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3. Lesson 10-1 Line and Angle Relationships
Lesson 10-2 Congruent Triangles
Lesson 10-3 Transformations on the Coordinate Plane
Lesson 10-4 Quadrilaterals
Lesson 10-5 Area: Parallelograms, Triangles, and Trapezoids
Lesson 10-6 Polygons
Lesson 10-7 Circumference and Area: Circles
Lesson 10-8 Area: Irregular Figures
Contents

4. Example 1 Find Measures of Angles
Example 2 Find a Missing Angle Measure
Example 3 Find Measures of Angles
Example 4 Apply Angle Relationships
Lesson 1 Contents

5. In the figure, m || n and t is a transversal. If find m2 and m8.
Since are alternate exterior angles, they are congruent. So, .
Answer:
Since are corresponding angles, they are congruent. So, .
Answer:
Example 1-1a

6. In the figure, m || n and t is a transversal. If find m5 and m1.
Answer:
Example 1-1b

7. Since are complementary, .
Multiple-Choice Test Item If and D and E are complementary, what is mE?
A 53° B 37° C 127° D 7°
Read the Test Item
Since are complementary, .
Example 1-2a

8. Subtract 53 from each side.
Solve the Test Item
Complementary angles
Replace with 53°.
Subtract 53 from each side.
Answer: The answer is B.
Example 1-2a

9. Multiple-Choice Test Item
If and G and H are supplementary, what is mh?
A 76° B 104° C 83° D 14°
Answer: A
Example 1-2b

10. Angles PQR and STU are supplementary. If. and
Angles PQR and STU are supplementary. If and , find the measure of each angle.
Step 1 Find the value of x.
Supplementary angles
Substitution
Combine like terms.
Add 80 to each side.
Divide each side by 2.
Example 1-3a

11. Step 2 Replace x with 130 to find the measure of each angle.
Answer:
Example 1-3a

12. Angles ABC and DEF are complementary. If. and
Angles ABC and DEF are complementary. If and , find the measure of each angle.
Answer:
Example 1-3b

13. Since are corresponding angles, they are congruent.
Transportation A road crosses railroad tracks at an angle as shown. If find m6 and m5.
Since are corresponding angles, they are congruent.
Answer:
Since are supplementary angles, the sum of their measures is 180°. 180 – 131 = 49
Answer:
Example 1-4a

14. Transportation Main Street crosses Broadway Boulevard and Maple Avenue at an angle as shown. If m1 = 48°, find m3 and m4.
Answer:
Example 1-4b

15. End of Lesson 1

16. Example 1 Name Corresponding Parts Example 2 Use Congruence Statements
Example 3 Find Missing Measures
Lesson 2 Contents

17. HGI  ?
Name the corresponding parts in the congruent triangles shown. Then complete the congruence statement.
Answer: Corresponding Angles
Corresponding Sides
One congruence statement is .
Example 2-1a

18. Name the corresponding parts in the congruent triangles shown
Name the corresponding parts in the congruent triangles shown. Then complete the congruence statement.
ABC  ?
Answer:
Example 2-1b

19. If , complete each congruence statement.
Explore You know the congruence statement. You need to find the corresponding parts.
Plan Use the order of the vertices in to identify the corresponding parts.
Solve
Example 2-2a

20. M corresponds to Q, and N corresponds to P so
Answer:
M corresponds to Q, and N corresponds to P so
N corresponds to P, and O corresponds to R so
M corresponds to Q, and O corresponds to R so
Example 2-2a

21. Examine. Draw the triangles, using arcs and slash marks
Examine Draw the triangles, using arcs and slash marks to show the congruent angles and sides.
Example 2-2a

22. If , complete each congruence statement.
Answer:
Example 2-2b

23. F and C are corresponding angles. So, they are congruent. Since .
Construction A brace is used to support a tabletop. In the figure, . What is the measure of F ?
F and C are corresponding angles. So, they are congruent. Since .
Answer:
Example 2-3a

24. corresponds to . So, and are congruent. Since , .
What is the length of ?
corresponds to . So, and are congruent. Since , .
Answer:
Example 2-3a

25. a. What is the measure of B?
Art In the figure, .
a. What is the measure of B?
b. What is the length of ?
Answer: 44°
Answer: 22 in.
Example 2-3b

26. End of Lesson 2

27. Example 1 Translation in a Coordinate Plane
Example 2 Reflection in a Coordinate Plane
Example 3 Rotations in a Coordinate Plane
Lesson 3 Contents

