content.njctl.orgcontent.njctl.org/.../triangles-cw-and-hw-2014-12-05.docx · Web view2014/12/05 · If two triangles are similar, then their corresponding sides are proportional

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 AnglesHome 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 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.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 TheoremsClass WorkIn 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.

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


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.

PARCC type question59. 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.


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.

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 467. 1, 3, and 468. 5, 6, and 769. 16, 8, and 770. 20, 10, and 1071. 8x, 7x, and 14x

Given the lengths of two sides of a triangle, what lengths could the third side, x, have?72. 12 and 1473. 15 and 674. 22 and 2275. 9 and 1276. 8y and 10y

Inequalities in TrianglesHomeworkFor each triangle list the sides from greatest to smallest #77-79.77. 78. 79.


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


86. 11, 31, and 4487. 8, 6, and 588. 12, 5, and 789. 20, 30, and 1190. 9x, 17x, and 26x

Given the lengths of two sides of a triangle, what lengths could the third side, x, have?91. 10 and 2192. 19 and 893. 30 and 3094. 5 and 1595. 4y and 14y

Similar TrianglesClasswork

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 Reasons1. 1.2. ∠D≅∠G 2.3. ∠DEH ≅∠GEF 3.


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~

4. ∆ DEH ∆GEF 4.

98.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. 99. Given: , Prove: ∆ ABC ∆≝¿

A

B

C D

E

F

Statements Reasons1. , 1.2. ∆ ABC ∆≝¿ 2.

100. Given: FDCA

EFBC

DEAB

Prove: ∆ ABC ∆≝¿

A

B

C D

E

F

Statements Reasons


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

Reasons Bank


1. FDCA

EFBC

DEAB

1.

2. ∆ ABC ∆≝¿ 2.

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 BD║ AE

Prove CECD

CACB


A E

C

B D

Reasons Bank



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

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

5. CECD

CACB

5.

Similar TrianglesHomework

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

c.


Reasons Bank



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

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

109. Given: , , , Prove: ∆ PQR ∆ ABC

Statements Reasons1. , ,

,

1.

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

110. Given: FDCA

DEAB

,

Prove: ∆ ABC ∆≝¿

A

B

C D

E

F

Statements Reasons


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


1. FDCA

DEAB

,

1.

2. ∆ ABC ∆≝¿ 2.

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

113.

PARCC type questions:Solve for y.

114. 115.

116.


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 CECD

CACB

Prove BD║ AE

Statements Reasons

1. CECD

CACB

1.

2. ∠C≅∠C 2.3. ∆CBD ∆CAE 3.4. ∠CBD≅∠CAE 4.5. BD║ AE 5.


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.

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

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 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 5. Scalene6. Right7. Equiangular & acute8. Obtuse9. Acute10.Obtuse11.Never12.Never13.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 24. 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. (PROBLEM MISSING)49.X=76°, y=104°, z=55°50.X=61°51.X=3152.X=3753.X=2754.X=3255.X=96°, z=151°, y=6856.X=66°, z=88°, y=79°57.Z=90°, y=43°, x=71°58.X=95°, y=51°, z=39°

59.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

60.

61.

62. 63. <K, <L, <J64. <N, <M, <O65. <P, <Q, <R66. yes67. no68. yes69. no70. no71. yes72. 2 < x < 2673. 9 < x < 2174. 0 < x < 4475. 3 < x < 2176. 2y < x < 18y


77.

78.

79. 80. <H <I, <G81. <K, <J, <L82. <N, <M, <O

83.

84. 85. yes86. no87. yes88. no89. yes90. no91. 11< x < 3192. 11 < x < 2793. 0 < x < 6094. 10 < x < 2095. 10y < x < 18y96.a. not similar

b. yes by SASc. not similar

97. Statements Reasons

1. 1. c

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

98.Statements Reasons1. , ,

, , and

1. b

2. PRCA

=96=3

2, QRBA

=7.24.8

=32

2. d

3. PRCA

=QRBA

3. a

4. 4. e99.

Statements Reasons1. , 1. d2. 2. a

100.Statements Reasons

1. FDCA

EFBC

DEAB

1. d

2. 2. c101. yes 102. no103. no104. 12105. 11.36106.

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

5. CECD

CACB

5. g

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

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

109.Statements Reasons1. ,

, ,

1. b

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

110.Statements Reasons

1. FDCA

DEAB

,

1. d

2. 2. b


111. yes112. yes113. no114. 10115. 11.25116. 10117.

Statements Reasons

1. CECD

CACB

1. c

2. ∠C≅∠C 2. d

3. ∆CBD≅ ∆CAE 3. f4. ∠CBD≅∠CAE 4. h5. BD║ AE 5. b

Unit Review Answer Key

1. c2. b3. c4. b5. b6. d

7. c8. c9. a10. a11. d

12. x = 50 & z = 9013. x = 18 & y = 22.514.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-cw-and-hw-2014-12-05.docx · Web view2014/12/05 · If two triangles are similar, then their corresponding sides are proportional