1 Alan S. Tussy R. David Gustafson Prealgebra Second Edition Copyright © 2002 Wadsworth Group.

2 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

3 Points, lines, and planes

4 Line segment

5 Midpoint

6 Ray

7 Angles

8 Measure in degrees

9 Classification of angles

10 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.

11 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.

12 Vertical angles

13 EXAMPLE 3 Evaluating angles. In Figure 9-12, find x.

14 Complementary and supplementary angles

15 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.

16 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.

Parallel and Perpendicular Lines In this section, you will learn about Parallel and perpendicular lines Transversals and angles Properties of parallel lines

18 Parallel and perpendicular lines

19 Parallel lines

20 Transversals and angles

21 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.

22 Properties of parallel lines

23 EXAMPLE 2 Evaluating angles. See Figure 9-24 on the next page. If l 1 || l 2 and m(angle 3) = 120 o, find the measures of the other angles.

24 EXAMPLE 3 Identifying congruent angles. See Figure If AB || DE, which pairs of angles are congruent?

25 EXAMPLE 4 Using algebra in geometry. In Figure 9-26, l 1 || l 2. Find x.

26 EXAMPLE 5 Using algebra in geometry. In Figure 9-27, l 1 || l 2. Find x.

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

28 Polygons

29 Triangles

30 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.

31 EXAMPLE 2 Determining whether a triangle is isosceles. Is the triangle in Figure 9-30 an isosceles triangle?

32 The sum of the measures of the angles of a triangle

33 EXAMPLE 3 Sum of the angles of a triangle. See Figure Find x.

34 EXAMPLE 4 Vertex angle of an isosceles triangle. See Figure If one base angle of an isosceles triangle measures 70°, how large is the vertex angle?

35 Quadrilaterals

36 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.

37 EXAMPLE 5 Squaring a foundation. A carpenter intends to build a shed with an 8-by-12- foot base. How can he make sure that the rectangular foundation is “square”?

38 EXAMPLE 6 Properties of rectangles and tritriangles. In rectangle ABCD (Figure 9- 35), the length of AC is 20 centimeters. Find each measure: a. m(BD), b. m(angle 1), and c. m(angle 2).

39 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.

40 The sum of the measures of the angles of a polygon

Properties of Triangles In this section, you will learn about Congruent triangles Similar triangles The Pythagorean theorem

42 Congruent triangles

43 EXAMPLE 1 Corresponding parts of congruent triangles. Name the corresponding parts of the congruent triangles in Figure 9-38.

44 SSS property

45 SAS property

46 ASA property

47 SSA

48 EXAMPLE 2 Determining whether triangles are congruent. Explain why the triangles in Figure 9-43 are congruent.

49 Similar triangles

50 Property of similar triangles

51 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.

52 The Pythagorean theorem

53 EXAMPLE 4 Constructing a high- ropes 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?

54 Finding the width of a television screen

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

56 Perimeter of a square

57 Perimeter of a rectangle

58 EXAMPLE 3 Converting units. Find the perimeter of the rectangle in Figure 9- 49, in meters.

59 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.

60 Perimeter of a figure

61 Areas of polygons

62 Areas of polygons

63 EXAMPLE 6 Number of square feet in 1 square yard. Find the number of square feet in 1 square yard. (See Figure 9-55.)

64 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 Give the answer in square feet.

65 EXAMPLE 8 Area of a parallelogram. Find the area of the parallelogram in Figure 9-57.

66 EXAMPLE 9 Area of a triangle. Find the area of the triangle in Figure 9-58.

67 EXAMPLE 10 Area of a triangle. Find the area of the triangle in Figure 9-59.

68 EXAMPLE 11 Area of a trapezoid. Find the area of the trapezoid in Figure 9-60.

69 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 If carpet costs $29 per square yard, including pad and installation, how much will it cost to carpet the room? (Assume no waste.)

70 EXAMPLE 13 Area of one side of a tent. Find the area of one side of the tent in Figure 9-62.

Circles In this section, you will learn about Circles Circumference of a circle Area of a circle

72 Circles

73 Circumference of a circle

74 EXAMPLE 1 Circumference of a circle. Find the circumference of a circle that has a diameter of 10 centimeters. (See Figure 9-65.)

75 Calculating revolutions of a tire

76 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.

77 Area of a circle

78 EXAMPLE 3 Area of a circle. To the nearest tenth, find the area of the circle in Figure 9-68.

79 Painting a helicopter pad

80 EXAMPLE 4 Finding the area. Find the shaded area in Figure 9-69.

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

82 Volumes of solids

83 Volumes of solids

84 Volumes of solids

85 Volumes of solids

86 Height of a geometric solid

87 EXAMPLE 1 Number of cubic inches in one cubic foot. How many cubic inches are there in 1 cubic foot? (See Figure 9-74.)

88 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.

89 EXAMPLE 3 Volume of a triangular prism. Find the volume of the triangular prism in Figure 9-76.

90 Surface areas of rectangular solids

91 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.

92 Volumes and surface areas of spheres

93 Filling a water tank

94 Surface area of a sphere

95 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.)

96 Volumes of cylinders

97 EXAMPLE 6 Find the volume of the cylinder in Figure 9-83.

98 Volume of a silo

99 EXAMPLE 7 Machining a block of metal. See Figure Find the volume that is left when the hole is drilled through the metal block.

100 Volumes of cones

101 EXAMPLE 8 Volume of a cone. To the nearest tenth, find the volume of the cone in Figure 9-87.

102 Volumes of pyramids

103 Volume of a pyramid.