Download 2 Dimensional Geometry – Chapter Review

2D Geometry: Chapter Questions
1.
What are some of the special relationships between angles?
2.
What are perimeter and circumference and how are they relevant in daily life?
3.
How is the formula for area of a parallelogram related to area of a rectangle?
4.
How is the formula for area of a triangle related to area of a rectangle?
5.
How do you find the area of irregular shapes?
6.
How do you find the area of shaded regions?
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2D Geometry: Chapter Problems
Special Pairs of Angles
Classwork
1. If two angles have the same measure are they congruent?
2. Angle 1 and Angle 2 are complementary. If ∠1 has a measure of 42o what would the measure of ∠2
be?
3. Determine the measure of ∠3 in the figure below.
63°
4.
5.
6.
7.
3
What is the complement of 72o?
If Angle 1 measures 48o and Angle 2 measures 42o are these angles complementary?
Angle R and S are supplementary. If Angle R measures 115o what is the measure of Angle S?
What is the measure of angle 4?
4 52°
8.
9.
10.
11.
What is the supplement of 72o?
If Angle X measures 53o and Angle Y measures 127o are these angles supplementary?
What is the complement of 68o? Supplement?
Using what you know about vertical angles, find the measure of the missing angles. You must be
able to explain why.
3
65°
Angle
Measurement
Reason
∠1
65°
Given
∠2
2 4
∠3
∠4
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12. Parallel lines m and n are cut by a transversal l. If the measure of angle 5 is 37o determine the
measures of the remaining angles. You must be able to explain why.
Angle
Measurement
Reason
∠1
∠2
∠3
∠4
∠6
∠7
∠8
Homework
13. Angle M has a measure of 36o and Angle N has a measure of 24o. Are these two angles congruent?
14. Angle R and Angle S are complementary. If ∠𝑆 has a measure of 26o what would be the measure of
∠𝑅?
15. Determine the measure of the unknown angle 5 in the figure below.
18°
5
16.
17.
18.
19.
What is the complement of 9o?
If Angle X measures 53o and Angle Y measures 127o are these angles complimentary?
Angle A and B are supplementary. If ∠𝐴 measures 97o what is the measure of ∠𝐵?
What is 𝑚∠6?
6 110°
20. What is the supplement of 9o?
21. If 𝑚∠1 = 48o and 𝑚∠2 = 42o are these angles supplementary?
22. What is the complement of 84o? Supplement?
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23. Using what you know about vertical angles, find the measure of the missing angles. You must be
able to explain why.
2
3
40°
1
24. Parallel lines m and n are cut by a transversal l. If the measure of angle 5 is 68o determine the
measures of the remaining angles. You must be able to explain why.
Perimeter and Circumference
Classwork
Find the perimeter of each figure.
25.
26.
27.
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28.
In #29-33, use 3.14 as your value of 𝜋.
29. Determine the circumference of a circle with radius 5 cm.
30. Determine the circumference of a circle with radius 2.3 m.
31. Determine the circumference of a circle whose diameter is 8 m.
32. Determine the circumference of a circle whose diameter is 24.6 cm.
33. What is the diameter of a circle whose circumference is 47.1 m?
In #34-38, leave your answer in terms of 𝜋.
34. Determine the circumference of a circle with radius of 4 m.
35. Determine the circumference of a circle with radius of 3.6 cm.
36. Determine the circumference of a circle whose diameter is 10 m.
37. Determine the circumference of a circle whose diameter is 31.5 cm.
38. What is the radius of a circle whose circumference is 10.5𝜋 in?
Homework
Find the perimeter of each figure.
39.
40.
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41.
42.
In #43-47, use 3.14 as your value of 𝜋.
43. Determine the circumference of a circle whose radius is 3.5 m.
44. Determine the circumference of a circle whose radius is 32 inches.
45. Determine the circumference of a circle whose diameter is 32 inches.
46. Determine the circumference of a circle with diameter 8 feet.
47. What is the radius of a circle whose circumference is 13.188 cm?
In #48-52, leave your answer in terms of 𝜋.
48. Determine the circumference of a circle with radius of 5.5 m.
49. Determine the circumference of a circle with radius of 24 in.
50. Determine the circumference of a circle whose diameter is 24 m.
51. Determine the circumference of a circle whose diameter is 12 feet.
52. What is the radius of a circle whose circumference is 11.4𝜋 in?
Rectangles & Parallelograms
Classwork
Find the area of each figure.
53.
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54.
55.
56.
57.
58. The length of a rectangle is 10 in and the width is 3 in. What is its area?
59. How do you find the area of a rectangle?
60. How do you find the area of a parallelogram?
61. What is true of the base and height of a parallelogram?
62. What is the base of a parallelogram whose height is 4 cm and whose area is 44 cm 2?
Homework
Find the area of each figure.
