

3. 5.

4. 6.

answers 597

1. 2π(1.75)(1.9) + π(1.752) + π(1.75)(4.2); 53.6 m2

2. 2( 1 )(16)(6) + 2(10)(17) + 2(17)(11) + 2(16)(11) + 16(17); 1,434 ft2
2

3. 23(25)(19) + 1 (12)(23)(19); 13,547 mm3
2

4. 1 ( 4 )π(63) + 1 π(62)(8); 754.0 cm3
2 3 3

5. 4.6(7.3)(6.8) - 1(1)(7.3); 221.0 m3

6. 1 ( 4 )π(63) - π(1.52)(5.8); 411.2 in.3
2 3

598

Chapter 55

SOLIDS OF
REVOLUTION

A SOLID OF REVOLUTION is the solid formed when

a two-dimensional object is rotated about a line, called
the AXIS.

Examples of a solid of revolution:

Rotating the triangle (2-D) about

line l forms a cone (3-D).

599

Rotating the semicircle (2-D) about line l forms a sphere (3-D).

EXAMPLE: Find the volume

of the solid formed when the

triangle is rotated about line l .

The solid formed is a cone.
The hypotenuse of the triangle
becomes the slant height of the

cone, so l = 9 in. The 6-in. leg of

the triangle becomes the radius of
the base of the cone, so r = 6 in.

To use the formula for the volume of a Pcoynteh,aVgo=re31anπrT2hh,efoirresmt .
find the height of the cone, using the

lh2 + r2 = 2 9• 5 =3 5

h2 + 62 = 92
h2 + 36 = 81
h2 = 45
h = 45 =

600

Then insert the solution into the formula:

V= 1 πr2h
3

= 1 π(6)2(3 5)
3

= 36 5 π

The volume of the cone is 36 5 π in.3.

EXAMPLE: Find the volume of the solid formed when

the rectangle is rotated about line l .

The solid formed is a cylinder.
The 10-cm side of the rectangle
becomes the height of the cylinder.
The 4-cm side of the rectangle becomes
the radius of the base of the cylinder.

The volume is:

V = πr2h
= π(4)2(10)
= 160π

The volume of the cylinder is 160π cm3.

601

EXAMPLE: Find the surface area of

the solid formed when the semicircle is

rotated about line l .

The solid formed is a sphere with
a radius of 7 feet.

The surface area is:

SA = 4πr2
= 4π(7)2
= 196π

The surface area is 196π ft2.

SOLIDS OF REVOLUTION ON
A COORDINATE PLANE

A two-dimensional figure rotated around the x- or y-axis
(or another line in the plane) also forms a three-dimensional
object.

Rotating a figure around the y-axis rotates the figure
horizontally (left and right). Rotating a figure around the

x-axis rotates the figure vertically (up and down).

602

EXAMPLE: Find the surface area of the triangle rotated

about the y-axis.

The solid formed is a cone
with a height of 3 units and
base radius of 4 units.

In order to use the surface
area formula for a cone,

SA = πr2 + πrl , we must find
l , the slant height.

Since the hypotenuse of
the triangle becomes the
slant height, we can use
Pythagorean triples 3, 4, 5
(or the Pythagorean
Theorem) to find slant

height, l = 5.

The surface area:

SA = πr2 + πrl

= π(4)2 + π(4)(5)
= 36π

The surface area of the cone is 36π units2.

603

EXAMPLE: Find the volume

of the solid formed by
rotating the shaded figure
around the x-axis.

The solid formed by each
semicircle is a sphere. The
portion between the spheres
(the shaded part) is the
volume we need to find.

Volume of shaded portion = Volume of larger sphere -
Volume of the smaller sphere.

= 4 πr3 - 4 πr3
3 3

= 4 π(6)3 - 4 π(2)3
3 3

= 277.3π

The volume of the shaded portion is 277.3π units3.

604

w

For questions 1-3, name the solid formed when the shaded

figure is rotated about line l .

1. 2. 3.

For questions 4 and 5, find the volume of the solid formed

when the shaded figure is rotated about line l . Round

answer to the nearest tenth.
4. 5.

More questions 605

6. Find the surface area of
the solid formed when the

rectangle is rotated about line l .

Leave answer in terms of pi.
7. Find the volume of the solid

formed when the figure is
rotated about the x-axis. Leave
answer in terms of pi.

