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Parallel Lines Chapter Problems Lines: Intersecting, parallel & skew Class Work – Use image 1 ̅̅̅̅: 1. Name all segments parallel to ̅̅̅̅ 2. Name all segments skew to : 3. Name all segments intersecting with ̅̅̅̅ : ̅̅̅̅ and ̅̅̅̅ coplanar? Explain your answer. 4. Are segments 5. Are segments ̅̅̅̅ and ̅̅̅̅ coplanar? Explain your answer. Is each statement true always, sometimes, or never? 6. Two intersecting lines are skew. 7. Two parallel lines are coplanar. 8. Two lines in the same plane are parallel. 9. Two lines that do not intersect are parallel. 10. Two skew lines are coplanar Lines: Intersecting, parallel & skew Homework -Use Image 1 11. Name all segments parallel to ̅̅̅̅ : ̅̅̅̅ : 12. Name all segments skew to 13. Name all segments intersecting with ̅̅̅̅ : ̅̅̅̅ ̅̅̅̅ 14. Are segments and coplanar? Explain your answer. ̅̅̅̅ and ̅̅̅̅ coplanar? Explain your answer. 15. Are segments Image 1 State whether the following statements are always, sometimes, or never true: 16. Two coplanar lines are skew. 17. Two intersecting lines are in the same plane. 18. Two lines in the same plane are parallel. Lines & Transversals Classify each pair of angles as alternate interior, alternate exterior, same-side interior, sameside exterior, corresponding angles, or none of these. 19. ∠11 and ∠16 are 20. ∠12 and ∠2 are 21. ∠14 and ∠8 are 22. ∠6 and ∠16 are 23. ∠7 and ∠14 are 24. ∠3 and ∠16 are Geometry – Parallel Lines ~1~ NJCTL.org Classify each pair of angles as alternate interior, alternate exterior, same-side interior, sameside exterior, corresponding angles, or none of these. 25. ∠7 and ∠12 26. ∠3 and ∠6 27. ∠6 and ∠11 28. ∠7 and ∠11 29. ∠4 and ∠10 30. ∠14 and ∠16 31. ∠2 and ∠3 32. ∠2 and ∠10 Parallel Lines & Proofs Classwork Match each expression/equation with the property used to make the conclusion. 33. AB = AB 36. If DE = FG, then FG = DE. a) Substitution Property of Equality 34. If m∠A = m∠B and m∠B = m∠C, then b) Transitive Property of Equality m∠A = m∠C. c) Reflexive Property of Equality 35. If x + y = 9 and y = 5, then x + 5 = 9. d) Symmetric Property of Equality PARCC type question: 37. Alternate Exterior Angles Proof: Complete the proof by filling in the missing reasons with the “reasons bank” below. Given: line m || line k Prove: ∠2 ≅ ∠8 Statements 1. line m || line k 2. ∠2 ≅ ∠6 Reasons 1. 2. 3. ∠6 ≅ ∠8 3. 4. ∠2 ≅ ∠8 4. nce nding Geometry – Parallel Lines ~2~ NJCTL.org PARCC type question: 38. Same-Side Interior Angles Proof: Complete the proof by filling in the missing reasons with the “reasons bank” below. Some reasons may be used more than once. Given: line m || line k Prove: ∠5 & ∠4 are supplementary Statements 1. line m || line k 2. ∠1 ≅ ∠5 3. m∠1 = m∠5 4. ∠1 & ∠4 are supplementary 5. m∠1 + m∠4 = 180 6. m∠5 + m∠4 = 180 7. ∠5 & ∠4 are supplementary Reasons 1. 2. 3. 4. 5. 6. 7. a) b) c) d) e) f) Parallel Lines & Proofs Homework For #39-42 match the description on the left to 39. ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C. 40. If bc = 77 and b = 11, then 11c = 77. 41. If ∠P ≅ ∠M, then ∠M ≅ ∠P. 42. QR = QR Reasons Bank Angles that form a linear pair are supplementary. Substitution Property of Equality Definition of supplementary angles If 2 parallel lines are cut by a transversal, then the corresponding angles are congruent. Definition of congruent angles Given the name of the property on the right. a) Substitution Property of Equality b) Transitive Property of Congruence c) Reflexive Property of Equality d) Symmetric Property of Congruence PARCC type question: Geometry – Parallel Lines ~3~ NJCTL.org 43. Alternate Interior Angles Proof: Complete the proof by filling in the missing reasons with the “reasons bank” below. Given: line m || line k Prove: ∠3 ≅ ∠5 Statements 1. line m || line k 2. ∠3 ≅ ∠7 3. ∠7 ≅ ∠5 4. ∠3 ≅ ∠5 Reasons 1. 