Triangles Chapter Problems

Transcription

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 Homework 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. A 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). Geometry: Triangles ~3~ NJCTL.org

4 Triangle Sum & xterior Angle Theorems Homework In the given triangles, solve for the missing variable(s). 50. x 51. (3x-14) (2x+15) (x-7) (x-9) 53. (x+4) (3x-22) 54. (2x+27) (3x-18) (6x-23) x (y-13) >> z >> x y 57. x y z z z x 34 y Geometry: Triangles ~4~ NJCTL.org

5 PARCC type question 60. 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 1. j k 1. DB is a straight angle 2. m DB = m 3 + m 4 + m 5 = m DB m 3 + m 4 + m 5 = , m 3 = m 1, m 5 = m m 1 + m 4 + m 7 = Inequalities in Triangles Class work For each triangle list the sides from greatest to smallest # A 65 B 30 D C 112 For each triangle list the angles in order from greatest to smallest # J L K M 43 N O C Q P F H G I R 7 6 A Bank a) Angle Addition Postulate b) Substitution Property of quality 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 angles are congruent. e) Given f) Definition of congruent angles. g) Definition of a straight angle. j Geometry: Triangles ~5~ NJCTL.org

6 Will the three lengths given make a triangle? 67. 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 # B 41 A 95 C D F G H 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 Geometry: Triangles ~6~ NJCTL.org

7 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 , 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 Class work 97. Determine if the triangles are similar. If so, write a similarity statement & state the similarity postulate or theorem. a. b. c. Geometry: Triangles ~7~ NJCTL.org

8 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. 98. Given: DH FG Prove: DH~ GF Bank D F H 1. DH FG D G DH GF DH~ GF 4. G 99. Given: ÐA, PR = 9, QR = 7.2, AB = 4.8, and AC = 6 Prove: QPR~ BCA 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) Vertical angles are congruent e) AA~ f) SAS~ g) SSS~ 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~ 100. Given: ÐD, Ð Prove: ABC~ DF Bank A B C D F a) AA~ b) SAS~ c) SSS~ d) Given 1. ÐD, Ð ABC~ DF 2. Geometry: Triangles ~8~ NJCTL.org

9 AB BC CA 101. Given: D F FD Prove: ABC~ DF B a) AA~ b) SAS~ c) SSS~ d) Given Bank A C AB BC CA D F FD 2. ABC~ DF 2. D F In problems , determine if D BC PARCC type questions: Solve for y Geometry: Triangles ~9~ NJCTL.org

10 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 Prove BA = D A B C D 1. BD A C C CBD CA CBD~ CA CA = C 6. CA = CB + BA C = CD + D 7. CB+BA CB = CD+D CD 8. CB + BA = CD + D CB CD BA = 1 + D 10. BA = D 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) If two triangles are similar, then their corresponding sides are proportional. h) SSS~ i) Substitution Property of quality j) Subtraction Property of quality k) Addition Property of quality l) Segment Addition Postulate m) Addition Property of Fractions (e. g = = 3 5 ) n) Simplifying the quation Similar Triangles Homework 108. 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~ 110. 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 111. 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. 1., ÐD D FD 2. ABC~ DF 2. Geometry: Triangles ~11~ NJCTL.org

12 In problems , determine if D BC PARCC type questions: Solve for y Geometry: Triangles ~12~ NJCTL.org

13 PARCC type question: 118. 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 BA = D C Bank Prove BD A 1. BA = D BA = 1 + D 3. CB + BA = CD + D CB CD 4. CB+BA = CD+D 5. CA = CB + BA C = CD + D 6. CA = C C C CBD~ CA CBD CA BD A 10. A 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. i) Substitution Property of quality j) Subtraction Property of quality k) Addition Property of quality l) Segment Addition Postulate m) Addition Property of Fractions (e. g = = 3 5 ) n) Simplifying the quation Applications Class work 119. 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 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. Geometry: Triangles ~13~ NJCTL.org

14 121. 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 d 15 ft 20 ft Applications Homework 122. 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 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 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. 28 ft 56 ft 15 ft d Geometry: Triangles ~14~ NJCTL.org

15 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 ~15~ NJCTL.org

16 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 ~16~ NJCTL.org

17 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 ~17~ NJCTL.org

18 14. To find the distance d across a stream, you locate the points as shown. Find the value of d. 200 ft d 21 ft 28 ft xtended 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 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 ~18~ NJCTL.org

19 Answers 1. Scalene 2. Isosceles 3. Isosceles 4. quilateral and isosceles 5. Scalene 6. Right 7. quiangular & 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. quilateral and isosceles 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= X=76, y=104, z= a = 54, b = 36, c = 36, d = 116, e = 152, f = 28, g = 98, h = 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= a = 143, b = 37, c = 37, d = 121, e = 158, f = 22, g = 108, h = 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 65. <N, <M, <O 66. <P, <Q, <R 67. yes 68. no 69. yes 70. no 71. no 72. yes < x < < x < 21 Geometry: Triangles ~19~ NJCTL.org

20 75. 0 < x < < x < y < x < 18y <I, <G 82. <K, <J, <L 83. <N, <M, <O 84. BC,AB, AC F,D, DF GI, HI,GH UV,TV,UT,ST,SU FJ,GF,GJ,GH, HJ, HI, JI yes 87. no 88. yes 89. no 90. yes 91. no < x < < x < < x < < x < y < x < 18y 97. 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 101. AB BC CA 1. d 1. D F FD 2. DABC : DDF 2. c 102. yes 103. no 104. no BD A 1. c 2. C C 2. d 3. CBD CA 3. b 4. CBD~ CA 4. e 5. g 5. CA = C 6. CA = CB + BA C = CD + D 7. CB+BA CB = CD+D CD 8. CB + BA = CD + D CB CD BA = 1 + D 10. BA = D 6. l 7. i 8. m 9. n 10. j 108. 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 Geometry: Triangles ~20~ NJCTL.org

21 111. AB CA 1. d 1., ÐD D FD 2. DABC : DDF 2. b 112. yes 113. yes 114. no BA = D BA = 1 + D 3. CB + BA = CD + D CB CD 4. CB+BA = CD+D 5. CA = CB + BA C = CD + D 6. CA CB = C CD 1. c 2. k 3. n 4. m 5. l 6. i 7. C C 7. d 8. CBD~ CA 8. f 9. CBD CA 9. h 10. BD A 10. b feet ft ft feet ft ft 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 = ft BD A 1. f 2. C C 2. c 3. CBD CA 3. b 4. BCD~ AC 4. d Geometry: Triangles ~21~ NJCTL.org