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Triangles Chapter Problems

Classify the Triangles by Sides or AnglesClass 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 cm2. Side lengths: 3 cm, 3 cm, 4 cm3. Side lengths: 2 cm, 3 cm, 2 cm4. Side lengths: 5 cm, 5 cm, 5 cm5. Side lengths: 2 cm. 3 cm, 4 cm6. 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°

Geometry: Triangles ~1~ NJCTL.org

Classify the Triangles by Sides or AnglesHomework

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 cm22. Side lengths: 2 cm, 2 cm, 3 cm23. Side lengths: 3 cm, 3 cm, 3 cm24. Side lengths: 3 cm, 4 cm, 4 cm25. Side lengths: 4 cm, 3 cm, 2 cm26. 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.A 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 TheoremsClass WorkIn the given triangles, solve for the missing variable(s).


Triangle Sum & Exterior Angle TheoremsHomeworkIn the given triangles, solve for the missing variable(s).


PARCC type question66. 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 angleProve: m∠1+m∠4+m∠7=180 °

Statements Reasons1. 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.

Inequalities in TrianglesClass workFor 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.


Reasons Bank

a) Angle Addition Postulateb) Substitution Property of Equalityc) 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 angles are congruent.

e) Givenf) Definition of congruent angles.g) Definition of a straight angle.

Will the three lengths given make a triangle?73. 2, 3, and 474. 1, 3, and 475. 5, 6, and 776. 16, 8, and 777. 20, 10, and 1078. 8x, 7x, and 14x

Given the lengths of two sides of a triangle, what lengths could the third side, x, have?79. 12 and 1480. 15 and 681. 22 and 2282. 9 and 1283. 8y and 10y

Inequalities in TrianglesHomeworkFor 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 4993. 11, 31, and 4494. 8, 6, and 595. 12, 5, and 796. 20, 30, and 1197. 9x, 17x, and 26x

Given the lengths of two sides of a triangle, what lengths could the third side, x, have?98. 10 and 2199. 19 and 8100. 30 and 30101. 5 and 15102. 4y and 14y

Similar TrianglesClass work

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

a.

b. c.


A

BC

D

EF

K

L

G

J

HQ

R

N

PM

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: ∆ DEH ∆GEF

Statements Reasons1. 1.2. ∠D≅∠G 2.3. ∠DEH ≅∠GEF 3.4. ∆ DEH ∆GEF 4.

105. Given: , , , , and Prove: ∆QPR ∆BCA

Statements Reasons1. , , ,

, and

1.

2. PRCA

=96=3

2, QRBA

=7.24.8

=32

2.

3. PRCA

=QRBA

3.

4. ∆QPR ∆BCA 4.

106. Given: , Prove: ∆ ABC ∆≝¿

A

B

C D

E

F

Statements Reasons1. , 1.


Reasons Bank

a) If 2 parallel lines are cut by a transversal, then the alternate interior angles are congruent.

b) If 2 parallel lines are cut by a transversal, then the corresponding angles are congruent.

c) Givend) Vertical angles are congruente) AA~f) SAS~g) SSS~

Reasons Bank

a) If the scale factors are the same, then the sides are proportional.

b) Givenc) 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

2. ∆ ABC ∆≝¿ 2.

107. Given: FDCA

EFBC

DEAB

Prove: ∆ ABC ∆≝¿

A

B

C D

E

F

Statements Reasons

1. FDCA

EFBC

DEAB

1.

2. ∆ ABC ∆≝¿ 2.

In problems 108-110, determine if .108. 109.

110.

PARCC type questions:Solve for y.

111. 112.


Reasons Bank


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 BD║ AE

Prove BACB

=DECD

Statements Reasons1. BD║ AE 1.2. ∠C≅∠C 2.3. ∠CBD≅∠CAE 3.4. ∆CBD ∆CAE 4.

5. CACB

=CECD

5.

6. CA = CB + BA CE = CD + DE

6.

7. CB+BACB

=CD+DECD

7.

8. CBCB

+BACB

=CDCD

+ DECD

8.

9. 1+BACB

=1+ DECD

9.

10. BACB

=DECD

10.

Similar Triangles


A E

C

B D

Reasons Bank



c) Givend) Reflexive Property of Congruencee) AA~f) SAS~g) If two triangles are similar, then their

corresponding sides are proportional.h) SSS~i) Substitution Property of Equalityj) Subtraction Property of Equalityk) Addition Property of Equalityl) Segment Addition Postulatem) Addition Property of Fractions

(e .g . 25 + 15=2+1

5=3

5 )n) Simplifying the Equation

Homework114. 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: ∆ ACE ∆BCD

Statements Reasons1. BD║ AE 1.2. ∠C≅∠C 2.3. ∠CDB≅∠CEA 3.4. ∆ ACE ∆BCD 4.