2. 3. 4. a) b) c) d) Geometry – Parallel Lines ~4~ Reasons Bank Vertical Angles are congruent. Given Transitive Property of Congruence If 2 parallel lines are cut by a transversal, then the corresponding angles are congruent. NJCTL.org PARCC type question: 44. Same-Side Exterior Angles Proof: Complete the proof by filling in the missing reasons with the “reasons bank” below. Some reasons may be used more than once. Given: line m || line k Prove: ∠1 & ∠8 are supplementary Statements 1. line m || line k 2. ∠1 ≅ ∠5 3. m∠1 = m∠5 4. ∠5 & ∠8 are supplementary 5. m∠5 + m∠8 = 180 6. m∠1 + m∠8 = 180 7. ∠1 & ∠8 are supplementary Reasons 1. 2. 3. 4. 5. 6. 7. Reasons Bank a) Definition of supplementary angles b) If 2 parallel lines are cut by a transversal, then the corresponding angles are congruent. c) Given d) Definition of congruent angles e) Angles that form a linear pair are supplementary. f) Substitution Property of Equality Properties of Parallel Lines Classwork Use the given diagram to answer problems #45-53. If m∠9 = 54°, then find the measure the following angles: 45. m∠1= 46. m∠2= 47. m∠4= 48. m∠5= 49. m∠15= Geometry – Parallel Lines ~5~ NJCTL.org If m∠2 = (12x-54)° and m∠10 = (7x+26)°, then find the measure the following angles: 50.m∠6= 51. m∠11= 52. m∠9= 53. m∠16= Find the values of the unknown variables in each figure. (# 54-58) 54. 55. 56. 57. Geometry – Parallel Lines 58. ~6~ NJCTL.org Find measure of the following angles: 59. m∠1= 60. m∠2= 61. m∠3= 62. m∠4= 63. m∠5= State which segments (if any) are parallel. 64. 65. 66. Solve for the unknowns 67. 68. Geometry – Parallel Lines ~7~ NJCTL.org Properties of Parallel Lines Homework If m∠9 = 62°, then find the measure the following angles: 69. m∠1= 70. m∠2= 71. m∠4= 72. m∠5= 73. m∠15= If m ∠2 = (14x-24)° and m ∠10 = (6x+72)°, then find the measure the following angles: 74. m∠6= 75. m∠11= 76. m∠9= 77. m∠16= Find the values of the unknown variables in each figure. (#78-82) 78. 81. Geometry – Parallel Lines 79. 80. 82. ~8~ NJCTL.org Find measure of the following angles: 83. m∠1= 84. m∠2= 85. m∠3= 86. m∠4= 87. m∠5= State which segments (if any) are parallel. 88. D C 124° 124° A B 90. 89. 91. Geometry – Parallel Lines 92. ~9~ NJCTL.org Constructing Parallel Lines Class Work 93. Construct a line m that is parallel to line l that passes thru point C using the stated method. Corresponding Angles 94. Error Analysis: A person was constructing the line n thru point D such that it was parallel to line l using the alternate interior angles method. Using their markings, state their mistake. 95. Use paper- folding techniques to construct parallel lines. Geometry – Parallel Lines ~10~ NJCTL.org Constructing Parallel Lines Homework 96. Error Analysis: A person was constructing the line n thru point D such that it was parallel to line l using the alternate exterior angles method. Using their markings, state their mistake. 97. Construct parallel lines using a straightedge and compass using alternate interior angles. 98. Construct parallel lines using a straightedge and compass using alternate exterior angles. Geometry – Parallel Lines ~11~ NJCTL.org PARCC type question: 99. The figure shows line j, points C and B are on line j, and point A is not on line j. Also shown is line AB. A j B C Part A: A j C F G B Consider the partial construction of a line parallel to j through point A. What would be the final step in the construction? a) Draw a line through points B and F b) Draw a line through points C and F c) Draw a line through points A and F d) Draw a line through points A and G Part B: Once the construction is complete, which of the following reasons listed contribute to providing the validity of the construction? a) If two parallel lines are cut by a transversal, then the corresponding