28. A(1, 2)  (4, –5)  A(–3, 7) B(3, –5)  (4, –5)  B(–1, 0) C(9, 0)
The vertices of ABC are A(–3, 7), B(–1, 0), and C(5, 5). Graph the triangle and the image of ABC after a translation 4 units right and 5 units down.
This translation can be written as the ordered pair (4, –5). To find the coordinates of the translated image, add 4 to each x-coordinate and add –5 to each y-coordinate.
translation
4 right, 5 down
vertex
A(1, 2) 
(4, –5) 
A(–3, 7)
B(3, –5) 
(4, –5) 
B(–1, 0)
C(9, 0) 
(4, –5) 
C(5, 5)
Example 3-1a

29. The coordinates of the vertices of  ABC are A(1, 2), B(3, –5), and C(9, 0). Graph  ABC and  ABC.
Answer:
Example 3-1a

30. The vertices of DEF are D(–1, 5), E(–3, 1), and F(4, –4)
The vertices of DEF are D(–1, 5), E(–3, 1), and F(4, –4). Graph the triangle and the image of DEF after a translation 3 units left and 2 units up.
Answer:
Example 3-1b

31. M(8, 6)  M(–8, 6) N(–5, 9)  N(5, 9) O(–2, 1)  O(2, 1) P(10, 3)
The vertices of a figure are M(–8, 6), N(5, 9), O(2, 1), and P(–10, 3). Graph the figure and the image of the figure after a reflection over the y-axis.
To find the coordinates of the vertices of the image after a reflection over the y-axis, multiply the x-coordinate by –1 and use the same y-coordinate.
reflection
vertex
M(8, 6) 
M(–8, 6)
N(–5, 9) 
N(5, 9)
O(–2, 1) 
O(2, 1)
P(10, 3) 
P(–10, 3)
Example 3-2a

32. The coordinates of the vertices of the reflected figure are M(8, 6), N(–5, 9), O(–2, 1) and P(10, 3). Graph the figure and its image.
Answer: –8 –4 4 8
Example 3-2a

33. The vertices of a figure are Q(–2, 4), R(–3, 1), S(3, –2), and T(4, 3)
The vertices of a figure are Q(–2, 4), R(–3, 1), S(3, –2), and T(4, 3). Graph the figure and the image of the figure after a reflection over the y-axis.
Answer:
Example 3-2b

34. A(4, –5)  A(–4, 5) B(2, –4)  B(–2, 4) C(1, –2)  C(–1, 2)
A figure has vertices A(–4, 5), B(–2, 4), C(–1, 2), D(–3, 1), and E(–5, 3). Graph the figure and the image of the figure after a rotation of 180°.
To rotate the figure, multiply both coordinates of each point by –1.
A(4, –5) 
A(–4, 5)
B(2, –4) 
B(–2, 4)
C(1, –2) 
C(–1, 2)
D(3, –1) 
D(–3, 1)
E(5, –3) 
E(–5, 3)
Example 3-3a

35. The coordinates of the vertices of the rotated figure are A(4, –5), B(2, –4), C(1, –2), D(3, –1), and E(5, –3). Graph the figure and its image.
Answer:
Example 3-3a

36. A figure has vertices A(2, –1), B(3, 4), C(–3, 4), D(–5, –1), and E(1, –4). Graph the figure and the image of the figure after a rotation of 180°.
Answer:
Example 3-3b

37. End of Lesson 3

38. Example 1 Find Angle Measures Example 2 Classify Quadrilaterals
Lesson 4 Contents

39. Find the value of x. Then find each missing angle measure.
Words The sum of the measures of the angles is 360°.
Variable Let mQ, mR, mS, and mT represent the measures of the angles.
Example 4-1a

40. Angles of a quadrilateral
Equation
Angles of a quadrilateral
Substitution
Combine like terms.
Subtract 185 from each side.
Simplify.
Divide each side by 5.
Answer: The value of x is 35. So, and .
Example 4-1a

41. Find the value of x. Then find each missing angle measure.
Answer:
Example 4-1b

42. Classify the quadrilateral using the name that best describes it.
The quadrilateral has one pair of opposite sides parallel.
Answer: It is a trapezoid.
Example 4-2a

43. Classify the quadrilateral using the name that best describes it.
The quadrilateral has both pairs of opposite sides parallel and congruent.
Answer: It is a parallelogram.
Example 4-2a

44. Classify the quadrilateral using the name that best describes it.
The quadrilateral has four congruent sides and four right angles.
Answer: It is a square.
Example 4-2a

45. Classify each quadrilateral using the name that best describes it.
Answer: rectangle
Answer: trapezoid
Answer: parallelogram
Example 4-2b

