triangles chapter problems 58° 58° 43°...

Geometry: Triangles ~1~ NJCTL.org

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.

11. An isosceles triangle is ___________ a scalene triangle. 12. An equilateral triangle is __________ an isosceles triangle. 13. An isosceles triangle is ___________ an equilateral triangle. 14. An acute triangle is ___________ an equiangular triangle. 15. An isosceles triangle is __________ a right triangle.

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

135°

20.19.

106°

37°

37°

18.

43°

22°

115°

17.16.

64°

58°58°


Classify the Triangles by Sides or Angles Home Work In problems #21-30, choose the most appropriate description for the given triangle. (Equilateral, Scalene, Isosceles, Obtuse, Acute, Right, Equiangular)

21. Side lengths: 5 cm, 6 cm, 7 cm 22. Side lengths: 2 cm, 2 cm, 3 cm 23. Side lengths: 3 cm, 3 cm, 3 cm 24. Side lengths: 3 cm, 4 cm, 4 cm 25. Side lengths: 4 cm, 3 cm, 2 cm

26. Angle Measures: 60°, 60°, 60° 27. Angle Measures: 60°, 30°, 90° 28. Angle Measures: 33°, 52°, 95° 29. Angle Measures: 37°, 43°, 100° 30. Angle measures: 25°, 67°, 88°

Complete the statement using ALWAYS, SOMETIMES, and NEVER.

31. A scalene triangle is ___________ an equilateral triangle. 32. An equilateral triangle is __________ an obtuse triangle. 33. An isosceles triangle is ___________ an acute triangle. 34. An equiangular triangle is ___________ a right triangle. 35. An right triangle is __________ an isosceles triangle.

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

46°46°

40.39.

106°

47°27°

38.

36°

36°

108°

37.36.

51°

51°78°


Triangle Sum & Exterior Angle Theorems Class Work In the given triangles, solve for the missing variable(s).

Triangle Sum & Exterior Angle Theorems Home Work In the given triangles, solve for the missing variable(s).

z°y°x°

21°

39°

65°

x°

(3x-18)°

(4x-13)°

2x° (3x+23)°

49.

47.

z°

x°

(y+10)°

65° 36°

>>

>>

46.

x°

65°85°

45.44.

(4x+21)°

(x-10)°(2x+8)°

43.

x°

57°42.

x°

93°

29°

41.


PARCC type question 59. 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, ∠𝐷𝐵𝐸 is a straight angle Prove: 𝑚∠1 + 𝑚∠4 + 𝑚∠7 = 180° Statements Reasons

1. j || k

∠𝐷𝐵𝐸 is a straight angle

1.

2. 𝑚∠𝐷𝐵𝐸 = 180° 2.

3. 𝑚∠3 + 𝑚∠4 + 𝑚∠5 = 𝑚∠𝐷𝐵𝐸

3.

4. 𝑚∠3 + 𝑚∠4 + 𝑚∠5 = 180° 4.

5. ∠3 ≅ ∠1, ∠5 ≅ ∠7 5.

6. 𝑚∠3 = 𝑚∠1, 𝑚∠5 = 𝑚∠7 6.

7. 𝑚∠1 + 𝑚∠4 + 𝑚∠7 = 180° 7.

z°

34°

y°x°

23°

62°19° 47°z°

26°y°

35°

x°66°

58.

z°

y°x°

(x+4)°(3x-22)°

(3x-18)°

(2x+27)° (6x-23)°

57.56.

z°

x°

(y-13)°

55° 29°

>>

>>55.

54.53.

(3x-14)°

(x-7)°(2x+15)°

52

(x-9)°

62°51.

x°

87°

32°

50.

j

k

7 6

543

2 1

B

C A

D E

Reasons Bank

a) Angle Addition Postulate

b) Substitution Property of Equality

c) If 2 parallel lines are cut by a

transversal, then the alternate interior

angles are congruent.

d) If 2 parallel lines are cut by a

transversal, then the corresponding


e) Given

f) Definition of congruent angles.

g) Definition of a straight angle.


Inequalities in Triangles Classwork For each triangle list the sides from greatest to smallest #60-62. 60. 61. 62.

For each triangle list the angles in order from greatest to smallest #63-65. 63. 64. 65.

Will the three lengths given make a triangle?

66. 2, 3, and 4 67. 1, 3, and 4 68. 5, 6, and 7 69. 16, 8, and 7 70. 20, 10, and 10 71. 8x, 7x, and 14x

Given the lengths of two sides of a triangle, what lengths could the third side, x, have? 72. 12 and 14 73. 15 and 6 74. 22 and 22 75. 9 and 12 76. 8y and 10y

Inequalities in Triangles Homework For each triangle list the sides from greatest to smallest #77-79. 77. 78. 79.