116. Given: , , , Prove: ∆ PQR ∆ ABC

Statements Reasons1. , ,

,

1.

2. m∠R=77 ° 2.3. ∠Q≅∠B ,∠R≅∠C 3.4. ∆ PQR ∆ ABC 4.

117. Given: FDCA

DEAB

,

Prove: ∆ ABC ∆≝¿


Reasons Bank



c) Givend) Reflexive Property of Congruencee) AA~f) SAS~g) SSS~

Reasons Bank

a) Triangle Sum Theoremb) Givenc) AA~d) SAS~e) Definition of congruent anglesf) SSS~

Reasons Bank


A

BC

D

E

F

KL

G

J

H Q

N

PM

A

B

C D

E

F

Statements Reasons

1. FDCA

DEAB

,

1.

2. ∆ ABC ∆≝¿ 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 BACB

=DECD

Prove BD║ AE

Statements Reasons

1. BACB

=DECD

1.

2. 1+BACB

=1+ DECD

2.

3. CBCB

+BACB

=CDCD

+ DECD

3.

4. CB+BACB

=CD+DECD

4.


5.

6. CACB

=CECD

6.

7. ∠C≅∠C 7.8. ∆CBD ∆CAE 8.9. ∠CBD≅∠CAE 9.10. BD║ AE 10.

ApplicationsClass work


A E

C

B D

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) Givend) Reflexive Property of Congruencee) AA~f) SAS~g) SSS~h) If two triangles are similar, then the

corresponding angles are congruent.i) Substitution Property of Equalityj) Subtraction Property of Equalityk) Addition Property of Equalityl) Segment Addition Postulatem) Addition Property of Fractions

(e .g . 25 + 15=2+1

5=3

5 )n) Simplifying the Equation

125. 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.

126. 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.

127. 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.

120 ft

15 ft

20 ft

d

ApplicationsHomework

128. 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.

129. 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.


130. 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.

15 ft

56 ft

28 ft

d


Triangles ReviewMultiple Choice

1. Identify the triangles by sides and angles a. scalene, acute b. isosceles, obtusec. scalene, obtused. 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 - 1c. 32 m∠B=¿ x + 9d. 30 m∠C=¿ 3x + 4

3. Classify the triangle by sides and angles.a. scalene, obtuseb. isosceles, acutec. scalene, acuted. isosceles, obtuse

4. Using the figure at right, list the segments from least to greatest. a. BC , AC , BC , AD ,DCb. AB ,BC , AC , AD, DCc. AD, DC , AC , AB ,BCd. cannot be determined

5. Use the diagram to find the value of x. a. 130b. 125c. 120d. 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. 18b. 20c. 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. 8b. 4.5c. 6d. 12


11.Solve for x

a. 4b. 4.8c. 10d. 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


2x-10

x

40

z

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 BD║ AEProve ∆ BCD ∆ ACE

Statements Reasons1. BD║ AE 1.2. ∠C≅∠C 2.3. ∠CBD≅∠CAE 3.4. ∆ BCD ∆ ACE 4.


A E

C

B D

Reasons Bank



c) Reflexive Property of Congruenced) AA~e) SAS~f) Giveng) SSS~

Answers1. Scalene2. Isosceles3. Isosceles4. Equilateral and isosceles5. Scalene6. Right7. Equiangular & acute8. Obtuse9. Acute10.Obtuse11.Never12.Always13.Sometimes14.Sometimes15.Sometimes16.Sides: Isosceles, Angles: Acute17.Sides: Scalene, Angles: Obtuse18.Sides: Isosceles, Angles: Obtuse19.Sides: Scalene, Angles: Right20.Sides: Isosceles, Angles: Obtuse21.Scalene22. Isosceles23.Equilateral and isosceles24. Isosceles25.Scalene26.Equiangular & acute27.Right28.Obtuse29.Obtuse30.Acute31.Never32.Never33.Sometimes34.Never35.Sometimes36.Sides: Scalene, Angles: Right37.Sides: Scalene, Angles: Obtuse38.Sides: Isosceles, Angles: Obtuse39.Sides: Isosceles, Angles: Acute40.Sides: Isosceles, Angles: Acute41.X=58°42.X=33°43.X=23°44.X=30°45.X=79°, z=144°, y=55°46.X=12