46. End of Lesson 4

47. Example 1 Find Areas of Parallelograms
Example 2 Find Areas of Triangles
Example 3 Find Area of a Trapezoid
Example 4 Use Area to Solve a Problem
Lesson 5 Contents

48. Find the area of the parallelogram.
The base is 3 meters. The height is 3 meters.
Area of a parallelogram
Replace b with 3 and h with 3.
Multiply.
Answer: The area is 9 square meters.
Example 5-1a

49. Find the area of the parallelogram.
The base is 4.3 inches. The height is 6.2 inches.
Area of a parallelogram
Replace b with 4.3 and h with 6.2.
Multiply.
Answer: The area is square inches.
Example 5-1a

50. Find the area of each parallelogram.b.
Answer: 12 cm2
Answer: 1.95 ft2
Example 5-1b

51. Find the area of the triangle.
The base is 3 meters. The height is 4 meters.
Area of a triangle
Replace b with 3 and h with 4.
Multiply.
Multiply.
Answer: The area of the triangle is 6 square meters.
Example 5-2a

52. Find the area of the triangle.
The base is 3.9 feet. The height is 6.4 feet.
Area of a triangle
Replace b with 3.9 and h with 6.4.
Example 5-2a

53. Answer: The area of the triangle is 12.48 square feet.
Multiply.
Multiply.
Answer: The area of the triangle is square feet.
Example 5-2a

54. Find the area of each triangle.b.
Answer:
Answer:
Example 5-2b

55. a with and b with . Find the area of the trapezoid.
The height is 6 meters.
The bases are meters and meters.
Area of a trapezoid
Replace h with 6 and
a with and b with .
Example 5-3a

56. Divide out the common factors.
Simplify.
Answer: The area of the trapezoid is square meters.
Example 5-3a

57. Find the area of the trapezoid.
Answer:
Example 5-3b

58. Painting A wall that needs to be painted is 16 feet wide and 9 feet tall. There is a doorway that is 3 feet by 8 feet and a window that is 6 feet by feet. What is the area to be painted?
To find the area to be painted, subtract the areas of the door and window from the area of the entire wall.
Example 5-4a

59. Answer: The area to be painted is 144 – 24 – 33 or 87 square feet.
Area of the wall
Area of the door
Area of the window
Answer: The area to be painted is 144 – 24 – 33 or 87 square feet.
Example 5-4a

60. Gardening A garden needs to be covered with fresh soil
Gardening A garden needs to be covered with fresh soil. The garden is 12 feet wide and 15 feet long. A rectangular concrete path runs through the middle of the garden and is 3 feet wide and 15 feet long. Find the area of the garden which needs to be covered with fresh soil.
Answer:
Example 5-4b

61. End of Lesson 5

62. Example 1 Classify Polygons Example 2 Measures of Interior Angles
Example 3 Find Angle Measure of a Regular Polygon
Lesson 6 Contents

63. Answer: It is a pentagon.
Classify the polygon.
This polygon has 5 sides.
Answer: It is a pentagon.
Example 6-1a

64. Answer: It is a heptagon.
Classify the polygon.
This polygon has 7 sides.
Answer: It is a heptagon.
Example 6-1a

65. Classify each polygon. a. b. Answer: hexagon Answer: heptagon
Example 6-1b

66. A quadrilateral has 4 sides. Therefore, .
Find the sum of the measures of the interior angles of a quadrilateral.
A quadrilateral has 4 sides. Therefore, .
Replace n with 4.
Simplify.
Answer: The sum of the measures of the interior angles of a quadrilateral is 360°.
Example 6-2a

67. Find the sum of the measures of the interior angles of a pentagon.
Answer: 540°
Example 6-2b

68. The sum of the measures of the interior angles is 1080°.
Traffic Signs A stop sign is a regular octagon. What is the measure of one interior angle in a stop sign?
Step 1 Find the sum of the measures of the angles. An octagon has 8 sides. Therefore, .
Replace n with 8.
Simplify.
The sum of the measures of the interior angles is 1080°.
Step 2 Divide the sum by 8 to find the measure of one angle.
Answer: So, the measure of one interior angle in a stop sign is 135°.
Example 6-3a

69. Picnic Table A picnic table in the park is a regular hexagon
Picnic Table A picnic table in the park is a regular hexagon. What is the measure of one interior angle in the picnic table?
Answer: 120°
Example 6-3b

70. End of Lesson 6

71. Example 1 Find the Circumference of a Circle
Example 2 Use Circumference to Solve a Problem
Example 3 Find Areas of Circles
Lesson 7 Contents