606

For questions 8 and 9, find the volume of the solid formed
when the shaded figure is rotated about the y-axis. Leave
answers in terms of pi.
8.

9.

answers 607

1. Cone

2. Cylinder

3. Hemisphere

4. 1 π(102) 44 ; 694.6 m3
3

5. 4 π(83) - 4 π(33); 2,031.6 ft3
3 3

6. 2π(142) + 2π(14)(9); 644π in.2

7. π(22)(7); 28π units3

8. 1 π(32)(7); 21π units3
3

9. 4 π(13); 4 π units3
3 3

608

INDEX

A inscribed angles, 455-459 of regular polygons, 521
interior angles, 90, 123-125, of rhombuses, 521
absolute value, 367 of sectors, 514-515
acute angle, 19 129-131, 129-135, of trapezoids, 520
acute triangle, 121, 359-360 220-222 of triangles, 493-495, 520
addition-subtraction property interiors of, 18 axis, 599
measure of, 18-19, 442
of equality, 70 measures of in polygons, B
adjacent angles, 20, 27 219-225
adjacent arcs, 433 naming, 17-18 base, 124
adjacent leg, definition of, 400 non-adjacent angles, 20 base angles, 124
alternate exterior angles, 91 obtuse angles, 19 bases, 533
alternate exterior angles proving special angle pairs, biconditional statements,
99-105
theorem, 102-103, 105 right angles, 19 60-62, 65
alternate interior angles, 91 same-side interior angles, 91, bisectors
alternate interior angles 103-104, 105
straight angles, 20 altitude, 171
theorem, 101, 105, 156 supplementary angles, 30-31 angle bisector theorem,
altitude, 171, 173 transversal angle pairs,
angle addition postulates, 91-92 333-335
types of, 19-20 angle bisectors, 33-37
21-22, 34 vertical, 27 centroid, 168-170
angle bisector theorem, 333-334 vertical angles, 27-28 of chords, 448, 451
angle bisectors, 33-37, 172 angle-side-angle (ASA) circumcenter, 165-167
angle of rotation, 251-252, congruence, 153-156, 159 constructing angle, 47-48
apex, 545 incenter, 167-168
256-259 apothem, 504-506 median, 168-170
angle pairs, 27-40, 91-92, 99-105 arc addition postulate, 433-434 orthocenter, 171
angle-angle (AA) similarity arc length formula, 436 perpendicular bisectors,
arcs
postulate, 319-322, 325 adjacent arcs, 433 35-37, 163-165, 172,
angle-angle-side (AAS) congruent arcs, 435, 447 234-235
definition of, 430 points of concurrencies,
congruence, 157-158, 159 intercepted arcs, 455-456, 172-173
angles 471 segment bisectors, 12-13
length of, 434-437 triangle bisectors, 163-174
acute angles, 19 major, 432
adjacent angles, 20, 27 measure of, 431-434, 435-437 C
base angles, 124 minor arcs, 432
basic angles, 19 area Cavalieri’s principle, 564-565
central angles, 430, 434, of circles, 511-515, 521 center of gravity, 170
of composite figures, center of rotation, 251-252,
435-437 519-527
classifying triangles by, of a figure, 490 260-261, 275
of kites, 521 center point, 418
121-122 of other polygons, 499-507 central angles, 430-434, 435-437
comparing, 177-181 of parallelograms, 490-493, centroid, 168-170, 172
complementary angles, 29, 520 centroid theorem, 169-170
of rectangles, 490-491, 520 chords
31-32
congruence and, 139-148 congruent, 447-448
congruent angles, 22-23, 28, definition of, 418, 447
inscribed angles and, 455
33-34 theorems about, 447-451
constructing, 46-47 circles
corresponding, 92, 139, arcs and chords, 447-451

