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4. Contents Lesson 3-1Parallel Lines and Transversals Lesson 3-2Angles and Parallel Lines Lesson 3-3Slopes of Lines Lesson 3-4Equations of Lines Lesson 3-5Proving Lines Parallel Lesson 3-6Perpendiculars and Distance

5. Lesson 1 Contents Example 1Identify Relationships Example 2Identify Transversals Example 3Identify Angle Relationships

6. Example 1-1a Name all planes that are parallel to plane AEF. Answer: plane BHG

7. Example 1-1b Name all segments that intersect Answer:

8. Example 1-1c Name all segments that are parallel to Answer:

9. Example 1-1d Name all segments that are skew to Answer:

10. Example 1-1e Use the figure to name each of the following. Answer: plane XYZ a. Name all planes that are parallel to plane RST. b. c. d. Answer:

11. Example 1-2a BUS STATION Some of a bus station’s driveways are shown. Identify the sets of lines to which line v is a transversal. Answer: If the lines are extended, line v intersects lines u, w, x, and z.

12. Example 1-2b BUS STATION Some of a bus station’s driveways are shown. Identify the sets of lines to which line y is a transversal. Answer: lines u, w, x, z

13. Example 1-2c BUS STATION Some of a bus station’s driveways are shown. Identify the sets of lines to which line u is a transversal. Answer: lines v, x, y, z

14. Example 1-2d BUS STATION Some of a bus station’s driveways are shown. Identify the sets of lines to which line w is a transversal. Answer: lines v, x, y, z

15. Example 1-2e HIKING A group of nature trails is shown. Identify the sets of lines to which each given line is a transversal. Answer: lines c, d, e, f Answer: lines a, b, d, e, f a. line a b. line b c. line c d. line d Answer: lines a, b, c, e, f

16. Example 1-3a Answer: corresponding Identify as alternate interior, alternate exterior, corresponding, or consecutive interior angles.

17. Example 1-3b Answer: alternate exterior Identify as alternate interior, alternate exterior, corresponding, or consecutive interior angles.

18. Example 1-3c Answer: corresponding Identify as alternate interior, alternate exterior, corresponding, or consecutive interior angles.

19. Example 1-3d Answer: alternate exterior Identify as alternate interior, alternate exterior, corresponding, or consecutive interior angles.

20. Example 1-3e Answer: alternate interior Identify as alternate interior, alternate exterior, corresponding, or consecutive interior angles.

21. Example 1-3f Answer: consecutive interior Identify as alternate interior, alternate exterior, corresponding, or consecutive interior angles.

22. Example 1-3g Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. Answer: consecutive interior Answer: corresponding Answer: alternate exterior a. b. c.

23. Example 1-3g Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. Answer: alternate interior Answer: corresponding Answer: alternate exterior d. e. f.

24. End of Lesson 1

25. Lesson 2 Contents Example 1Determine Angle Measures Example 2Use an Auxiliary Line Example 3Find Values of Variables

26. Answer: Example 2-1a Corresponding Angles Postulate Vertical Angles Theorem Transitive Property Definition of congruent angles Substitution In the figure and Find

27. Example 2-1b Answer: In the figure and Find

28. Example 2-2a Grid-In Test Item What is the measure of  RTV? Read the Test Item Be sure to identify it correctly on the figure.

29. Example 2-2b Solve the Test Item Alternate Interior Angles Theorem Definition of congruent angles Substitution

30. Example 2-2c Angle Addition Postulate Write each digit of 125 in a column of the grid. Then shade in the corresponding bubble in each column. Definition of congruent angles Substitution Alternate Interior Angles Theorem

31. Example 2-2c Answer:

32. Example 2-2d Grid-In Test Item What is the measure of  IGE? Answer:

33. Example 2-3e If ALGEBRAand find x and y. Find x. by the Corresponding Angles Postulate.

34. Example 2-3f Definition of congruent angles Substitution Subtract x from each side and add 10 to each side. Definition of congruent angles Substitution Find y. by the Alternate Exterior Angles Theorem.

35. Example 2-3g Simplify. Add 100 to each side. Divide each side by 4. Answer:

36. Example 2-3h Answer: ALGEBRA If and find x and y.

37. End of Lesson 2

38. Lesson 3 Contents Example 1Find the Slope of a Line Example 2Use Rate of Change to Solve a Problem Example 3Determine Line Relationships Example 4Use Slope to Graph a Line

39. From (–3, 7) to (–1, –1), go down 8 units and right 2 units. Example 3-1a Find the slope of the line. Answer: – 4

40. Example 3-1b Use the slope formula. Answer: undefined Find the slope of the line. Let be and be.

41. Example 3-1c Find the slope of the line. Answer:

42. Example 3-1d Find the slope of the line. Answer: 0

43. Example 3-1e a. Find the slope of the line. Answer:

44. Example 3-1f b. Find the slope of the line. Answer: 0

45. Example 3-1g c. Find the slope of the line. Answer: 2

46. Example 3-1h d. Find the slope of the line. Answer: undefined

47. Example 3-2a RECREATION For one manufacturer of camping equipment, between 1990 and 2000, annual sales increased by $7.4 million per year. In 2000, the total sales were $85.9 million. If sales increase at the same rate, what will be the total sales in 2010? Slope formula

48. Example 3-2b Multiply each side by 10. Add 85.9 to each side. Simplify. The coordinates of the point representing the sales for 2010 are (2010, 159.9). Answer: The total sales in 2010 will be about $159.9 million.

