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Parallel and Perpendicular Lines Angles and Parallel Lines Class Work 1. Name all segments parallel to : 2. Name all segments skew to : 3. Name all segments intersecting with : 4. Are segments and coplanar? Explain your answer. 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 Classify each pair of angles as alternate interior, same-side interior, corresponding angles, or none of these. 11. 12. 13. 14. 15. 16. Angles and Parallel Lines Homework 17. Name all segments parallel to : 18. Name all segments skew to : 19. Name all segments intersecting with : 20. Are segments and coplanar? Explain your answer. 21. Are segments and coplanar? Explain your answer. Classify each pair of angles as alternate interior, same-side interior, corresponding angles, or none of these. 22. 7 and 12 23. 3 and 6 24. 6 and 11 25. 7 and 11 26. 4 and 10 27. 14 and 16 28. 2 and 3 29. 2 and 10 Geometry Parallel & Perpendicular ~1~ NJCTL.org State whether the following statements are always, sometimes, or never true: 30. Two coplanar lines are skew. 31. Two intersecting lines are in the same plane. 32. Two lines in the same plane are parallel. Properties of Parallel Lines Class Work Use the given diagram to answer problems #33-41. If m 9=540, then find the measure the following angles: 33. 34. 35. 36. 37. If m 38. 39. 40. 41. 2=12x-54 and m 10=7x+26, then find the measure the following angles: Find the values of the unknown variables in each figure. (# 42-46) 42. 43. 44. 45. Geometry Parallel & Perpendicular 46. ~2~ NJCTL.org Find measure of the following angles: 47. 48. 49. 50. 51. State which segments (if any) are parallel. 52. 53. 54. Properties of Parallel Lines Homework If m 9=620, then find the measure the following angles: 55. 56. 57. 58. 59. If m 60. 61. 62. 63. 2=14x-24 and m 10=6x+72, then find the measure the following angles: Geometry Parallel & Perpendicular ~3~ NJCTL.org Find the values of the unknown variables in each figure. (#64-68) 64. 65. 67. 66. 68. Find measure of the following angles: 69. 70. 71. 72. 73. State which segments (if any) are parallel. 74. D C 124° 124° A B 75. Geometry Parallel & Perpendicular ~4~ NJCTL.org 76. Equations of Lines and Slope Class Work Identify the slope of the line containing the given points: 77. (-2,11), (-5, 2) 78. (-4,11), (-4, -7) 79. (-2,22), (-5, 22) Write the equation of the line in slope-y intercept form containing the given information: 80. m= 4; b=4 81. m= -2; (0,1) 82. (3, -12), (-5, 4) 83. (-16,-2), (-4, 7) Write the equation of the line in standard form containing the following information: 84. (-7,4), (-7, 32) 85. (1,3), (6,-2) 86. x-intercept=8; y-intercept= -4 Write the equation of the line in point-slope form containing the following information: 87. m= -3, (-8,-2) 88. (5,2), (-3,4) 89. (-11,12), (-10, 11) 90. The cost (C) of a cab ride is $4 to start and $0.25 per tenth of a mile (t) a. Write the equation that models the information. b. How much will a 3 mile ride cost? c. If you have $10, what is the greatest distance you can go? 91. The median house price (p) in Smallville in 2010 was $250,000. That value has grown constantly at $5000 per year. a. Write an equation that models the information. b. How much will the median house be worth in 2020? c. If this growth rate continues when will the median value reach $350,000? 