63.
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64.
65.
66.
67.
68. The length of a rectangle is 17 m and the width is 6 m. What is its area?
69. How do you find the area of a rectangle?
70. How do you find the area of a parallelogram?
71. What is true of the base and height of a parallelogram?
72. What is the base of a parallelogram whose height is 7 cm and whose area is 63 cm 2?
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Triangles & Trapezoids
Classwork
Find the area of each figure.
73.
74.
75.
76.
77.
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78. Find the height of a triangle whose area is 20 in2 and whose base is 8 in.
Homework
Find the area of each figure.
79.
80.
81.
82.
83.
84. Find the height of a triangle whose area is 25 in2 and whose base is 5 in.
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Circles
Classwork
Find the area of each figure. Use 3.14 as your value of 𝜋.
85.
86.
In #87-90, use 3.14 as your value of 𝜋.
87. A goat is tied to a 5 m rope. What is the area of the grass he can eat?
88. What is the area of a circle with a radius of 7 cm?
89. What is the area of a circle with a diameter of 16 yd?
90. How do you find area of a circle when given radius?
Find the area of each figure. Leave your answer in terms of 𝜋.
91.
9 cm
92.
24 in.
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In #93-95, leave your answer in terms of 𝜋.
93. A goat is tied to an 8 m rope. What is the area of the grass he can eat?
94. What is the area of a circle with a radius of 10 cm?
95. What is the area of a circle with a diameter of 22 yd?
Homework
Find the area of each figure. Use 3.14 as your value of 𝜋.
96.
97.
In #98-101, use 3.14 as your value of 𝜋.
98. A circular sprinkler has a radius of 4 m. How much grass can the sprinkler water?
99. What is the area of a circle with a radius of 5 in?
100.
What is the area of a circle with a diameter of 14 m?
101.
How do you find area of a circle when given diameter?
Find the area of each figure. Leave your answer in terms of 𝜋.
102.
5 cm
103.
30 in.
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In #104-106, leave your answer in terms of 𝜋.
104.
A circular sprinkler has a radius of 8 m. How much grass can the sprinkler water?
105.
What is the area of a circle with a radius of 13 in?
106.
What is the area of a circle with a diameter of 18 m?
Irregular Figures
Classwork
Find the area of each figure.
107.
108.
109.
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110.
Homework
Find the area of each figure.
111.
112.
113.
Use 3.14 as your value of 𝜋.
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114.
Shaded Figures
Classwork
Find the area of each shaded region. If needed, use 3.14 as your value of 𝜋.
115.
116.
117.
Homework
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Find the area of each shaded region. If needed, use 3.14 as your value of 𝜋.
118.
119.
120.
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2 Dimensional Geometry – Chapter Review
1. Match each vocabulary word with its definition given to the right.
_____ Vertical Angles
_____ Alternate Interior Angles
_____ Alternate Exterior Angles
_____ Adjacent Angles
_____ Corresponding Angles
_____ Same-side Interior Angles
a) Two angles that are next to each other and have a
common ray
b) When two lines are crossed by a transversal, the pairs of
angles on opposite sides of the transversal and outside of
the two lines.
c) When two lines are crossed by a transversal, the pairs of
angles on opposite sides of the transversal and inside of
the two lines.
d) When two lines are crossed by a transversal, the pairs of
angles that are on the same side of the transversal and in
the same location at each intersection.
e) When two lines are crossed by a transversal, the pairs of
angles on the same side of the transversal and inside of
the two lines.
f) Two angles that are opposite each other when two lines
intersect.
2. ABC measures 62. Its complementary angle measures:
a. 28
b. 62
c. 208
d. 298
3. In the figure below, j || k and 𝑚∠3 = 82°. Find the measurements of each remaining angle.
7
j
8
6
1
2 5 3
4
k
4. What is the perimeter of an equilateral triangle with 6-inch sides?
a. 18 inches
b. 24 inches
c. 27 inches
d. 36 inches
5. A soccer field has a length of 100 yards and a width of 60 yards. What is the perimeter of the field?
a. 160 yards
b. 220 yards
c. 320 yards
d. 6000 square yards
6. What is the area of a rectangle with length 36 cm and width 4 cm?
a. 72 cm
b. 144 cm
c. 144 cm2
d. 72 cm2
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7. In which picture(s) below are ∠1 & ∠2 considered to be congruent angles? Circle all that apply.
a.
d.
m || n
2
1
n
2
1
e.
m
b.
m || n
1 2
1
m
n
2
f.
c.