72. [  ] 12 Find the circumference of the circle to the nearest tenth.
Circumference of a circle
Replace d with 12.
Simplify. This is the exact circumference.
To estimate the circumference, use a calculator.
ENTER
2nd
[  ] 12
Answer: The circumference is about 37.7 meters.
Example 7-1a

73. Find the circumference of the circle to the nearest tenth.
Circumference of a circle
Replace r with 7.1.
Simplify. Use a calculator.
Answer: The circumference is about 44.6 meters.
Example 7-1a

74. Find the circumference of each circle to the nearest tenth.b.
Answer: ft
Answer: cm
Example 7-1b

75. Circumference of a circle
Landscaping A landscaper has a tree whose roots form a ball-shaped bulb with a circumference of about 110 inches. How wide will the landscaper have to dig the hole in order to plant the tree?
Explore You know the circumference of the roots of the tree. You need to know the diameter of the hole to be dug.
Plan Use the formula for the circumference of a circle to find the diameter.
Solve
Circumference of a circle
Replace C with 110.
Divide each side by .
Example 7-2a

76. Simplify. Use a calculator.
Answer: The diameter of the hole should be at least 35 inches.
Examine Check the reasonableness of the solution by replacing d with 35 in .
Circumference of a circle
Replace d with 35.
Simplify. Use a calculator.
The solution is reasonable.
Example 7-2a

77. Swimming Pool A circular swimming pool has a circumference of 24 feet
Swimming Pool A circular swimming pool has a circumference of 24 feet. Matt must swim across the diameter of the pool. How far will Matt swim?
Answer: about 7.6 ft
Example 7-2b

78. Find the area of the circle. Round to the nearest tenth.
Area of a circle
Replace r with 11.
Evaluate .
Use a calculator.
Answer: The area is about square feet.
Example 7-3a

79. Find the area of the circle. Round to the nearest tenth.
Area of a circle
Replace r with 4.15.
Evaluate .
Use a calculator.
Answer: The area is about 54.1 square centimeters.
Example 7-3a

80. Find the area of each circle. Round to the nearest tenth.b. a.
Answer:
Answer:
Example 7-3b

81. End of Lesson 7

82. Example 1 Find Area of Irregular Figures
Example 2 Use Area of Irregular Figures
Lesson 8 Contents

83. Find the area of the figure to the nearest tenth.
Explore You know the dimensions of the figure. You need to find its area.
Plan Solve a simpler problem. First, separate the figure into a triangle, square, and a quarter-circle. Then find the sum of the areas of the figure.
Example 8-1a

84. Find the area of the figure to the nearest tenth.
Solve Area of Triangle
Area of a triangle
Replace b with 2 and h with 4.
Simplify.
Example 8-1a

85. Find the area of the figure to the nearest tenth.
Solve Area of Square
Area of a square
Replace b and h with 2.
Simplify.
Example 8-1a

86. Find the area of the figure to the nearest tenth.
Solve Area of Quarter-circle
Area of a quarter-circle
Replace r with 2.
Simplify.
Answer: The area of the figure is or about 11.1 square inches.
Example 8-1a

87. Find the area of the figure to the nearest tenth.
Answer:
Example 8-1b

88. Step 1 Find the area to be carpeted. Area of Rectangle
Carpeting Carpeting costs $2 per square foot. How much will it cost to carpet the area shown?
Step 1 Find the area to be carpeted.
Area of Rectangle
Area of a rectangle
Replace b with 14 and h with 10.
Simplify.
Example 8-2a

89. Carpeting Carpeting costs $2 per square foot
Carpeting Carpeting costs $2 per square foot. How much will it cost to carpet the area shown?
Area of Square
Area of a square
Replace b and h with 3.
Simplify.
Example 8-2a

90. Replace b with 14 and h with 12.
Carpeting Carpeting costs $2 per square foot. How much will it cost to carpet the area shown?
Area of Triangle
Area of a triangle
Replace b with 14 and h with 12.
Simplify.
The area to be carpeted is or 233 square feet.
Example 8-2a

91. Step 2 Find the cost of the carpeting.
Carpeting Carpeting costs $2 per square foot. How much will it cost to carpet the area shown?
Step 2 Find the cost of the carpeting.
Answer: So, it will cost $466 to carpet the area shown.
Example 8-2a

92. Painting One gallon of paint is advertised to cover 100 square feet of wall surface. About how many gallons will be needed to paint the wall shown below?
Answer: about 4 gallons
Example 8-2b

93. End of Lesson 8

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