310-311 609
definition of, 3, 17
exterior angles, 90, 129-135
exteriors of, 18
included angles, 144

area of, 511-515, 521 side-side-side (SSS), 142-143 using Pythagorean theorem,
central angles and arcs, congruence statement, 11 379-380
congruence transformation, 230
430- 437 congruent angles, 22-23, 28, using slope formula for,
circumference (C) of, 420-425 377-378
concentric, 422-423 33-34
congruent, 435 congruent arcs, 435, 447 writing, 373-374
on the coordinate plane, congruent chords, 447-448 coplanar points, 6
congruent circles, 435 corollary, definition of, 332
477- 481 congruent line segments, 10-12 corollary to the triangle
definition of, 418 congruent tangents, 464
equations of, 477-484 conjectures, 53- 55 proportionality theorem,
fundamentals of, 418-425 constants, definition of, 350 332-333
inscribed angles and, constructions corresponding angles, 92, 139,
310-311
455- 459 angle bisectors, 47-48 corresponding angles postulate,
parts of, 418-419 angles, 46-47 99, 105
radian measure, 442-444 parallel lines, 44-45 corresponding sides, 139, 310-311
secants, 471-474 perpendicular lines, 42-43 cosine (cos), 401-402
semicircle, 431 tools for, 41 cosines, law of, 412-414
tangents and, 463-467 converse, 58- 62 counterexamples, 54-55
circumcenter, 165-167, 172 converse of alternate exterior cross products, 289-291
circumcenter theorem, 166-167 angles theorem, 109-110 cube, 7
circumference (C) converse of alternate interior cubic units, 557
definition of, 418 angles theorem, 109, 111-112 cylinders
formula for, 420-425 converse of corresponding
collinear points, 5 angles postulate, 109-110 oblique, 563-565
common tangents, 463-464 converse of isosceles triangle surface area of, 539- 541
compass, 41 theorem, 125 volume of, 562- 565, 590
complementary angles, 29, 31-32 converse of perpendicular
completing the square, 481-484 bisectors theorem, 164 D
composite figures converse of same-side interior
area of, 519- 527 angles theorem, 109-110, 113 decagon, number of sides in, 219
volume of, 589- 596 converting degrees and radians, deductive reasoning, 62-65
compositions 444 degrees, 18-19
glide reflections, 269-270 coordinate plane degrees and radians, converting,
of reflections, 271-273 circles on, 477-481
symmetry and, 274-275 dilations on, 302-304 444
of translations, 267-268 distance on, 368-369 detachment, law of, 62-63, 65
concentric circles, 422-423 midpoint on, 364-366 diameter (d)
conclusions, 56-57 reflections on, 233-239
concurrent, definition of, 165 rotations on, 257-259 as bisector of chord, 448, 451
conditional statements, 56-60, 65 solids of revolution on, definition of, 419
cones formula for, 420-422
definition of, 551 602- 604 of spheres, 579
surface area of, 551- 553 solving problems with, dilations
volume of, 572- 574, 589 on the coordinate plane,
congruence 147-148
angle-angle-side (AAS), translations on, 244-247 302-304
coordinate quadrilateral proofs definition of, 295
157-158 using distance formula for, drawing, 300-301
angle-side-angle (ASA), finding scale factor,
390-394
153-156 using slope formula for, 297-299
basics of, 279-281 scale factor for, 296
definition of, 139 388-389 distance formula, 367-369,
properties of, 70-71 writing, 386-387 373-377, 387, 390-391
side-angle-side (SAS), coordinate triangle proofs distributive property, 71
using distance formula for, division property of equality, 70
144-148 drawing rotations, 253-255
374-377
610 E

edges, 532

endpoints, angles and, 17 I inductive reasoning example
enlargements, 295-296 using, 55
equality properties, 69-71 if-then statements, 56- 62
equiangular triangle, 122 image, 230-232, 295 lines
equidistant, definition of, 163 incenter, 167-168, 172 on the coordinate plane,
equilateral triangle, 121, 506 incenter theorem, 167-168 349-352
equivalent fractions, 289 included angle, 144 definition of, 2, 5
extended ratio, 288 included side, 153-154 finding slope of, 344-349
exterior angle measures, inductive reasoning, 53- 62, 65 intersection of, 7
inscribed angles, 455-459 naming, 5
223-225 inscribed shapes, 458-459 parallel, 4, 44-45, 88-90,
exterior angle space, 18 intercepted arcs, 455-456, 471 99-105, 109-113, 271-273
exterior angles, 90, 132-135 interior angle measures, perpendicular, 4, 35, 42-43,
347
F 220-222 as secants, 471-474
interior angle space, 18 slope of, 347
faces, 532 interior angles, 90, 129-131 as tangents, 463-464
flowchart proofs, 75-77 intersecting lines of reflection, vertical and horizontal, 352
frustum, volume of, 574- 575
271-273 lines of reflection, 232-239,
G intersection of lines and planes, 271-273