2
1
1
2
8. What is the area of the figure pictured to the right?
9. A triangle has an area of 36 yd2 and a base of 3 yd, what is the height of the triangle?
10. Find the area of the figure to the right.
11. Find the height of the parallelogram with a length of 50 in and an area of 275 in 2.
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12. Jewell wants to tile her kitchen floor with tiles that are 1.5 ft2. If her kitchen is 18 ft by 12 ft, how many
tiles will she need?
13. Find the area of the figure to the right.
14. Find the area of the shaded region to the right.
15. Find the length of the base of a triangle with an area of 20 cm 2 and a height of 5 cm.
16. A circular carpet covers an area of 201 cm 2. What is the radius of the carpet? Use 3.14 as your
value of 𝜋.
17. What is the diameter of a circle whose area is 38.04 in2?
18. In order to decorate for a party, you need to cover 3 rectangular tables with tablecloth. The tables
measure 36 in by 72 in. If you want 2 inches overhang on each side, how many square inches of
fabric would it take to cover the 3 tables?
19. The doors in your home need to be painted. Each door measures 8 ft x 3 ft. You are amazed that you
have 22 doors to paint! A can of paint can cover 400 square feet. How many cans of paint will you
need to purchase?
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20. The diagram below shows one side of Joe’s shed. In order for him to install siding on to both sides of
his shed, he needed to calculate its total area. Below is a series of steps that Joe took to complete
his calculation. Is he correct? If so, explain how all of his steps are correct. If not, explain what he
should have done calculate the area correctly.
Joe’s calculations:
𝑇𝑜𝑡𝑎𝑙 𝐴𝑟𝑒𝑎 = 𝑡𝑜𝑝 𝑎𝑟𝑒𝑎 + 𝑏𝑜𝑡𝑡𝑜𝑚 𝑎𝑟𝑒𝑎
1
𝑇𝑜𝑡𝑎𝑙 𝐴𝑟𝑒𝑎 = (12)(4) + 12(8)
2
𝑇𝑜𝑡𝑎𝑙 𝐴𝑟𝑒𝑎 = 24 + 96
𝑇𝑜𝑡𝑎𝑙 𝐴𝑟𝑒𝑎 = 120 𝑓𝑡 2
Extended Constructed Response - Solve the problem, showing all work. Partial credit may be given.
21. Your parents have offered to pay you to wallpaper the kitchen walls. The wallpaper that you want
costs $12 per square foot. The rectangular kitchen measures 16 ft x 12 ft, with a height of 8 ft. One
doorway measures 8 ft x 3 ft and two windows measures 3 ft x 5 ft.
a. Calculate the area of the kitchen walls, pretending there are no windows or doorways .
b. What is the area of the kitchen walls, when the doorways and windows are taken into
account?
c. What will be the minimum cost to wallpaper the kitchen?
22. You have been asked to design a flower garden with an area of 52 ft 2. The client has no
preference for the shape of the garden. It’s all up to you!
a. If it will cost $3/ft to put flowers in the garden, determine the total cost.
b. Provide two possible dimensions of the flower garden.
c. Identify the shape that has the smallest perimeter.
23. A decorator wants to install two picture windows, measuring 12 ft. x 8 ft. The windows will be
placed on a wall measuring 72 ft x 12 ft.
a. What is the area of the wall, without windows?
b. How much area will two windows cover?
c. Wallpaper, costing $6/square foot, will be purchased to cover the wall, once the windows
are installed. What will be the least amount of money spent on the wallpaper?
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24. A bag of grass seed can cover 120 square feet. Ms. Peabody wants to seed her front and back
yard. The front yard is rectangular in shape, measuring 10 ft x 8 ft. Her back yard has an irregular
shape and is pictured below.
