Triangles Chapter Problems

Classify the Triangles by Sides or Angles

Class Work

In problems #1-10, choose the most appropriate description for the given triangle. (Equilateral, Scalene, Isosceles, Obtuse, Acute, Right, Equiangular)

  1. Side lengths: 3 cm, 4 cm, 5 cm
  2. Side lengths: 3 cm, 3 cm, 4 cm
  3. Side lengths: 2 cm, 3 cm, 2 cm
  4. Side lengths: 5 cm, 5 cm, 5 cm
  5. Side lengths: 2 cm. 3 cm, 4 cm
  6. Angle Measures: 30, 60, 90
  7. Angle Measures: 60, 60, 60
  8. Angle Measures: 92, 37, 51
  9. Angle Measures: 88, 67, 25
  10. Angle measures: 37, 39, 104

Complete the statement using ALWAYS, SOMETIMES, and NEVER.

  1. An isosceles triangle is ______a scalene triangle.
  2. An equilateral triangle is ______an isosceles triangle.
  3. An isosceles triangle is ______an equilateral triangle.
  4. An acute triangle is ______an equiangular triangle.
  5. An isosceles triangle is ______a right triangle.

For #16-20, classify the triangles by Sides & Angles

Classify the Triangles by Sides or Angles

Homework

In problems #21-30, choose the most appropriate description for the given triangle. (Equilateral, Scalene, Isosceles, Obtuse, Acute, Right, Equiangular)

  1. Side lengths: 5 cm, 6 cm, 7 cm
  2. Side lengths: 2 cm, 2 cm, 3 cm
  3. Side lengths: 3 cm, 3 cm, 3 cm
  4. Side lengths: 3 cm, 4 cm, 4 cm
  5. Side lengths: 4 cm, 3 cm, 2 cm
  6. Angle Measures: 60, 60, 60
  7. Angle Measures: 60, 30, 90
  8. Angle Measures: 33, 52, 95
  9. Angle Measures: 37, 43, 100
  10. Angle measures: 25, 67, 88

Complete the statement using ALWAYS, SOMETIMES, and NEVER.

  1. A scalene triangle is ______an equilateral triangle.
  2. An equilateral triangle is ______an obtuse triangle.
  3. An isosceles triangle is ______an acute triangle.
  4. An equiangular triangle is ______a right triangle.
  5. A right triangle is ______an isosceles triangle.

For #36-40, classify the triangles by Sides & Angles

Triangle Sum & Exterior Angle Theorems

Class Work

In the given triangles, solve for the missing variable(s).

Triangle Sum & Exterior Angle Theorems

Homework

In the given triangles, solve for the missing variable(s).

PARCC type question

66. Proof of Triangle Sum Theorem: Complete the proof by filling in the missing reasons
with the “reasons bank” to the right. Some reasons may be used more than once, and some may not be used at all.

Given: j || k, is a straight angle

Prove:

Statements / Reasons
1. j || k
is a straight angle / 1.
2. / 2.
3. / 3.
4. / 4.
5. / 5.
6. / 6.
7. / 7.

Inequalities in Triangles

Classwork

For each triangle list the sides from greatest to smallest #67-69.

67.68.69.

For each triangle list the angles in order from greatest to smallest #70-72.

70.71.72.

Will the three lengths given make a triangle?

73. 2, 3, and 4

74. 1, 3, and 4

75. 5, 6, and 7

76. 16, 8, and 7

77. 20, 10, and 10

78. 8x, 7x, and 14x

Given the lengths of two sides of a triangle, what lengths could the third side, x, have?

79. 12 and 14

80. 15 and 6

81. 22 and 22

82. 9 and 12

83. 8y and 10y

Inequalities in Triangles

Homework

For each triangle list the sides from greatest to smallest #84-86.

84.85.86.

For each triangle list the angles in order from greatest to smallest #87-89.

87.88.89.

List the sides in order from shortest to longest #90-91.

90.91.