geometric proofs 7 lines of symmetry, 274
definition of, 69 inverse trigonometric functions, logic and reasoning, 53- 68
flowchart proofs, 75-77
paragraph proofs, 77-79 411 M
properties of equality irrational number, 357
and congruence for, isosceles right triangle, 403 magnifications, 295-296
69-71 isosceles trapezoid, 212-213, major arcs, 432
two-column proofs, 72-75 mapping of reflection, 232
215 median, 168-170, 172
geometry isosceles triangle, 121, 124-125, midpoint, 12
definition of, 2 midpoint formula, 363-366, 373
key terms for, 2-4 374-376 midsegments, 210
isosceles triangle theorem, 124 minor arcs, 432, 447
glide reflections, 269-270 multiplication property of
graphing linear equations, K
equality, 70
349-351 kites
great circle, 580 area of, 502- 503, 521 N
definition of, 214, 215
H negative reciprocals, 347,
L 377-378
hemisphere
definition of, 580 lateral area (LA), 534- 535, 540, negative rise, 343
surface area of, 582-583 547, 551, 589- 590 negative run, 343
volume of, 585, 590 negative slope, 342
lateral faces, 533 net, 534
heptagon law of cosines, 412-414 non-adjacent angles, 20
area of, 504 law of detachment, 62- 63, 65 nonagon, number of sides in, 219
number of sides in, 219 law of sines, 409-412 number line
law of syllogism, 62, 64, 65
hexagon laws of deductive reasoning, distance on, 367
area of, 505, 507 midpoint on, 363-364
number of sides in, 219 62- 65
legs, 124 O
horizontal lines, 352 line segment postulates, 8-10
hypotenuse line segments oblique cylinders, 563
oblique prisms, 563
definition of, 400 bisectors of, 12-13 observations, 53- 55
Pythagorean theorem and, congruent, 10-12 obtuse angle, 19
definition of, 3 obtuse triangle, 121, 360
355-356 linear equations, graphing, octagon, number of sides in, 219
hypotenuse-leg (HL) theorem, 349-351
linear pairs 611
158, 159 definition of, 30
hypothesis, 56-57

one-dimensional shapes, 5 polygons surface area of, 533- 538
opposite leg, definition of, 400 angle measures in, 219-225 triangular, 533, 538, 561- 562
opposite orientations, 232 area of, 499- 507 types of, 533
orthocenter, 171, 173 definition of, 120, 219 volume of, 558- 562, 590
exterior angle measures proofs
P and, 223-225 coordinate quadrilateral,
interior angle measures and,
paragraph proofs, 77-79 220-222 386-394
parallel lines polyhedrons and, 532 coordinate triangle, 373-380
regular, 224-225, 504- 507, definition of, 8
basics of, 88-90 521, 545 geometric, 69-79
constructing, 44-45 similar, 311-315 proportions
definition of, 4 types of, 219 basics of, 289-292
proving, 109-113 See also quadilaterals; for finding area of circle,
of reflection, 271-273 rectangles; squares;
slope of, 347 triangles 514- 515
special angle pairs and, in triangles, 329-335
polyhedron, 532 proving lines parallel, 109-113
99-105 positive rise, 343 proving special angle pairs,
parallel planes, 89 positive run, 343 99-105
parallelograms positive slope, 342 pyramids
postulates definition of, 545
area of, 490-493, 520 height of, 546
definition of, 187, 215 angle addition postulates, regular, 545, 547- 550
properties of, 188-189 21-22, 34 surface area of, 545- 550
theorems to prove, 190-193 types of, 546
See also quadilaterals; angle-angle (AA) similarity volume of, 569- 571, 590
postulate, 319-322, 325 Pythagorean theorem, 355-360,
rectangles; rhombuses; 379-380
squares angle-angle-side (AAS) Pythagorean triples, 359
pentagon, number of sides in, congruence postulate,
219 157-158 Q
perfect squares, 357
perimeter, 504, 507 angle-side-angle (ASA) quadilateral proofs, coordinate.
perpendicular bisectors, 35-37, congruence postulate, See coordinate quadrilateral
163-165, 172, 234-235 155-156 proofs
perpendicular bisectors theorem,
163 arc addition postulate, quadilaterals
perpendicular lines 433- 434 common, 187
constructing, 42-43 definition of, 186
definition of, 4, 35 converse of corresponding inscribed, 458-459
slope of, 347 angles postulate, 109-110 number of sides in, 219
pi (π), 419-420 types of, 215
planes corresponding angles See also kites; rectangles;
definition of, 6 postulate, 99, 105 rhombuses; squares;
intersection of, 7 trapezoids
naming, 6 definition of, 8
parallel, 89 line segment postulates, 8-10 quadratic equation, 481-484
point of tangency, 463 segment addition postulate,
points R
collinear, 5 8-10
coplanar, 6 side-angle-side (SAS) radians, 442-444
definition of, 2 radical sign, 357
intersections and, 7 congruence postulate, radius (r)
points of concurrencies 148
definition of, 165 side-side-side (SSS) definition of, 419
summary of, 172-173 congruence postulate, formula for, 420-422
polygon exterior angle-sum 142 of spheres, 579
theorem, 223 preimage, 230-232, 295 tangents and, 464
prime mark (‘), 231 ratio, 286-288
612 prisms ratios, trigonometric, 400-405
definition of, 533
oblique, 563-565
rectangular, 533, 536- 537,
558- 560