angles are congruent. b) If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. c) If two parallel lines are cut by a transversal, then the same-side interior angles are supplementary. d) If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Geometry – Parallel Lines ~12~ NJCTL.org PARCC type question: 100. The figure shows line p; points H, K, and M are on line p, and point J is not on line p. Also shown is line JK. J p H K M Part A: N p H J K M Consider the partial construction of a line parallel to p through point J. What would be the final step in the construction? a) Draw a line through points K and N b) Draw a line through points J and N c) Draw a line through points H and N d) Draw a line through points M and M Part B: Once the construction is complete, which of the following reasons listed contribute to providing the validity of the construction? a) If two parallel lines are cut by a transversal, then the corresponding angles are congruent. b) If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. c) If two parallel lines are cut by a transversal, then the same-side interior angles are supplementary. d) If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Geometry – Parallel Lines ~13~ NJCTL.org Parallel Lines Review Multiple Choice 1. Name the segment parallel to ̅̅̅̅ and skew to ̅̅̅̅ . ̅̅̅̅ a. ̅̅̅̅ b. ̅ c. ̅̅̅̅ d. ̅̅̅̅ and skew to . ̅̅̅̅ 2. Name the segment parallel to ̅̅̅̅ a. b. ̅̅̅̅ ̅ c. ̅̅̅̅ d. 3. Determine if the statement is always, sometimes, or never true: Two skew lines are coplanar. a. Always b. Sometimes c. Never 4. Determine if the statement is always, sometimes, or never true: Two intersecting lines are coplanar a. Always b. Sometimes c. Never 5. Determine if the statement is always, sometimes, or never true: Two lines that do not intersect are skew. a. Always b. Sometimes c. Never 6. Determine the relationship between ∠1 & ∠10. a. Alternate Interior b. Same-side Interior c. Corresponding Angles d. None of these 7. Determine the relationship between ∠5 & ∠15. a. Alternate Exterior b. Alternate Interior c. Same-side Interior d. None of these Geometry – Parallel Lines ~14~ NJCTL.org 8. Given in the diagram to the right, m∠2=3x-10 and m∠15=2x+30 , what is m∠12? a. 32o b. 40o c. 86o d. 110o 9. Given in the diagram to the right, m∠5= (7x+2)°and m∠11=(5x+14)°, what is m∠14? a. 6° b. 44° c. 46° d. 136° In 10-11, use the diagram at the right. 10. Given ∠2 ≅ ∠6, what justifies k || m. a. Converse Alternate Interior Angles Theorem b. Converse Alternate Exterior Angles Theorem c. Converse Corresponding Angles Theorem d. there is not enough info to state parallel 11. Given n || p , what justifies ∠1 ≅ ∠12 a. Alternate Interior Angles Theorem b. Alternate Exterior Angles Theorem c. Corresponding Angles Theorem d. there is not enough info to make this statement Extended Constructed Response 1. Complete the proof by filling in the missing reasons with the “reasons bank” to the right. Some reasons may be used more that once. Given: ∠1 ≅ ∠3; ̅̅̅̅̅ || ̅̅̅̅ Prove: ∠2≅∠3 M 1 3 P Statements 1. ∠1 ≅ ∠3 ̅̅̅̅ ̅̅̅̅̅ || 2. 3. ∠1 ≅ ∠2 4. ∠2≅∠3 Geometry – Parallel Lines Reasons 1. 2. 3. 4. N 2 Q Reasons Bank a) Transitive Property of Congruence b) If 2 parallel lines are cut by a transversal, then the alternate interior angles are congruent. c) Given ~15~ NJCTL.org 2. Complete the proof by filling in the missing reasons with the “reasons bank” to the right. Some reasons may be used more that once. Given: n || p, k || m Prove: ∠2 & ∠13 are supplementary Statements 1. n || p, k || m 2. ∠2 ≅ ∠12 3. ∠12 ≅ ∠14 4. ∠2 ≅ ∠14 5. m∠2 = m∠14 6. m∠13 & m∠14 are supplementary 7. m∠13 + m∠14 = 180° 8. m∠13 + m∠2 = 180° 9. ∠2 &∠13 are supplementary Reasons 1. 2. 3. 4. 5. 6. 7. 8. 