76°

75°

24°112°

30°

65°

A

B

C

D

E

F

H

G

I

6.75

5.5

3.8

62

6043

5.7 2.5

4J

L

K

N

OM Q

P R


For each triangle list the angles in order from greatest to smallest #80-82. 80. 81. 82.

List the sides in order from shortest to longest #83-84. 83. 84.

Will the three lengths given make a triangle?

85. 21, 34, and 49

18°

80°

63°

57°

41°

95°

B

A

C D

E

FG

H

I

27

18

33

11

10

85

7

7

G

H

I J

K

L

M N

O

80°

60°

40°

75°

40°110°

80°

40°

70°

75°

S

T

U

VG

H

J

I

F


86. 11, 31, and 44 87. 8, 6, and 5 88. 12, 5, and 7 89. 20, 30, and 11 90. 9x, 17x, and 26x

Given the lengths of two sides of a triangle, what lengths could the third side, x, have? 91. 10 and 21 92. 19 and 8 93. 30 and 30 94. 5 and 15 95. 4y and 14y

Similar Triangles Classwork

96. 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.

97. Given: Prove: ∆𝐷𝐸𝐻~∆𝐺𝐸𝐹

Statements Reasons

1. 1.

2. ∠𝐷 ≅ ∠𝐺 2.

3. ∠𝐷𝐸𝐻 ≅ ∠𝐺𝐸𝐹 3.

DH ||FG

E

H

D

G

F

DH ||FG

Reasons Bank

a) If 2 parallel lines are cut by a



b) If 2 parallel lines are cut by a



c) Given

d) Vertical angles are congruent

e) AA~

f) SAS~

g) SSS~


4. ∆𝐷𝐸𝐻~∆𝐺𝐸𝐹 4.

98. Given: , , , , and

Prove: ∆𝑄𝑃𝑅~∆𝐵𝐶𝐴

Statements Reasons

1. , , ,

, and

1.

2. 𝑃𝑅

𝐶𝐴=

9

6=

3

2,

𝑄𝑅

𝐵𝐴=

7.2

4.8=

3

2 2.

3. 𝑃𝑅

𝐶𝐴=

𝑄𝑅

𝐵𝐴 3.

4. ∆𝑄𝑃𝑅~∆𝐵𝐶𝐴 4. 99. Given: ,

Prove: ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹

Statements Reasons

1. , 1.

2. ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹 2.

100. Given:


Statements Reasons

1. 1.

ÐR @ ÐA PR = 9 QR = 7.2 AB = 4.8 AC = 6

ÐR @ ÐA PR = 9 QR = 7.2

AB = 4.8 AC = 6

ÐA @ ÐD ÐB @ ÐE

A

B

C D

E

F

ÐA @ ÐD ÐB @ ÐE

FD

CA

EF

BC

DE

AB

A

B

C D

E

F

FD

CA

EF

BC

DE

AB

Reasons Bank

a) If the scale factors are the same, then the

sides are proportional.

b) Given

c) AA~

d) Calculating the scale factor between the

sides of the triangles.

e) SAS~

f) SSS~

Reasons Bank

a) AA~

b) SAS~

c) SSS~

d) Given

Reasons Bank

a) AA~

b) SAS~

c) SSS~

d) Given


A E

C

B D


In problems 101-103, determine if . 101. 102.

103.

PARCC type questions: Solve for y.

104. 105.

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. 106. Prove the Side Splitter Theorem

Given ║

Prove

DE || BC

BD AE

CE

CD

CA

CB

Reasons Bank







c) Given

d) Reflexive Property of Congruence

e) AA~

f) SAS~

g) If two triangles are similar, then their

corresponding sides are proportional.

h) SSS~


Statements Reasons

1. ║ 1.

2. ∠𝐶 ≅ ∠𝐶 2.

3. ∠𝐶𝐵𝐷 ≅ ∠𝐶𝐴𝐸 3.

4. ∆𝐶𝐵𝐷~∆𝐶𝐴𝐸 4.

5. 5.

Similar Triangles Homework

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

c.

BD AE

CE

CD

CA

CB


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.

108. Given:

Prove: ∆𝐴𝐶𝐸~∆𝐵𝐶𝐷

Statements Reasons

1. ║ 1.

2. ∠𝐶 ≅ ∠𝐶 2.

3. ∠𝐶𝐷𝐵 ≅ ∠𝐶𝐸𝐴 3.

4. ∆𝐴𝐶𝐸~∆𝐵𝐶𝐷 4.

109. Given: , , ,

Prove: ∆𝑃𝑄𝑅~∆𝐴𝐵𝐶

Statements Reasons

1. , ,

,

1.

2. 𝑚∠𝑅 = 77° 2.