92. The price(p) for the Sunnyside Day Camp to go to the movies is $7 for each adult and $5 for each camper. a. Write an equation that models the information. b. How much will it cost for 5 adults and 20 campers? c. If for every 6 campers, there needs to be an adult, how much will it cost for 40 campers? Geometry Parallel & Perpendicular ~5~ NJCTL.org Equations of Lines and Slope Homework Identify the slope of the line containing the given points: 93. (-6,12), (-2, 5) 94. (14,11), (-14, 11) 95. (-2,17), (-2, 18) Write the equation of the line in slope-y intercept form containing the given information: 96. m= -3; b=4 97. m= 6, (0,2) 98. (-2, -11), (3, 4) 99. (16,-2), (4, 6) Write the equation of the line in standard form containing the following information: 100. (-7,41), (-9, 41) 101. (-5,12), (-5, 11) 102. x-intercept= -2; y-intercept= 6 Write the equation of the line in point-slope form containing the following information: 103. m= 5, (7,-1) 104. (6,2), (8,-4) 105. (-3,0), (-6,2) 106. The cost (C) of a cab ride is $5 to start and $0.30 per tenth of a mile (t) a. Write the equation that models the information. b. How much will a 4 mile ride cost? c. If you have $20, what is the greatest distance you can go? 107. The median house price (p) in Largeton in 2010 was $300,000. That value has grown constantly at $6000 per year. a. Write an equation that models the information. b. How much will the median house be worth in 2015? c. If this growth rate continues when will the median value reach $360,000? 108. The price for the Sunnyside Day Camp to go to the movies is $8 for each adult and $6 for each camper. a. Write an equation that models the information. b. How much will it cost for 4 adults and 15 campers? c. If for every 5 campers, there needs to be an adult, how much will it cost for 36 campers? Slopes of Parallel Lines Class Work 109. Find an equation of the line in point-slope form passing through point (-2,5) and parallel to the line whose equation is 4x -2y = -5 110. Two lines are represented by equations: 2x +4y =21 and y=kx -12. What value of k will make lines parallel? 111. Find an equation of the line in slope-intercept form passing through point (-4,6) and parallel to the line whose equation is y = -¾ x+11 112. The sides of a quadrilateral lie on the lines y= 4x +5, y= 1/3x +7, 8x - 2y= 1, and x β 3y =2. Is the quadrilateral a parallelogram? Justify your answer. Geometry Parallel & Perpendicular ~6~ NJCTL.org 113. Is the following system of equations parallel? Justify your answer. A B Slopes of Parallel Lines Homework 114. Find an equation of the line in point-slope form passing through point (-3,2) and parallel to the line whose equation is 6x -2y = 7. 115. Two lines are represented by equations: -4x +12y =21 and y=kx -12. What value of k will make lines parallel? 116. Find an equation of the line in point-slope form passing through point (-8,3) and parallel to the line whose equation is y = -¾ x+11. 117. The sides of a quadrilateral lie on the lines 3x+ y= 7, x + y= 12, 6x β 2y =2, and x - y=2. Is the quadrilateral a parallelogram? Justify your answer. 118. Is the following system of equations parallel? Justify your answer. A B Geometry Parallel & Perpendicular ~7~ NJCTL.org Perpendicular Lines and Angles Class Work Solve for the unknowns 119. 120. In exercises 121-122 make a conclusion, how lines a and d are related based on given information.( Lines a, b, c, and d are distinct and in the same plane. ) 121. 122. Perpendicular Lines and Angles Homework 123. 124. Lines a, b, c, and d are distinct lines in the same plane. In exercises 125-126 make a conclusion, how lines b and d are related based on given information. 125. 126. Slopes of Perpendicular Lines Class Work 127. Find an equation of the line passing through point (4,-5) and perpendicular to the line whose equation is 3x -6y = -11. Geometry Parallel & Perpendicular ~8~ NJCTL.org 128. Two lines are represented by equations: -3x +6y =21 and y=kx +5. What value of k will make lines perpendicular? 129. Find an equation of the line passing through point (8,-2) and perpendicular to the line whose equation is y = 4x+11. 