Will the three lengths given make a triangle?

92. 21, 34, and 49

93. 11, 31, and 44

94. 8, 6, and 5

95. 12, 5, and 7

96. 20, 30, and 11

97. 9x, 17x, and 26x

Given the lengths of two sides of a triangle, what lengths could the third side, x, have?

98. 10 and 21

99. 19 and 8

100. 30 and 30

101. 5 and 15

102. 4y and 14y

Similar Triangles

Classwork

103.Determine if the triangles are similar. If so, write a similarity statement & state the similarity postulate or theorem.

a.

b. c.

PARCC type questions: Complete the proof by filling in the missing reasons with the “reasons bank” to the right. Some reasons may not be used at all.

104.Given:

Prove:

Statements / Reasons
1. / 1.
2. / 2.
3. / 3.
4. / 4.

105.Given: , , , , and

Prove:

Statements / Reasons
1. , , , , and / 1.
2. / 2.
3. / 3.
4. / 4.

106. Given: ,

Prove:

Statements / Reasons
1. , / 1.
2. / 2.

107. Given:

Prove:

Statements / Reasons
1. / 1.
2. / 2.

In problems 108-110, determine if .

108.109.

110.

PARCC type questions:

Solve for y.

111.112.

PARCC type question:Complete the proof by filling in the missing reasons with the “reasons bank” to the right. Some reasons may not be used at all.

113. Prove the Side Splitter Theorem

Given ║

Prove

Statements / Reasons
1. ║ / 1.
2. / 2.
3. / 3.
4. / 4.
5. / 5.
6. CA = CB + BA
CE = CD + DE / 6.
7. / 7.
8. / 8.
9. / 9.
10. / 10.

Similar Triangles

Homework

114. Determine if the triangles are similar. If so, write a similarity statement & state the similarity postulate or theorem.

a.b. c.

PARCC type questions:Complete the proof by filling in the missing reasons with the “reasons bank” to the right. Some reasons may not be used at all.

115. Given:

Prove:

Statements / Reasons
1. ║ / 1.
2. / 2.
3. / 3.
4. / 4.

116. Given: , , ,

Prove:

Statements / Reasons
1. , , , / 1.
2. / 2.
3. / 3.
4. / 4.

117. Given: ,

Prove:

Statements / Reasons
1. , / 1.
2. / 2.

In problems 118-120, determine if .

118.119.

120.

PARCC type questions:

Solve for y.

121.122.

123.

PARCC type question:

124. Prove the Converse to the Side Splitter Theorem: Complete the proof by filling in the missing reasons with the “reasons bank” to the right. Some reasons may not be used at all.

Given

Prove ║

Statements / Reasons
1. / 1.
2. / 2.
3. / 3.
4. / 4.
5. CA = CB + BA
CE = CD + DE / 5.
6. / 6.
7. / 7.
8. / 8.
9. / 9.
10. ║ / 10.

Applications

Classwork

  1. You want to know the approximate height of your school building. You place a mirror on the ground and stand where you can see the top of the building in the mirror. How tall is your school? The mirror is 30 feet from the base of the school. You are 36 inches from the mirror and your eyes are 5 feet above the ground.
    Round your answer to the nearest whole number.
  1. You want to know the approximate height of a very tall pine tree. You place a mirror on the ground and stand where you can see the top of the tree in the mirror. How tall is the tree? The mirror is 24 feet from the base of the tree. You are 24 inches from the mirror and your eyes are 6 feet above the ground. Round your answer to the nearest tenth.
  1. To find the distance d across a lake, you locate the points as shown. Find the value of d. Round your answer to the nearest tenth.