4
A. Calculate the square footage of Ms. Peabody’s front yard.
B. Calculate the square footage of Ms. Peabody’s back yard.
C. How many bags of grass seed will Ms. Peabody need to purchase?
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Answer Key
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
Yes
48°
27°
18°
Yes
65°
128°
108°
Yes
Complement = 22°, Supplement = 112°
Multiple answers for the reasons. Some
examples:
∠2 = 65° since it's vertical to a 65° angle
and vertical angles are congruent
∠3 = 115° since it's supplemental to ∠2 and
to the 65° angle
∠4 = 115° since it's vertical to ∠3 or
supplemental to ∠2
Multiple answers for the reasons. Some
examples:
∠1 = 37° since it's an alternate exterior
angle to ∠5 and alternate exterior angles are
congruent
∠2 = 143° since it's supplemental to ∠1
∠3 = 37° since it's vertical to ∠5
∠4 = 143° since it's supplemental to ∠3 and
∠5 or since it corresponds to ∠2
∠6 = 143° since it's vertical to ∠4 or since
it's an alternate interior angle to ∠2
∠7 = 37° since it's vertical to ∠1 or since it's
an alternate interior angle to ∠3
∠8=143° since it's vertical to ∠2 or an
alternate exterior angle to ∠4 or corresponds
to ∠6
No; not congruent
64°
72°
81°
No, they are supplementary
83°
70°
171°
No, they are complementary
Complement = 6°, Supplement = 96°
Multiple answers for the reasons. Some
examples:
∠2 = 40° since it's vertical to a 40° angle and
vertical angles are congruent
∠3 = 140° since it's the supplement of ∠2
∠1 = 140° since it's vertical to ∠3 or
supplement of the 40° angle
Multiple answers for the reasons. Some
examples:
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25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
∠1 = 68° since it's an alternate exterior
angle to ∠5
∠2 = 112° since it's supplemental to ∠1
∠3 = 68° since it's vertical to ∠5 or
corresponds to ∠1
∠4 = 112° since it's supplemental to ∠3 and
∠5
∠6 = 112° since it's supplemental to ∠5 or
vertical to ∠4 or an alternate interior angle to
∠2
∠7 = 68° since it's vertical to ∠1 or an
alternate interior angle to ∠3
∠8 = 112° since it's vertical to ∠2 or an
alternate exterior angle to ∠4
10 m
16 ft
16 in
24 m
31.4 cm
14.444m
25.12 m
77.244 cm
15 m
8𝜋 m
7.2𝜋 cm
10𝜋 m
31.5𝜋 cm
𝑟 = 5.25 𝑖𝑛.
42 cm
100 m
18 ft
64 m
21.98 m
200.96 in
100.48 in
25.12 ft
2.1 cm
11𝜋 m
48𝜋 in.
24𝜋 m
12𝜋 ft
𝑟 = 5.7 𝑖𝑛.
24 cm²
121 m²
84 in²
108 m²
80 cm²
30 in²
Area of a rectangle = length x width
Area of a parallelogram = base x height
The base and height always form a right
angle
11 cm
28 cm²
81 m²
50 in²
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66.
67.
68.
69.
70.
71.
91 m²
143 cm²
102 m²
Area of a rectangle = length x width
Area of a parallelogram = base x height
The base and height always form a right
angle
72. 9 cm
73. 60 ft²
74. 78 m²
75. 30 in²
76. 84 in²
77. 195 m²
78. 5 in
79. 90 ft²
80. 91 m²
81. 56 in²
82. 105 in²
83. 135 m²
84. 10 in
85. 314 m²
86. 379.94 in²
87. 78.5 m²
88. 153.86 cm²
89. 200.96 yd²
90. Area of a circle = π x radius²
91. 81𝜋 cm2
92. 144𝜋 in2
93. 64𝜋 m2
94. 100𝜋 cm2
95. 121𝜋 yd2
96. 254.34 m²
97. 254.34 in²
98. 50.24 m²
99. 78.5 in²
100.
153.86 m²
101.
Divide the diameter by 2 to find the
radius. Then, area of a circle = π x radius²
102.
25𝜋 cm2
103.
225𝜋 in2
104.
64𝜋 m2
105.
169𝜋 in2
106.
81𝜋 m2
107.
10,500 units²
108.
14,300 units²
109.
88 cm²
110.
85 mm²
111.
12,200 units²
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112.
113.
114.
115.
116.
117.
118.
119.
120.
15,650 units²
238.3125 in²
58 ft²
28.26 in²
43 units²
40 ft²
42.14 cm²
43 units²
96 m²
Unit Review Answers
1. F, C, B, A, D, E
2. A
3. 𝑚∠1 = 𝑚∠4 = 𝑚∠5 = 𝑚∠8 = 98°
𝑚∠2 = 𝑚∠6 = 𝑚∠7 = 82°
4. A
5. C
6. C
7. B, C, and F
8. 12 𝑢𝑛𝑖𝑡𝑠 2
9. 24 yd.
10. 27 𝑐𝑚2
11. 5.5 in.
12. 144 tiles
13. 44 𝑐𝑚2
14. 20 𝑐𝑚2
15. 8 cm
16. 8 in
17. 6.96 in
18. 9120 in2
19. 2
20. No, Joe is not correct. Since 2 sides of the
shed need to have siding installed, the total
area is 240 ft2, which is twice Joe’s answer.
He didn’t multiply by 2.
21. A. 448 ft2
B. 394 ft2
C. $4728
22. A. $156
B. Answers will vary
C. Answers will vary
23. A. 864 ft.
B. 192 ft.
C.$4,032
24. A. 80 ft2
B. 57.57 ft2
C. 2 bags
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