130. The sides of a quadrilateral lie on the lines y= 4x +5, y= 1/3x +7, x + 4y= 1, and x β 3y =2, is the quadrilateral a rectangle? Justify your answer. 131. Is the following system of equations perpendicular? Justify your answer. A B Slopes of Perpendicular Lines Homework 132. Find an equation of the line passing through point (-6,2) and perpendicular to the line whose equation is 4x +6y = -1 133. Two lines are represented by equations: 10x -15y =21 and y=kx +5. What value of k will make lines perpendicular? 134. Find an equation of the line passing through point (8,-2) and perpendicular to the line whose equation is y = -2x+11 135. The sides of a quadrilateral lie on the lines 4x - y= 5, x + 4y=7, 8x β 2y= 1, and 3x + 12y =2. Is the quadrilateral a rectangle? Justify your answer. 136. Is the following system of equations perpendicular? Justify your answer. A B Geometry Parallel & Perpendicular ~9~ NJCTL.org Proofs Involving Parallel and Perpendicular Lines Class Work 137. Fill in the blanks for the following two-column proof. Given: Prove : πδ«π Statements 1. p||n; <9δ<5 2. <9δ<1 3. <5δ<1 4. k || m Reasons 1. 2. 3. 4. 138. Fill in the blanks for the following paragraph proof. 0 Given: Prove : πδ«π Given k || m, therefore m<12 + m<13 = 180°due to __1___. It is also given that m<12+m<5 = 180°, therefore by ___2__ , m<12 + m<13 = m<12 + m<5. If we subtract the m<12 from both sides by ___3__ we arrive at m<13 = m<5. If m<13=m<5 then <13 @ <5 by __4___. If the <13 @ <5 then by __5___ p || n. Proofs Involving Parallel and Perpendicular Lines Homework X H 1 4 139. Write a 2 column proof. Given: The diagram to the right Prove: Vertical Angles Theorem 2 3 G 140. Write suur asuuflow r chart proof. Given: AB || CD Prove: m<4+m<2+m<5 = 180° Y B A 1 4 C Geometry Parallel & Perpendicular ~10~ 2 3 5 D NJCTL.org Constructing Parallel Lines Class Work 141. Construct a line m that is parallel to line l that passes thru point C using the stated method. Corresponding Angles 142. 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. 143. Use paper- folding techniques to construct parallel lines. Constructing Parallel Lines Homework 144. 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. Geometry Parallel & Perpendicular ~11~ NJCTL.org 145. Construct parallel lines using a straightedge and compass using alternate interior angles. 146. Construct a 60° using a straightedge and compass. 147. Construct a regular hexagon inscribed in a circle using a straightedge and compass. Constructing Perpendiculars Class Work 148. Using a compass and straightedge, construct a perpendicular line to m through C. 149. A builder wants to put in the shortest access road possible to his new project. If P represents his project and Elm Street is the road he is connecting to, where should he build the road? 150. What did the person do incorrectly when constructing the perpendicular to m thru C? Geometry Parallel & Perpendicular ~12~ NJCTL.org Constructing Perpendiculars Homework Construct a perpendicular line to m that passes thru point C. 151. Use a compass and straightedge. 152. Use paper folding to make the construction. 153. Rafee launches a toy rocket (R) straight up during his science lab. Since there was no wind at that time, the rocket came back down to the same exact spot. Construct the path of the rocket using a compass and straightedge. 