3. ∠𝑄 ≅ ∠𝐵, ∠𝑅 ≅ ∠𝐶 3.

4. ∆𝑃𝑄𝑅~∆𝐴𝐵𝐶 4.

110. Given: ,


Statements Reasons

1. , 1.

BD || AE

BD AE

mÐP = 48° mÐQ = 55° mÐB = 55° mÐC = 77°

mÐP = 48° mÐQ = 55°

mÐB = 55° mÐC = 77°

FD

CA

DE

AB ÐA @ ÐD

A

B

C D

E

F

FD

CA

DE

AB ÐA @ ÐD

Reasons Bank







c) Given


e) AA~

f) SAS~

g) SSS~

Reasons Bank

a) Triangle Sum Theorem

b) Given

c) AA~

d) SAS~

e) Definition of congruent angles

f) SSS~

Reasons Bank

a) AA~

b) SAS~

c) SSS~

d) Given



In problems 111-113, determine if . 111. 112.

113.

PARCC type questions: Solve for y.

114. 115.

116.

DE || BC


A E

C

B D

PARCC type question: 117. 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. ∠𝐶 ≅ ∠𝐶 2.

3. ∆𝐶𝐵𝐷~∆𝐶𝐴𝐸 3.

4. ∠𝐶𝐵𝐷 ≅ ∠𝐶𝐴𝐸 4.

5. ║ 5.

CE

CD

CA

CB

BD AE

CE

CD

CA

CB

BD AE

Reasons Bank

a) If 2 lines are cut by a transversal and the

alternate interior angles are congruent,

then the lines are parallel.

b) If 2 lines are cut by a transversal and the

corresponding angles are congruent,

then the lines are parallel.

c) Given


e) AA~

f) SAS~

g) SSS~

h) If two triangles are similar, then the

corresponding angles are congruent.


Triangles Review Multiple Choice

1. Identify the triangles by sides and angles

a. scalene, acute

b. isosceles, obtuse

c. scalene, obtuse

d. equilateral, equiangular

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

a. 24 A triangle’s angles are:

b. 28 m∠𝐴 = 2x - 1

c. 32 m∠𝐵 = x + 9

d. 30 m∠𝐶 = 3x + 4

3. Classify the triangle by sides and angles.

a. scalene, obtuse

b. isosceles, acute

c. scalene, acute

d. isosceles, obtuse

4. Using the figure at right, list the segments from least to greatest.

a. 𝐵𝐶̅̅ ̅̅ , 𝐴𝐶̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅ , 𝐴𝐷̅̅ ̅̅ , 𝐷𝐶̅̅ ̅̅

b. 𝐴𝐵̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅ , 𝐴𝐶̅̅ ̅̅ , 𝐴𝐷̅̅ ̅̅ , 𝐷𝐶̅̅ ̅̅

c. 𝐴𝐷̅̅ ̅̅ , 𝐷𝐶̅̅ ̅̅ , 𝐴𝐶̅̅ ̅̅ , 𝐴𝐵̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅

d. cannot be determined

5. Use the diagram to find the value of x.

a. 130

b. 125

c. 120

d. 110

6. Which of the following values cannot be the third side of a triangle if two of the sides are 14

and 20?

a. 18

b. 20

c. 32.5

d. 34

8

12

10


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

DABC : DDEF

DABC : DDFE

DABC : DFDE

D E

B

A

C


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. ∆𝐽𝐾𝐿~∆𝑁𝑃𝑀. Find the values if x & y

x

6

10

8

E

F

D

G

H

y

15

x

12

13.59

NM

PK

JL

z

40

x

2x-10


A E

C

B D

Extended Constructed Response – Write the correct answer for each question. Partial credit will be given. 14. 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 ∆𝐵𝐶𝐷~∆𝐴𝐶𝐸

Statements Reasons

1. ║ 1.

2. ∠𝐶 ≅ ∠𝐶 2.

3. ∠𝐶𝐵𝐷 ≅ ∠𝐶𝐴𝐸 3.

4. ∆𝐵𝐶𝐷~∆𝐴𝐶𝐸 4.

BD AE

BD AE

Reasons Bank







c) Reflexive Property of Congruence

d) AA~

e) SAS~

f) Given

g) SSS~


Answers 1. Scalene 2. Isosceles 3. Isosceles 4. Equilateral 5. Scalene 6. Right 7. Equiangular & acute 8. Obtuse 9. Acute 10. Obtuse 11. Never 12. Never 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 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. (PROBLEM MISSING) 49. X=76°, y=104°, z=55° 50. X=61° 51. X=31 52. X=37 53. X=27 54. X=32 55. X=96°, z=151°, y=68 56. X=66°, z=88°, y=79° 57. Z=90°, y=43°, x=71° 58. X=95°, y=51°, z=39°

59.

