Download EXAMPLE 2 Evaluating angles.

Alan S. Tussy
R. David Gustafson
Prealgebra
Second Edition
Copyright © 2002 Wadsworth Group.
1
9.1 Some Basic Definitions
In this section, you will learn about
• Points, lines, and planes • Angles •
Adjacent and vertical angles
• Complementary and supplementary angles
2
Points, lines, and planes
3
Line segment
4
Midpoint
5
Ray
6
Angles
7
Measure in degrees
8
Classification of angles
9
EXAMPLE 1 Classifying angles.
Classify each angle in Figure 9-8 as an acute
angle, a right angle, an obtuse angle, or a
straight angle.
10
Adjacent and vertical angles
EXAMPLE 2 Evaluating angles.
Two angles with measures of x°
and 35° are adjacent angles. Use
the information in Figure 9-9 to
find x.
11
Vertical angles
12
EXAMPLE 3 Evaluating angles.
In Figure 9-12, find x.
13
Complementary and
supplementary angles
14
EXAMPLE 5 Complementary and
supplementary angles.
a. Angles of 60° and 30° are
complementary angles, because
the sum of their measures is 90°.
Each angle is the complement of
the other.
b. Angles of 130° and 50° are
supplementary, because the sum
of their measures is 180°. Each
angle is the supplement of the
other.
15
EXAMPLE 6 Finding the complement
and supplement of an angle.
a. Find the complement of
a 35° angle.
b. Find the supplement of a
105° angle.
16
9.2 Parallel and Perpendicular
Lines
In this section, you will learn about
• Parallel and perpendicular lines •
Transversals and angles
• Properties of parallel lines
17
Parallel and perpendicular lines
18
Parallel lines
19
Transversals and angles
20
EXAMPLE 1 Identifying angles.
In Figure 9-18, identify a.
all pairs of alternate
interior angles, b. all pairs
of corresponding angles,
and c. all interior angles.
21
Properties of parallel lines
22
EXAMPLE 2 Evaluating angles.
See Figure 9-24 on the
next page. If l1 || l2 and
m(angle 3) = 120o, find
the measures of the
other angles.
23
EXAMPLE 3 Identifying
congruent angles.
See Figure 9- 25. If AB || DE, which pairs of
angles are congruent?
24
EXAMPLE 4 Using algebra in
geometry.
In Figure 9-26, l1 || l2. Find x.
25
EXAMPLE 5 Using algebra in
geometry.
In Figure 9-27, l1 || l2. Find x.
26
9.3 Polygons
In this section, you will learn about
• Polygons • Triangles • Properties of
isosceles triangles
• The sum of the measures of the angles of
a triangle • Quadrilaterals
• Properties of rectangles • The sum of the
measures of the angles of a polygon
27
Polygons
28
Triangles
29
Properties of isosceles triangles
1. Base angles of an isosceles triangle are
congruent.
2. If two angles in a triangle are congruent, the
sides opposite the angles have the same length,
and the triangle is isosceles.
30
EXAMPLE 2 Determining
whether a triangle is isosceles.
Is the triangle in Figure 9-30 an isosceles
triangle?
31
The sum of the measures of the
angles of a triangle
32
EXAMPLE 3 Sum of the angles of
a triangle.
See Figure 9-31. Find x.
33
EXAMPLE 4 Vertex angle of an
isosceles triangle.
See Figure 9-32. If one base angle of an
isosceles triangle measures 70°, how
large is the vertex angle?
34
Quadrilaterals
35
Properties of rectangles
1. All angles of a rectangle are right angles.
2. Opposite sides of a rectangle are parallel.
3. Opposite sides of a rectangle are of equal
length.
4. The diagonals of a rectangle are of equal
length.
5. If the diagonals of a parallelogram are of equal
length, the parallelogram is a rectangle.
36
EXAMPLE 5 Squaring a foundation.
A carpenter intends to build
a shed with an 8-by-12foot base. How can he
make sure that the
rectangular foundation is
“square”?
37
EXAMPLE 6 Properties of
rectangles and tritriangles.
In rectangle ABCD (Figure 935), the length of AC is 20
centimeters. Find each
measure: a. m(BD), b.
m(angle 1), and c. m(angle 2).
38
EXAMPLE 7 Cross section of a
drainage ditch.
A cross section of a drainage
ditch (Figure 9-36) is an
isosceles trapezoid with
AB || CD. Find x and y.
39
The sum of the measures of the
angles of a polygon
40
9.4 Properties of Triangles
In this section, you will learn about
• Congruent triangles • Similar triangles
• The Pythagorean theorem
41
Congruent triangles
42
EXAMPLE 1 Corresponding parts
of congruent triangles.
Name the corresponding parts of the congruent
triangles in Figure 9-38.
43
SSS property
44
SAS property
45
ASA property
46
SSA
47
EXAMPLE 2 Determining
whether triangles are congruent.
Explain why the triangles in Figure 9-43 are
congruent.
48
Similar triangles
49
Property of similar triangles
50
EXAMPLE 3 Finding the height of
a tree.