154. What did the person do incorrectly when constructing the perpendicular to m thru C? Geometry Parallel & Perpendicular ~13~ NJCTL.org Multiple Choice 1. Name the segment parallel to a. b. c. d. 2. Given the diagram a. 32 b. 40 c. 86 d. 110 and skew to and , what is 3. Write the equation of the line in slope-y intercept form with m= 3 and b= -2. a. y= -3x +2 b. y= -2x + 3 c. y= 3x + 2 d. y= 3x -2 4. Write the equation of the line in standard form with x-intercept=6 and y-intercept= -4. a. 6x β 4y = 1 b. 2x β 3y = 12 c. x β y = 4 d. -4x + 6y = 1 5. Write the equation of the line in point-slope form that passes thru (5,2) and (-3,4). a. y+4= ¼ (x-3) b. y- 4= ¼ (x+3) c. y-2= - ¼ (x+5) d. y-2= - ¼ (x-5) 6. What is the equation of the line parallel to 2x+8y=10 and passes thru (-1,5)? a. y β 5 = - ¼( x-1) b. y β 5 = - ¼( x +1) c. y β 5 = 4(x β 1) d. y β 5 =4(x + 1) 7. What is the equation of the line perpendicular to y β 3 = 2/3(x + 3) and has an x-intercept of 6? a. y= 2/3x +4 b. y= 2/3x β 4 c. y= -3/2x + 6 d. y= -3/2x + 9 Geometry Parallel & Perpendicular ~14~ NJCTL.org In 8-9, use the diagram at the right. 8. Given , what justifies πδ«π? 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 9. Given πδ«π, what justifies a. Alternate Interior Angles Theorem b. Alternate Exterior Angles Theorem c. Corresponding Angles Theorem d. there is not enough info to make this statement 10. The cost © of school supplies is $3 per notebook (n) and $4 per packet of pens(p). Which equation represents the total cost for pens and notebooks? a. C= 4n + 3p b. C= 7np c. C= 4n + 3 d. C= 4p + 3n 11. The Fashion Club sells boosters for their spring show. Half page adds cost $12 and full page adds $20. The printer donates the program so any ads sold are profit. If the program made the club $188 and there were 7 full page ads, how many ½ page ads were sold? a. 4 b. 5 c. 6 d. 7 12. The distance from a point to a line can be found a. along the arc of a circle b. along the perpendicular to the line thru the point c. calculating the opposite reciprocal of the y-intercept d. constructing a circle centered at the point. 13. How many lines can be drawn through a point not on a line perpendicular to the given line? a. none b. one c. two d. infinitely many Geometry Parallel & Perpendicular ~15~ NJCTL.org Extended Response 1. Create a proof for: Given: Prove: M N 1 3 P 2 Q 2. Draw the line m on the graph provided so that m passes thru (1,4) and (5,-3) a. What is the equation of the line in point slope form? b. Construct a parallel line n that contains (3,7) c. What is the slope of line n? Geometry Parallel & Perpendicular ~16~ NJCTL.org 3. Draw the line m on the graph provided so that m passes thru (1,4) and (5,-3) d. What is the equation of the line in standard form? e. Construct a line p that is perpendicular that contains A(3,7)? f. What is the slope of line p? g. State how you would find the distance from A to m? 4. Using a compass and straightedge, construct parallel lines. 5. Using a compass and straightedge, construct a 60° angle. Geometry Parallel & Perpendicular ~17~ NJCTL.org Answers 35. 126° 36. 54° 37. 54° 38. 138° 39. 42° 40. 42° 41. 138° 42. x= 144° 43. x= 64° and y= 49/4 44. x=6; z=2 45. x=24, y=11; z=22/5 46. x=33; y=2 47. 44° 48. 107° 49. 29° 50. 29° 51. 136° 52. Segments and are parallel 53. Segments and are parallel 54. None of these 55. 62° 56. 118° 57. 118° 58. 62° 59. 62° 60. 144° 61. 36° 62. 36° 63. 144° 64. x=55° 65. x=86° and y=7 66. x=9; y=6; z=7 67. x=15; y=10; z=8 68. x=25; y=3 69. 41° 70. 106° 71. 33° 1. 2. 3. 4. Segments , , , Segments Segments Yes, because these segments are parallel 5. No, these lines are skew, so they are not coplanar. 6. Never 7. Always 8. Sometimes 9. Sometimes 10. Never 11. Same side interior 12. None of these 13. Alternate interior 14. Corresponding 15. Same-side interior 16. None of these 17. Segments and 18. Segments and 19. Segments 20. Yes, because they are parallel 21. No, these lines are skew, so they are not coplanar 22. Corresponding 23. Same-side 24. Alternate interior 25. Corresponding 26. Corresponding 27. Same-side interior 28. None of these 29. None of these 30. Never 31. Always 32. Sometimes 33. 