ray, definition of, 3 rotations space figures, 532
rays, angles and, 17 basics of, 251-252 special right triangles, 403-405
reciprocals, 347, 377-378 on the coordinate plane, spheres
rectangles 257-259
drawing, 253-255 definition of, 579
area of, 490-491, 520 finding angle of, 256-257 surface area of, 580- 583
basics of, 202-204 finding center of, 260-261 volume of, 583- 585, 590
definition of, 187, 215 properties of, 270 square roots, 357-358
rectangular prisms, 533, 536- 537, symmetry and, 275 squares
558- 560 as type of rigid motion, 230 basics of, 204-205
reductions, 295-296 definition of, 187, 215
reflections rotations, center of, 251-252, rotating, 254
compositions of, 271-273 260-261, 275 standard form, converting to,
on the coordinate plane, 481- 484
run, 340-343 straight angle, 20
233-239 straight line, 343
glide, 269-270 S straightedge, 41
properties of, 270 substitution property of equality,
rigid motions, 230-232 same-side interior angles, 91 71
symmetry and, 274 same-side interior angles supplementary angles, 30-31
reflections, mapping and, 232 surface area
reflexive property of theorem, 103-104, 105 basics of, 532
congruence, 156 scale factor, 296-299, 312-313 of composite figures,
reflexive property of scalene triangle, 121, 376-377
equality-congruence, 70 secants, 471-474 591- 593
regular polygons, 224-225, sectors, 430, 514- 515, 521 of cones, 551-553
504- 507, 521, 545 segment addition postulate, 8-10 of cylinders, 539- 541
regular pyramid, 545, 547- 550 segment bisectors, 12-13 formulas for, 589- 590
revolution, solids of. See solids semicircle, 431 of hemisphere, 582- 583
of revolution shapes of prisms, 533-538
rhombuses of pyramids, 545- 550
area of, 502- 503, 521 inscribed, 458-459 of solids of revolution, 602,
basics of, 197-199 one-dimensional, 5
definition of, 187, 215 two-dimensional, 6 603
theorems to prove, 200-202 side-angle-side (SAS) of spheres, 580-583
right angle, 19 congruence, 144-148, 159 syllogism, law of, 62, 64, 65
right triangle side-angle-side (SAS) similarity symmetric property of
classifying triangles as, 122 theorem, 322, 325 congruence, 71
hypotenuse-leg (HL) theorem side-side-side (SSS) congruence, symmetric property of equality,
142-143, 159 71
and, 158 side-side-side (SSS) similarity, symmetry, 274
Pythagorean theorem and, 324-325
sides, comparing, 177-181 T
355-356, 379-380 similar figures, 310-315
rules for, 359-360 similar triangles, 319-326 tangency, point of, 463
slope formula and, 377-378 sine (sin), 401-402 tangent (tan), 401-402, 463-467
special, 403-405 sines, law of, 409-412 tangents, secants and, 474
terms for, 400 skew lines, 89 theorems
trigonometric functions and, slant height, 546
slope, 340-349 about chords, 447-451
401- 402 slope formula, 374, 377-378, alternate exterior angles
rigid motions 387-390
solids, 532 theorem, 102-103, 105
basics of, 230-232 solids of revolution alternate interior angles
congruence and, 279-281 on the coordinate plane,
properties of, 270 theorem, 101, 105, 156
See also reflections; 602- 604 angle bisector theorem,
definition of, 599- 604
rotations; translations surface area of, 602, 603 333-335
rise, 340-343 volume of, 600- 601, 604 centroid theorem, 169-170
rotational symmetry, 275 circumcenter theorem,