47.X=2748.X=76°, y=104°, z=55°49. j = 47, k = 180, m = 155, n = 25,

p = 94, q = 6150. r = 35, t = 145, u = 35, v = 90,

w = 12551.a = 54, b = 36, c = 36, d = 116,

e = 152, f = 28, g = 98, h = 5452. j = 128, k = 52, m = 52, n = 31,

p = 149, q = 31, r = 49, t = 13153.X=61°54.X=3155.X=3756.X=2757.X=3258.X=96°, z=151°, y=6859.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 = 11163.g = 49, h = 117, j = 49, k = 76,

m = 5564.a = 143, b = 37, c = 37, d = 121,

e = 158, f = 22, g = 108, h = 5065. r = 90, t = 38, u = 38, v = 34,

w = 146, x = 34, y = 30, z = 15066.

Statements Reasons1. j || k ∠DBE is a straight angle

1. e

2. m∠DBE=180 ° 2. g3. 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. c6. m∠3=m∠1 ,m∠5=m∠7

6. f

7. m∠1+m∠4+m∠7=180 °

7.b

67.

68.

69.


70. <K, <L, <J71. <N, <M, <O72. <P, <Q, <R73. yes74. no75. yes76. no77. no78. yes79. 2 < x < 2680. 9 < x < 2181. 0 < x < 4482. 3 < x < 2183. 2y < x < 18y

84.

85.

86. 87. <H <I, <G88. <K, <J, <L89. <N, <M, <O

90.

91. 92. yes93. no94. yes95. no96. yes97. no98. 11< x < 3199. 11 < x < 27100. 0 < x < 60101. 10 < x < 20102. 10y < x < 18y103. a. not similar

b. yes ∆GHJ ∆ LJK by SASc. not similar

104. Statements Reasons

1. 1. c

2. ∠D≅∠G 2. a3. ∠DEH ≅∠GEF 3. d4. 4. e

105.Statements Reasons1. , ,

, , and

1. b

2. PRCA

=96=3

2, QRBA

=7.24.8

=32

2. d

3. PRCA

=QRBA

3. a

4. 4. e

106.Statements Reasons1. , 1. d2. 2. a

107.Statements Reasons

1. FDCA

EFBC

DEAB

1. d

2. 2. c

108. yes 109. no110. no111. 12112. 11.36

113. Statements Reasons1. BD║ AE 1. c2. ∠C≅∠C 2. d3. ∠CBD≅∠CAE 3. b4. ∆CBD ∆CAE 4. e

5. CACB

=CECD

5. g


6. l

7. CB+BACB

=CD+DECD

7. i

8. CBCB

+BACB

=CDCD

+ DECD

8. m


9. 1+BACB

=1+ DECD

9. n

10. BACB

=DECD

10. j

114. a. yes, ∆ ABD ∆ ACD by AA~ or SAS~

b. not similarc. yes, ∆ KLM ∆QNP by SSS~

115.Statements Reasons1. BD║ AE 1. c2. ∠C≅∠C 2. d3. ∠CDB≅∠CEA 3. b4. ∆CBD≅ ∆CAE 4. e

116.Statements Reasons1. ,

, ,

1. b

2. m∠R=77 ° 2. a3. ∠Q≅∠B ,∠R≅∠C 3. e4. 4. c


1. FDCA

DEAB

,

1. d

2. 2. b

118. yes119. yes120. no121. 10122. 11.25123. 10


1. BACB

=DECD

1. c

2. 1+BACB

=1+ DECD

2. k

3. CBCB

+BACB

=CDCD

+ DECD

3. n

4. CB+BACB

=CD+DECD

4. m


5. l

6. CACB

=CECD

6. i

7. ∠C≅∠C 7. d8. ∆CBD ∆CAE 8. f9. ∠CBD≅∠CAE 9. h10. BD║ AE 10. b

125. 50 feet126. 72 ft127. 90 ft128. 27.5 feet129. 40 ft130. 45 ft

Unit Review Answer Key

1. c2. b3. c4. b5. b6. d7. c

8. c9. a10. a11. d12. x = 50 & z = 9013. x = 18 & y = 22.514. 150 ft


15.Statements Reasons1. BD║ AE 1. f2. ∠C≅∠C 2. c3. ∠CBD≅∠CAE 3. b4. ∆ BCD ∆ ACE 4. d


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content.njctl.orgcontent.njctl.org/.../triangles-classwork-homework-2016-01-13.docx · Web viewIf 2 parallel lines are cut by a transversal, ... Given the lengths of two sides of