Statements Reasons

1. j || k

∠𝐷𝐵𝐸 is a straight angle

1. e

2. 𝑚∠𝐷𝐵𝐸 = 180° 2. g

3. 𝑚∠3 + 𝑚∠4 +𝑚∠5 = 𝑚∠𝐷𝐵𝐸

3. a

4. 𝑚∠3 + 𝑚∠4 +𝑚∠5 = 180°

4. b

5. ∠3 ≅ ∠1, ∠5 ≅ ∠7 5. c

6. 𝑚∠3 = 𝑚∠1, 𝑚∠5 =𝑚∠7

6. f

7. 𝑚∠1 + 𝑚∠4 +𝑚∠7 = 180°

7.b

60.

61.

62.

63. <K, <L, <J 64. <N, <M, <O 65. <P, <Q, <R 66. yes 67. no 68. yes 69. no 70. no 71. yes 72. 2 < x < 26 73. 9 < x < 21 74. 0 < x < 44 75. 3 < x < 21 76. 2y < x < 18y

AC,BC,AB

DF,EF,DE

GI,HI,HG


77.

78.

79.

80. <H <I, <G 81. <K, <J, <L 82. <N, <M, <O

83.

84.

85. yes 86. no 87. yes 88. no 89. yes 90. no 91. 11< x < 31 92. 11 < x < 27 93. 0 < x < 60 94. 10 < x < 20 95. 10y < x < 18y 96. a. not similar

b. yes by SAS c. not similar

97. Statements Reasons

1. 1. c

2. ∠𝐷 ≅ ∠𝐺 2. a

3. ∠𝐷𝐸𝐻 ≅ ∠𝐺𝐸𝐹 3. d 4. 4. e


1. , ,

, , and

1. b

2. 𝑃𝑅

𝐶𝐴=

9

6=

3

2,

𝑄𝑅

𝐵𝐴=

7.2

4.8=

3

2 2. d

3. 𝑃𝑅

𝐶𝐴=

𝑄𝑅

𝐵𝐴 3. a

4. 4. e


1. , 1. d 2. 2. a


1. 1. d

2. 2. c 101. yes 102. no 103. no 104. 12 105. 11.36 106.

Statements Reasons

1. ║ 1. c

2. ∠𝐶 ≅ ∠𝐶 2. d

3. ∠𝐶𝐵𝐷 ≅ ∠𝐶𝐴𝐸 3. b

4. ∆𝐶𝐵𝐷 ≅ ∆𝐶𝐴𝐸 4. e

5. 5. g

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


1. ║ 1. c

2. ∠𝐶 ≅ ∠𝐶 2. d

3. ∠𝐶𝐷𝐵 ≅ ∠𝐶𝐸𝐴 3. b

4. ∆𝐶𝐵𝐷 ≅ ∆𝐶𝐴𝐸 4. e 109.

Statements Reasons

1. ,

, ,

1. b

2. 𝑚∠𝑅 = 77° 2. a

3. ∠𝑄 ≅ ∠𝐵, ∠𝑅 ≅ ∠𝐶 3. e

4. 4. c


1. , 1. d

2. 2. b 111. yes 112. yes 113. no 114. 10 115. 11.25

BC,AB,AC

EF,DE,DF

GI,HI,GH@

UV,TV,UT ,ST ,SU

FJ,GF,GJ,GH ,HJ,HI,JI

DH ||FG

DDEH : DGEF

ÐR @ ÐA PR = 9QR = 7.2 AB = 4.8

AC = 6

DQPR : DBCA

ÐA @ ÐD ÐB @ ÐE

DABC : DDEF

FD

CA

EF

BC

DE

AB

DABC : DDEF

BD AE

CE

CD

CA

CB

BD AE

mÐP = 48°mÐQ = 55° mÐB = 55°

mÐC = 77°

DPQR : DABC

FD

CA

DE

AB ÐA @ ÐD

DABC : DDEF


116. 10 117.

Statements Reasons

1. 1. c

2. ∠𝐶 ≅ ∠𝐶 2. d

3. ∆𝐶𝐵𝐷 ≅ ∆𝐶𝐴𝐸 3. f

4. ∠𝐶𝐵𝐷 ≅ ∠𝐶𝐴𝐸 4. h

5. ║ 5. b

Unit Review Answer Key

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.

Statements Reasons

1. ║ 1. f

2. ∠𝐶 ≅ ∠𝐶 2. c

3. ∠𝐶𝐵𝐷 ≅ ∠𝐶𝐴𝐸 3. b

4. ∆𝐵𝐶𝐷~∆𝐴𝐶𝐸 4. d

CE

CD

CA

CB

BD AE

BD AE

triangles chapter problems 58° 58° 43°...

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