Applications

Homework

  1. You want to know the approximate height of your house. You place a mirror on the ground and stand where you can see the top of your house in the mirror. How tall is your house? The mirror is 25 feet from the base of your house. You are 60 inches from the mirror and your eyes are 5 feet 6 inches above the ground. Round your answer to the nearest tenth.
  1. You want to know the approximate height of a tall oak tree. You place a mirror on the ground and stand where you can see the top of the tree in the mirror. How tall is the tree? The mirror is 24 feet from the base of the tree. You are 36 inches from the mirror and your eyes are 5 feet above the ground. Round your answer to the nearest tenth.
  1. To find the distance d across a lake, you locate the points as shown. Find the value of d. Round your answer to the nearest tenth.


Triangles Review

Multiple Choice

  1. Identify the triangles by sides and angles

a.scalene, acute

b.isosceles, obtuse

c.scalene, obtuse

d.equilateral, equiangular

  1. Angle measures of a triangle are given, find the value of x.

a.24 A triangle’s angles are:

b.28 m 2x - 1

c.32 m x + 9

d.30 m 3x + 4

  1. Classify the triangle by sides and angles.

a.scalene, obtuse

b.isosceles, acute

c.scalene, acute

d.isosceles, obtuse

  1. Using the figure at right, list the segments from least to greatest.
  2. cannot be determined
  1. Use the diagram to find the value of x.
  1. 130
  2. 125
  3. 120
  4. 110
  1. Which of the following values cannot be the third side of a triangle if two of the sides are 14 and 20?
  1. 18
  2. 20
  3. 32.5
  4. 34

7.Decide whether the triangles are similar. If so, write a similarity statement.

a.Yes,

b.Yes,

c.Yes,

d.The triangles are not similar

8.Determine if the triangles are similar. If so, state the similarity postulate or theorem.

a.Yes, by AA~

b.Yes, by SSS~

c.Yes, by SAS ~

d.The triangles are not similar

9. Determine if the triangles are similar. If so, state the similarity postulate or theorem.

a.Yes, by AA~

b.Yes, by SSS~

c.Yes, by SAS ~

d.The triangles are not similar

10.Solve for y

a.8

b.4.5

c.6

d.12

11.Solve for x

a.4

b.4.8

c.10

d.12.8

Short Constructed Response – Write the correct answer for each question.

No partial credit will be given.

12.Find the values of x and z.

13. Find the values if x & y

14. To find the distance d across a stream, you locate the points as shown. Find the value
of d.

Extended Constructed Response – Write the correct answer for each question.

Partial credit will be given.

15. Complete the proof by filling in the missing reasons with the “reasons bank” to the right. Some reasons may not be used at all.

Given ║

Prove

Statements / Reasons
1. ║ / 1.
2. / 2.
3. / 3.
4. / 4.