A tree casts a shadow 18 feet
long at the same time as a
woman 5 feet tall casts a
shadow that is 1.5 feet
long. (See Figure 9-45.)
Find the height of the tree.
51
The Pythagorean theorem
52
EXAMPLE 4 Constructing a highropes adventure course.
A builder of a high-ropes
adventure course wants to
secure the pole shown in
Figure 9-46 by attaching a
cable from the anchor stake
8 feet from its base to a
point 6 feet up the pole.
How long should the cable
be?
53
Finding the width of a television
screen
54
9.5 Perimeters and Areas of
Polygons
In this section, you will learn how to find
• Perimeters of polygons • Perimeters of
figures that are combinations of polygons
• Areas of polygons • Areas of figures that
are combinations of polygons
55
Perimeter of a square
56
Perimeter of a rectangle
57
EXAMPLE 3 Converting units.
Find the perimeter of the rectangle in Figure 949, in meters.
58
EXAMPLE 4 Finding the base of
an isosceles triangle.
The perimeter of the isosceles triangle in Figure
9-50 is 50 meters. Find the length of its base.
59
Perimeter of a figure
60
Areas of polygons
61
Areas of polygons
62
EXAMPLE 6 Number of square
feet in 1 square yard.
Find the number of square
feet in 1 square yard. (See
Figure 9-55.)
63
EXAMPLE 7 Women’s sports.
Field hockey is a team sport in which players use
sticks to try to hit a ball into their opponents’
goal. Find the area of the rectangular field
shown in Figure 9-56. Give the answer in
square feet.
64
EXAMPLE 8 Area of a parallelogram.
Find the area of the parallelogram in Figure 9-57.
65
EXAMPLE 9 Area of a triangle.
Find the area of the triangle in Figure 9-58.
66
EXAMPLE 10 Area of a triangle.
Find the area of the triangle in Figure 9-59.
67
EXAMPLE 11 Area of a trapezoid.
Find the area of the trapezoid in Figure 9-60.
68
Areas of figures that are
combinations of polygons
EXAMPLE 12 Carpeting
a room. A living
room/dining room area has
the floor plan shown in
Figure 9-61. If carpet costs
$29 per square yard,
including pad and
installation, how much
will it cost to carpet the
room? (Assume no waste.)
69
EXAMPLE 13 Area of one side of
a tent.
Find the area of one side of the tent in Figure 9-62.
70
9.6 Circles
In this section, you will learn about
• Circles • Circumference of a circle • Area of
a circle
71
Circles
72
Circumference of a circle
73
EXAMPLE 1 Circumference of a
circle.
Find the circumference
of a circle that has a
diameter of 10
centimeters. (See
Figure 9-65.)
74
Calculating revolutions of a tire
75
EXAMPLE 2 Architecture.
A Norman window is
constructed by adding a
semicircular window to the
top of a rectangular window.
Find the perimeter of the
Norman window shown in
Figure 9-66.
76
Area of a circle
77
EXAMPLE 3 Area of a circle.
To the nearest tenth, find
the area of the circle in
Figure 9-68.
78
Painting a helicopter pad
79
EXAMPLE 4 Finding the area.
Find the shaded area
in Figure 9-69.
80
9.7 Surface Area and Volume
In this section, you will learn about
• Volumes of solids • Surface areas of
rectangular solids
• Volumes and surface areas of spheres •
Volumes of cylinders
• Volumes of cones • Volumes of pyramids
81
Volumes of solids
82
Volumes of solids
83
Volumes of solids
84
Volumes of solids
85
Height of a geometric solid
86
EXAMPLE 1 Number of cubic
inches in one cubic foot.
How many cubic
inches are there in
1 cubic foot? (See
Figure 9-74.)
87
EXAMPLE 2 Volume of an oil
storage tank.
An oil storage tank is in the
form of a rectangular solid
with dimensions of 17 by
10 by 8 feet. (See Figure
9-75.) Find its volume.
88
EXAMPLE 3 Volume of a
triangular prism.
Find the volume of the
triangular prism in
Figure 9-76.
89
Surface areas of rectangular solids
90
EXAMPLE 4 Surface area of an
oil tank.
An oil storage tank is in
the form of a rectangular
solid with dimensions of
17 by 10 by 8 feet. (See
Figure 9-78.) Find the
surface area of the tank.
91
Volumes and surface areas of
spheres
92
Filling a water tank
93
Surface area of a sphere
94
EXAMPLE 5 Manufacturing
beach balls.
A beach ball is to have a
diameter of 16 inches. (See
Figure 9-81.) How many
square inches of material
will be needed to make the
ball? (Disregard any
waste.)
95
Volumes of cylinders
96
EXAMPLE 6
Find the volume of the cylinder
in Figure 9-83.
97
Volume of a silo
98
EXAMPLE 7 Machining a block of
metal.
See Figure 9-85. Find
the volume that is left
when the hole is
drilled through the
metal block.
99
Volumes of cones
100
EXAMPLE 8 Volume of a cone.
To the nearest tenth, find the
volume of the cone in
Figure 9-87.
101
Volumes of pyramids
102
Volume of a pyramid.
103