Triangles Chapter Problems
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- Bertram Sullivan
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1 Classify the Triangles by Sides or Angles Class Work Triangles Chapter Problems In problems #1-10, choose the most appropriate description for the given triangle. (quilateral, Scalene, Isosceles, Obtuse, Acute, Right, quiangular) 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, Angle Measures: 60, 60, Angle Measures: 92, 37, Angle Measures: 88, 67, Angle measures: 37, 39, 104 Complete the statement using ALWAYS, SOMTIMS, and NVR. 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 Geometry: Triangles ~1~ NJCTL.org
2 Classify the Triangles by Sides or Angles Home Work In problems #21-30, choose the most appropriate description for the given triangle. (quilateral, Scalene, Isosceles, Obtuse, Acute, Right, quiangular) 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, Angle Measures: 60, 30, Angle Measures: 33, 52, Angle Measures: 37, 43, Angle measures: 25, 67, 88 Complete the statement using ALWAYS, SOMTIMS, and NVR. 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 Geometry: Triangles ~2~ NJCTL.org
3 Triangle Sum & xterior Angle Theorems Class Work In the given triangles, solve for the missing variable(s) (4x+21) (2x+8) (x-10) >> (y+10) z >> (4x-13) (3x+23) (3x-18) 65 y z Triangle Sum & xterior Angle Theorems Home Work In the given triangles, solve for the missing variable(s). Geometry: Triangles ~3~ NJCTL.org
4 (3x-14) (2x+15) (x-7) (x-9) 53. (x+4) (3x-22) 54. (2x+27) (3x-18) (6x-23) (y-13) >> z >> y 57. y z z z y 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, DB is a straight angle Prove: m 1 + m 4 + m 7 = 180 D B k 2 1 C 7 6 A 1. j k 1. Bank DB is a straight angle a) Angle Addition Postulate 2. m DB = b) Substitution Property of quality 3. m 3 + m 4 + m 5 = 3. c) If 2 parallel lines are cut by a m DB transversal, then the alternate interior 4. m 3 + m 4 + m 5 = angles are congruent , d) If 2 parallel lines are cut by a 6. m 3 = m 1, m 5 = m 7 6. transversal, then the corresponding 7. m 1 + m 4 + m 7 = angles are congruent. e) Given f) Definition of congruent angles. Geometry: Triangles ~4~ g) Definition of a straight NJCTL.org angle. j
5 Inequalities in Triangles Classwork For each triangle list the sides from greatest to smallest # A For each triangle list the angles in order from greatest to smallest # J 65 B L K M 43 D C N 112 Will the three lengths given make a triangle? 66. 2, 3, and , 3, and , 6, and , 8, and , 10, and x, 7x, and 14x Given the lengths of two sides of a triangle, what lengths could the third side, x, have? and and and and y and 10y Inequalities in Triangles Homework For each triangle list the sides from greatest to smallest # O Q P F H G I R Geometry: Triangles ~5~ NJCTL.org
6 B 57 H A C D 63 F G I For each triangle list the angles in order from greatest to smallest # H 5 8 K 10 L M N 27 G 7 I J 11 O List the sides in order from shortest to longest # S T U 80 V G 75 F J H I Will the three lengths given make a triangle? , 34, and 49 Geometry: Triangles ~6~ NJCTL.org
7 86. 11, 31, and , 6, and , 5, and , 30, and x, 17x, and 26x Given the lengths of two sides of a triangle, what lengths could the third side, x, have? and and and and y 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: DH FG Prove: DH~ GF Bank D F a) If 2 parallel lines are cut by a transversal, then the alternate interior angles are congruent. G b) If 2 parallel lines are cut by a H transversal, then the corresponding angles are congruent. 1. DH FG 1. c) Given 2. D G 2. d) Vertical angles are congruent 3. DH GF 3. e) AA~ f) SAS~ g) SSS~ Geometry: Triangles ~7~ NJCTL.org
8 4. DH~ GF Given: ÐA, PR = 9, QR = 7.2, AB = 4.8, and AC = 6 Prove: QPR~ BCA Bank 1. ÐA, PR = 9, QR = 7.2, AB = 4.8, and AC = 6 2. PR = 9 = 3, QR = 7.2 = 3 CA 6 2 BA PR = QR CA BA QPR~ BCA 4. 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~ 99. Given: ÐD, Ð Prove: ABC~ DF B a) AA~ b) SAS~ c) SSS~ d) Given Bank A C D F 1. ÐD, Ð ABC~ DF Given: AB BC CA D F FD Prove: ABC~ DF B a) AA~ b) SAS~ c) SSS~ d) Given Bank A C D F AB BC 1. D F CA FD 1. Geometry: Triangles ~8~ NJCTL.org
9 2. ABC~ DF 2. In problems , determine if D BC PARCC type questions: Solve for y 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 Prove the Side Splitter Theorem Bank Given BD A CB CD a) If 2 parallel lines are cut by a Prove transversal, then the alternate interior C CA C angles are congruent. b) If 2 parallel lines are cut by a transversal, then the corresponding angles are congruent. c) Given d) Reflexive Property of Congruence B Geometry: Triangles D ~9~ e) AA~ NJCTL.org A f) SAS~ g) If two triangles are similar, then their corresponding sides are proportional.