54° 34. 126° Geometry Parallel & Perpendicular ~18~ NJCTL.org 72. 33° 73. 129° 74. cannot be determined 75. Segments and are parallel 76. Segments and are parallel 77. 3 78. No slope (undefined) 79. 0 80. y= 4x +4 81. y= -2x+1 82. y= -2x-6 83. y= ¾x +10 84. x= -7 85. x +y =4 86. x β 2y = 8 87. y+2= -3(x+8) 88. y-2= -1/4(x-5) or y-4=-1/4(x+3) 89. y-12= -(x+11) or y-11=-(x+10) 90. a. C=.25t+4 b. $11.50 c. 2.4 mi 91. a. p-250000=5000(x-2010) b. $300,000 c. 2030 92. a. 7A+5C=P b. $135 c. $249 93. -7/4 94. 0 95. No slope (Undefined) 96. y= -3x+4 97. y=6x +2 98. y=3x-5 99. y=-2/3 x +26/3 100. y=41 101. x=-5 102. 3x β y= -6 103. y+1=5(x-7) 104. y-2= -3(x-6) or y+4=-3(x-8) 105. y= -2/3(x+3) or y-2=2/3(x+6) 106. a. C=.30t+5 b. $17 c. 5 miles 107. a. y-300,000=6000(x-2010) b. $330,000 c. 2020 Geometry Parallel & Perpendicular ~19~ 108. a. 8A + 6C = P b. $122 c. $280 109. y -5= 2(x+2) 110. k= -1/2 111. y= -3/4x+3 112. Yes,π¦ = 4π₯ + 5δ«8π₯ β 1 2π¦ = 1 ; π¦ = 3 π₯ + 7δ«π₯ β 3π¦ = 2 113. a. no; slope of m=2, slope of n = 4/3; b. yes; slope of n and m = -4/5 114. y -2 = 3(x+3) 115. k=1/3 116. y-3=-3/4(x+8) 117. no; 118. a. no; slope of m = 1, slope of n = 4/7; b. no, slope of m=-1/6, slope of n= -1/7 119. x=9 and y=8 and z=7 120. x=8 and y=7 121. a is perpendicular to d 122. a is parallel to d 123. x=6; y=12; z=7 124. x=18; y=7 125. b is perpendicular to d 126. b is perpendicular to d 127. y=-2x+3 128. k= -2 129. y=-1/4x 130. no, y=4x +5 and y =1/3x +7 are not perpendicular 131. a. no slope of m= -1/6; slope of n= 7/2; b. no slope of m= 1/5; slope of n= -1/10 132. y=3/2x+11 133. k=-3/2 134. y=1/2x-6 135. yes, slopes are m=-1/4 and 4 NJCTL.org 136. a. no slope of m= 3/2; slope of n= -1/2; b. no, slope of m=1/5; slope of n= -10 137. Statement P is parallel to n Angle 9 is congruent to angle 5 Angle 9 is congruent to angle 1 Angle 5 is congruent to angle 1 K is parallel to m Reasoning Given Statement <1 and <4 form a linear pair. <3 and <4 form a linear pair m<1+m<4=180 m<3+m<4=180 m<1+m<4= m<3+m<4 m<1=m<3 <1 @ <3 Corresponding angles postulate Substitution property 140. suur suur AB || CD <1 @ 4 <3 @ <5 m<1=m<4 m<3=m<5 138. 1. Same-side interior angles theorem 2. Substitution prop of equality 3. Subtraction prop of equality 4. Definition of congruent segments 5. Converse of corresponding angles postulate Reasoning Given Linear Pair Postulate Substitution property of = Subtraction prop of = Def. of congruent Segments Written as a flowchart. Statement converse of corresponding angles postulate Geometry Parallel & Perpendicular 139. m<1+m<2+m<3 =180 m<4+m<2+m<5 =180 Reasoning Given Alternate interior angles th. Def. of congruent segments Angle Addition Postulate Substition Prop Of =. 141. See student work 142. made same side interior the same 143. See student work 144. Made angles congruent that should be supplementary. 145. see student work 146. see student work 147. see student work 148. see student work ~20~ NJCTL.org b. construct δ 149. construct through P to Elm. 150. arcs werenβt drawn to intersect 151. see student work 152. see student work 153. see student work 154. arcs werenβt drawn to intersect c. -7/4 3. a. 7x + 4y = 23 b. construct c. 4/7 d. Find the equation of line p. Solve the system of equations for the point of intersection. Use the distance formula to find the distance from A to the point of intersection. REVIEW 1. c 2. c 3. d 4. b 5. d 6. b 7. d 8. c 9. d 10. d 11. a 12. b 13. b 4. See student work 5. See student work EXTENDED RESPONSE 1. Statement Reasoning Given Given Alternate Interior Angles Theorem Substitution 2. a. y β 4 = -7/4(x β 1) or y + 3 = -7/4(x β 5) Geometry Parallel & Perpendicular ~21~ NJCTL.org