9. Reasons Bank a) Transitive Property of Congruence b) Definition of supplementary angles c) If 2 parallel lines are cut by a transversal, then the alternate interior angles are congruent. d) Definition of Congruent Angles e) Given f) If 2 parallel lines are cut by a transversal, then the alternate exterior angles are congruent. g) Angles that form a linear pair are supplementary h) Substitution Property of Equality 3. Using a compass and straightedge, construct parallel lines. You can use any method of your choice. Geometry – Parallel Lines ~16~ NJCTL.org Answers ̅ , ̅̅̅̅ , ̅̅̅̅ 1. Segments , ̅̅̅̅ ̅̅̅̅ ̅̅̅ , ̅̅̅ , ̅̅̅̅, Segments ̅̅̅, ̅̅̅̅ , ̅̅̅̅ , 2. Segments ̅̅̅̅ , ̅̅̅̅ , ̅̅̅̅ 3. Yes, because these segments are parallel 4. No, these lines are skew, so they are not coplanar. 5. Never 6. Always 7. Sometimes 8. Sometimes 9. Never ̅̅̅̅ , ̅̅̅̅, ̅̅̅̅ ̅ , 10. Segments ̅̅̅, ̅̅̅̅ , ̅̅̅̅ , ̅̅̅̅ 11. Segments ̅̅̅̅ , ̅̅̅̅, ̅̅̅, ̅̅̅̅ ̅̅̅̅, 12. Segments , ̅̅̅̅ , ̅̅̅ 13. Yes, because they are parallel 14. No, these lines are skew, so they are not coplanar 15. Never 16. Always 17. Sometimes 18. Same side interior 19. None of these 20. Alternate interior 21. Corresponding 22. Same-side interior 23. None of these 24. Corresponding 25. Same-side 26. Alternate interior 27. Corresponding 28. Corresponding 29. Same-side interior 30. None of these 31. None of these Geometry – Parallel Lines 32. c. Reflexive Property of Equality 33. b. Transitive Property of Equality 34. a. Substitution Property of Equality 35. d. Symmetric Property of Equality 36. Proof reasons should be: Statements Reasons 1. line m || line k 1. d. 2. b. 2. ∠2 ≅ ∠6 3. c. 3. ∠6 ≅ ∠8 4. a. 4. ∠2 ≅ ∠8 37. Proof reasons should be: Statements Reasons 1. line m || line k 1. f. 2. d. 2. ∠1 ≅ ∠5 3. e. 3. m∠1 = m∠5 4. a. 4. ∠1 & ∠4 are supplementary 5. c. 5. m∠1 + m∠4 = 180° 6. b. 6. m∠5 + m∠4 = 180° 7. c. 7. ∠5 & ∠4 are supplementary 38. b. Transitive Property of Congruence 39. a. Substitution Property of Equality 40. d. Symmetric Property of Congruence 41. c. Reflexive Property of Equality 42. Proof reasons should be: Statements Reasons 1. line m || line k 1. b. ~17~ NJCTL.org 2. ∠3 ≅ ∠7 3. ∠7 ≅ ∠5 4. ∠3 ≅ ∠5 2. d. 3. a. 4. c. 43. Proof reasons should be: Statements Reasons 1. line m || line k 1. c. 2. b. 2. ∠1 ≅ ∠5 3. d. 3. m∠1 = m∠5 4. e. 4. ∠5 & ∠8 are supplementary 5. a. 5. m∠5 + m∠8 = 180 6. f. 6. m∠1 + m∠8 = 180 7. a. 7. ∠1 & ∠8 are supplementary 44. 54° 45. 126° 46. 126° 47. 54° 48. 54° 49. 138° 50. 42° 51. 42° 52. 138° 53. x= 144° 54. x= 64° and y= 49/4 55. x=6; z=2 56. x=24, y=11; z=22/5 57. x=33; y=2 58. 44° 59. 107° 60. 29° 61. 29° 62. 136° 63. Segments ̅̅̅̅ and ̅̅̅̅̅ are parallel Geometry – Parallel Lines NJCTL.org ̅̅̅̅and ̅̅̅̅are 64. Segments parallel 65. None of these 66. x=9 and y=8 and z=7 67. x=8 and y=7 68. 62° 69. 118° 70. 118° 71. 62° 72. 62° 73. 144° 74. 36° 75. 36° 76. 144° 77. x=55° 78. x=86° and y=7 79. x=9; y=6; z=7 80. x=15; y=10; z=8 81. x=25; y=3 82. 41° 83. 106° 84. 33° 85. 33° 86. 129° 87. cannot be determined 88. Segments ̅̅̅̅̅ and ̅̅̅̅ are parallel 89. Segments ̅̅̅̅ and ̅̅̅̅ are parallel 90. x=6; y=12; z=7 91. x=18; y=7 92. See student work 93. made same side interior the same 94. See student work 95. Made angles congruent that should be supplementary. 96. see student work 97. see student work ~18~ 98. Part A: c & Part B: d 99. Part A: b & Part B: b 3. See student work REVIEW 1. c 2. b 3. c 4. a 5. b 6. c 7. a 8. c 9. d 10. c 11. d EXTENDED CONSTRUCTED RESPONSE 1. Statements ∠1 ≅ ∠3 ̅̅̅̅̅ || ̅̅̅̅ ∠1 ≅ ∠2 ∠2≅∠3 Reasons c. Given c. Given b. Alternate Interior Angles Theorem a. Transitive Property of congruence Statements 1. n || p, k || m 2. ∠2≅∠12 3. ∠12≅∠14 4. ∠2≅∠14 5. m∠2+m∠14 6. ∠13 & ∠14 are supplementary 7. m∠13 = m∠14 = 180° 8. m∠13 + m∠2 = 180° 9. ∠2 & ∠13 are supplementary Geometry – Parallel Lines NJCTL.org Reasons 1. e 2. f 3. c 4. a 5. d 6. g 7. b 8. h 9. b ~19~