10 1. BD A C C CBD CA CBD~ CA 4. CB CD CA C 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. Geometry: Triangles ~10~ NJCTL.org
11 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 Given: BD A Bank Prove: AC~ BCD 1. BD A C C CDB CA AC~ BCD 4. 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) Given d) Reflexive Property of Congruence e) AA~ f) SAS~ g) SSS~ 109. Given: mðp = 48, mðq = 55, mðb = 55, mðc = 77 Prove: PQR~ ABC Bank 1. mðp = 48, mðq = 55, 1. mðb = 55, mðc = m R = Q B, R C PQR~ ABC 4. AB CA 110. Given:, ÐD D FD Prove: ABC~ DF a) Triangle Sum Theorem b) Given c) AA~ d) SAS~ e) Definition of congruent angles f) SSS~ a) AA~ b) SAS~ c) SSS~ d) Given Bank B A C D F AB CA 1., D FD ÐD 1. Geometry: Triangles ~11~ NJCTL.org
12 2. ABC~ DF 2. In problems , determine if D BC PARCC type questions: Solve for y Geometry: Triangles ~12~ NJCTL.org
13 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. CB CD Given Bank CA C C Prove BD A A CB CD CA C 2. C C CBD~ CA CBD CA BD A 5. B D 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 d) Reflexive Property of Congruence e) AA~ f) SAS~ g) SSS~ h) If two triangles are similar, then the corresponding angles are congruent. Geometry: Triangles ~13~ NJCTL.org
14 1. Identify the triangles by sides and angles a. scalene, acute b. isosceles, obtuse c. scalene, obtuse d. equilateral, equiangular Triangles Review Multiple Choice 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 Classify the triangle by sides and angles. a. scalene, obtuse b. isosceles, acute c. scalene, acute d. isosceles, obtuse Using the figure at right, list the segments from least to greatest. a. BC, AC, BC, AD, DC b. AB, BC, AC, AD, DC c. AD, DC, AC, AB, BC d. cannot be determined 5. Use the diagram to find the value of x. a. 130 b. 125 c. 120 d 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 d. 34 Geometry: Triangles ~14~ NJCTL.org
15 7. Decide whether the triangles are similar. If so, write a similarity statement. a. Yes, DABC : DDF b. Yes, DABC : DDF c. Yes, DABC : DFD 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. B C a. Yes, by AA~ b. Yes, by SSS~ c. Yes, by SAS ~ d. The triangles are not similar A D 10. Solve for y a. 8 b. 4.5 c. 6 d. 12 Geometry: Triangles ~15~ NJCTL.org
16 11. Solve for x 6 F D 10 8 H x G a. 4 b. 4.8 c. 10 d Short Constructed Response Write the correct answer for each question. No partial credit will be given. 12. Find the values of x and z. x z 2x JKL~ NPM. Find the values if x & y K 12 9 x L J 15 P 13.5 M y N Geometry: Triangles ~16~ NJCTL.org
17 xtended 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 A Bank Prove BCD~ AC C 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. B D c) Reflexive Property of Congruence d) AA~ e) SAS~ A f) Given g) SSS~ 1. BD A C C CBD CA BCD~ AC 4. Geometry: Triangles ~17~ NJCTL.org
18 Answers 1. Scalene 2. Isosceles 3. Isosceles 4. quilateral 5. Scalene 6. Right 7. quiangular & 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. quilateral 24. Isosceles 25. Scalene 26. quiangular & 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= X= X= X= X=79, z=144, y= X= X= (PROBLM MISSING) 49. X=76, y=104, z= X= X= X= X= X= X=96, z=151, y= X=66, z=88, y= Z=90, y=43, x= X=95, y=51, z= j k 1. e DB is a straight angle 2. m DB = g 3. m 3 + m a m 5 = m DB 4. m 3 + m b m 5 = , c 6. m 3 = m 1, m 5 = 6. f m 7 7. m 1 + m b m 7 = AC,BC,AB DF,F, D GI,HI,HG <K, <L, <J 64. <N, <M, <O 65. <P, <Q, <R 66. yes 67. no 68. yes 69. no 70. no 71. yes < x < < x < < x < < x < y < x < 18y Geometry: Triangles ~18~ NJCTL.org
19 <I, <G 81. <K, <J, <L 82. <N, <M, <O 83. BC,AB, AC F,D, DF GI, HI,GH UV,TV,UT,ST,SU FJ,GF,GJ,GH, HJ, HI, JI yes 86. no 87. yes 88. no 89. yes 90. no < x < < x < < x < < x < y < x < 18y 96. a. not similar b. yes by SAS c. not similar DH FG 1. c 2. D G 2. a 3. DH GF 3. d 4. DDH : DGF 4. e ÐA, PR = 9, 1. b QR = 7.2, AB = 4.8, and AC = 6 2. PR = 9 = 3, QR = 7.2 = 3 2. d CA 6 2 BA PR = QR 3. a CA BA 4. DQPR : DBCA 4. e ÐD, Ð 1. d 2. DABC : DDF 2. a 100. AB BC CA 1. d 1. D F FD 2. DABC : DDF 2. c 101. yes 102. no 103. no BD A 1. c 2. C C 2. d 3. CBD CA 3. b 4. CBD CA 4. e CB CD 5. g 5. CA C 107. a. yes, by AA~ or SAS~ b. not similar c. yes, by SSS~ BD A 1. c 2. C C 2. d 3. CDB CA 3. b 4. CBD CA 4. e mðp = 48, 1. b mðq = 55, mðb = 55, mðc = m R = a 3. Q B, R C 3. e 4. DPQR : DABC 4. c 110. AB CA 1. d 1., ÐD D FD 2. DABC : DDF 2. b 111. yes 112. yes 113. no Geometry: Triangles ~19~ NJCTL.org
20 CB CD c CA C 2. C C 2. d 3. CBD CA 3. f 4. CBD CA 4. h 5. BD A 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 = x = 18 & y = BD A 1. f 2. C C 2. c 3. CBD CA 3. b 4. BCD~ AC 4. d Geometry: Triangles ~20~ NJCTL.org
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