Geometry: Triangles~1~ NJCTL.org

Answers

1. Scalene

2. Isosceles

3. Isosceles

4. Equilateral and isosceles

5. Scalene

6. Right

7. Equiangular & acute

8. Obtuse

9. Acute

10.Obtuse

11.Never

12.Always

13.Sometimes

14.Sometimes

15.Sometimes

16.Sides: Isosceles, Angles: Acute

17.Sides: Scalene, Angles: Obtuse

18.Sides: Isosceles, Angles: Obtuse

19.Sides: Scalene, Angles: Right

20.Sides: Isosceles, Angles: Obtuse

21.Scalene

22.Isosceles

23.Equilateral and isosceles

24.Isosceles

25.Scalene

26.Equiangular & acute

27.Right

28.Obtuse

29.Obtuse

30.Acute

31.Never

32.Never

33.Sometimes

34.Never

35.Sometimes

36.Sides: Scalene, Angles: Right

37.Sides: Scalene, Angles: Obtuse

38.Sides: Isosceles, Angles: Obtuse

39.Sides: Isosceles, Angles: Acute

40.Sides: Isosceles, Angles: Acute

41.X=58°

42.X=33°

43.X=23°

44.X=30°

45.X=79°, z=144°, y=55°

46.X=12

47.X=27

48.X=76°, y=104°, z=55°

49.j = 47, k = 180, m = 155, n = 25,
p = 94, q = 61

50.r = 35, t = 145, u = 35, v = 90,
w = 125

51.a = 54, b = 36, c = 36, d = 116,
e = 152, f = 28, g = 98, h = 54

52.j = 128, k = 52, m = 52, n = 31,
p = 149, q = 31, r = 49, t = 131

53.X=61°

54.X=31

55.X=37

56.X=27

57.X=32

58.X=96°, z=151°, y=68

59.X=66°, z=88°, y=79°

60.Z=90°, y=43°, x=71°

61.X=95°, y=51°, z=39°

62.a = 127, b = 90, c = 143, d = 37,
e = 74, f = 111

63.g = 49, h = 117, j = 49, k = 76,
m = 55

64.a = 143, b = 37, c = 37, d = 121,
e = 158, f = 22, g = 108, h = 50

65.r = 90, t = 38, u = 38, v = 34,
w = 146, x = 34, y = 30, z = 150

66.

Statements / Reasons
1. j || k
is a straight angle / 1. e
2. / 2. g
3. / 3. a
4. / 4. b
5. / 5. c
6. / 6. f
7. / 7.b

67.

68.

69.

70. <K, <L, <J

71. <N, <M, <O

72. <P, <Q, <R

73. yes

74. no

75. yes

76. no

77. no

78. yes

79. 2 < x < 26

80. 9 < x < 21

81. 0 < x < 44

82. 3 < x < 21

83. 2y < x < 18y

84.

85.

86.

87. <H<I, <G

88. <K, <J, <L

89. <N, <M, <O

90.

91.

92. yes

93. no

94. yes

95. no

96. yes

97. no

98. 11< x < 31

99. 11 < x < 27

100. 0 < x < 60

101. 10 < x < 20

102. 10y < x < 18y

103.a. not similar
b. yes by SAS
c. not similar

104.

Statements / Reasons
1. / 1. c
2. / 2. a
3. / 3. d
4. / 4. e

105.

Statements / Reasons
1. , , , , and / 1. b
2. / 2. d
3. / 3. a
4. / 4. e

106.

Statements / Reasons
1. , / 1. d
2. / 2. a

107.

Statements / Reasons
1. / 1. d
2. / 2. c

108.yes

109.no

110.no

111.12

112.11.36

113.

Statements / Reasons
1. ║ / 1. c
2. / 2. d
3. / 3. b
4. / 4. e
5. / 5. g
6. CA = CB + BA
CE = CD + DE / 6. l
7. / 7. i
8. / 8. m
9. / 9. n
10. / 10. j

114.a. yes, by AA~ or SAS~
b. not similar
c. yes, by SSS~

115.

Statements / Reasons
1. ║ / 1. c
2. / 2. d
3. / 3. b
4. / 4. e

116.

Statements / Reasons
1. , , , / 1. b
2. / 2. a
3. / 3. e
4. / 4. c

117.

Statements / Reasons
1. , / 1. d
2. / 2. b

118.yes

119.yes

120.no

121.10

122. 11.25

123. 10

124.

Statements / Reasons
1. / 1. c
2. / 2. k
3. / 3. n
4. / 4. m
5. CA = CB + BA
CE = CD + DE / 5. l
6. / 6. i
7. / 7. d
8. / 8. f
9. / 9. h
10. ║ / 10. b

125.50 feet

126.72 ft

127.90 ft

128.27.5 feet

129.40 ft

130.45 ft

Geometry: Triangles~1~ NJCTL.org

Unit Review Answer Key

Geometry: Triangles~1~ NJCTL.org

1.c

2.b

3.c

4.b

5.b

6.d

7.c

8.c

9.a

10.a

11.d

12.x = 50 & z = 90

13.x = 18 & y = 22.5

14.150 ft

Geometry: Triangles~1~ NJCTL.org

15.

Statements / Reasons
1. ║ / 1. f
2. / 2. c
3. / 3. b
4. / 4. d

Geometry: Triangles~1~ NJCTL.org