166-167

613

converse of alternate transformation number of sides in, 219
exterior angles theorem, compositions of, 265-266 obtuse triangle, 121, 359-360
109-110 dilations, 295-304 proportions and, 329-335
as type of rigid motion, 230 right triangle, 122, 158,
converse of alternate
interior angles theorem, transitive property of 355-356, 359-360,
109, 111-112 congruence, 71, 74 377-380, 400-405
scalene triangle, 121, 376-377
converse of isosceles transitive property of equality, 71 similar, 319-326
triangle theorem, 125 translation vector, 244-247 special right, 403-405
translations types of, 120-125
converse of perpendicular triangular prisms, 533, 538,
bisectors theorem, 164 basics of, 243-247 561-562
compositions of, 267-268 trigonometric functions, 401-402
converse of same-side properties of, 270 trigonometric ratios, 400-405
interior angles theorem, as type of rigid motion, 230 trigonometry, definition of, 400
109-110, 113 transversal angle pairs, 91-92 two or more transversals, 93-95
transversals, 90-95 two-column proofs, 72-75
corollary to the triangle trapezoids two-dimensional shapes, 6
proportionality theorem, area of, 499-501, 520
332-333 basics of, 209-211 U
definition of, 187, 215
definition of, 8 isosceles, 212-213, 215 undefined slope, 342
hypotenuse-leg (HL) theorem, triangle angle-sum theorem, 129 units squared, 490
triangle bisectors
158, 159 altitude, 171 V
incenter theorem, 167-168 centroid, 168-170
isosceles triangle theorem, circumcenter, 165-167 vertex
incenter, 167-168 angles and, 17
124 median, 168-170 definition of, 3
perpendicular bisectors orthocenter, 171
perpendicular bisectors, vertical angles, 27
theorem, 163 vertical lines, 352
polygon exterior angle-sum 163-165 vertices, 532, 545
points of concurrencies, volume
theorem, 223
to prove parallelograms, 172-173 basics of, 557
triangle inequalities, 177-181 of composite figures, 593-596
190-193 triangle inequality theorem, 179 of cones, 572-574, 589
to prove rhombuses, triangle proofs, coordinate. of cylinders, 562-565, 590
formulas for, 589-590
200-202 See coordinate triangle proofs of frustum, 574-575
Pythagorean theorem, triangle proportionality theorem, of hemisphere, 585, 590
of prisms, 558-562, 590
355-360, 379-380, 329-331, 334-335 of pyramids, 569-571, 590
478- 480 triangles of solids of revolution,
same-side interior angles
theorem, 103-104, 105 acute triangle, 121, 359-360 600-601, 604
side-angle-side (SAS) angle measures in polygons of spheres, 583-585, 590
similarity theorem, 322,
325 and, 220-222 X
side-side-side (SSS) area of, 493-495, 520
similarity theorem, 324, classifying, 121-125 x-axis line of reflection, 238-239
325 congruence and, 139-148,
triangle angle-sum theorem, Y
130, 141 153-160
triangle inequality theorem, congruence summary for, y = x line of reflection, 238-239
179 y-axis line of reflection, 238-239
triangle proportionality 159-160 y-intercept, 349-352
theorem, 329-331, definition of, 4, 120
334-335 equiangular triangle, 122 Z
theta, , definition of, 400 equilateral triangle, 121, 506
three-dimensional (3-D) isosceles triangle, 121, zero slope, 342
composite figures, 589
three-dimensional (3-D) figures, 124-125, 374-376
532 naming, 120
tick marks, 11

614