49. Example 3-2c CELLULAR TELEPHONES Between 1994 and 2000, the number of cellular telephone subscribers increased by an average rate of 14.2 million. In 2000, the total subscribers were 109.5 million. If the number of subscribers increase at the same rate, how many subscribers will there be in 2005? Answer: about 180.5 million

50. Example 3-3a Determine whether and are parallel, perpendicular, or neither.

51. Example 3-3b The slopes are not the same, The product of the slopes is are neither parallel nor perpendicular. Answer:

52. Example 3-3c Answer: The slopes are the same, so are parallel. Determine whether and are parallel, perpendicular, or neither.

53. Example 3-3d Answer: perpendicular Answer: neither a. b. Determine whether and are parallel, perpendicular, or neither.

54. Example 3-4a Graph the line that contains Q(5, 1) and is parallel to with M(–2, 4) and N(2, 1). Substitution Simplify. Slope formula

55. Example 3-4b The slopes of two parallel lines are the same. Graph the line. Answer: The slope of the line parallel to Start at (5, 1). Move up 3 units and then move left 4 units. Label the point R.

56. Example 3-4c Graph the line that contains R(2, –1) and is parallel to with O(1, 6) and P(–3, 1). Answer:

57. End of Lesson 3

58. Lesson 4 Contents Example 1Slope and y-Intercept Example 2Slope and a Point Example 3Two Points Example 4One Point and an Equation Example 5Write Linear Equations

59. Example 4-1a Write an equation in slope-intercept form of the line with slope of 6 and y-intercept of –3. Answer: The slope-intercept form of the equation of the line is Slope-intercept form

60. Example 4-1b Write an equation in slope-intercept form of the line with slope of –1 and y-intercept of 4. Answer:

61. Example 4-2a Answer: Write an equation in point-slope form of the line whose slope is that contains (–10, 8). Simplify. Point-slope form

62. Example 4-2b Answer: Write an equation in point-slope form of the line whose slope is that contains (6, –3).

63. Example 4-3a Write an equation in slope-intercept form for a line containing (4, 9) and (–2, 0). Find the slope of the line. Slope formula Simplify.

64. Example 4-3b Now use the point-slope form and either point to write an equation. Point-slope form Add 9 to each side. Using (4, 9): Distributive Property

65. Example 4-3c Point-slope form Distributive Property Using (–2, 0): Simplify. Answer:

66. Example 4-3d Write an equation in slope-intercept form for a line containing (3, 2) and (6, 8). Answer:

67. Example 4-4a Write an equation in slope-intercept form for a line containing (1, 7) that is perpendicular to the line the slope of a line perpendicular to it is 2.

68. Example 4-4b Point-slope form Distributive Property Add 7 to each side. Answer:

69. Example 4-4c Answer: Write an equation in slope-intercept form for a line containing (–3, 4) that is perpendicular to the line

70. Example 4-5a RENTAL COSTS An apartment complex charges $525 per month plus a $750 security deposit. Write an equation to represent the total annual cost A for r months of rent. For each month of rent, the cost increases by $525. So the rate of change, or slope, is 525. The y-intercept is located where 0 months are rented, or $750. Answer: The total annual cost can be represented by the equation Slope-intercept form

71. Example 4-5b RENTAL COSTS An apartment complex charges $525 per month plus a $750 security deposit. Compare this rental cost to a complex which charges a $200 security deposit but $600 per month for rent. If a person expects to stay in an apartment for one year, which complex offers the better rate? First complex: Second complex: Simplify. Answer: The first complex offers the better rate: one year costs $7050 instead of $7400.

72. Example 4-5d Answer: RENTAL COSTS A car rental company charges $25 per day plus a $100 deposit. a. Write an equation to represent the total cost C for d days of use. b. Compare this rental cost to a company which charges a $50 deposit but $35 per day for use. If a person expects to rent a car for 9 days, which company offers the better rate? Answer: The first company offers the better rate. Nine days cost $325 instead of $365.

73. End of Lesson 4

74. Lesson 5 Contents Example 1Identify Parallel Lines Example 2Solve Problems with Parallel Lines Example 3Prove Lines Parallel Example 4Slope and Parallel Lines

75. Example 5-1a Determine which lines, if any, are parallel. consecutive interior angles are supplementary. So, consecutive interior angles are not supplementary. So, c is not parallel to a or b. Answer:

76. Example 5-1b Determine which lines, if any, are parallel. Answer:

77. Example 5-2a ALGEBRA Find x and m  ZYN so that Explore From the figure, you know that and You also know that are alternate exterior angles.

78. Example 5-2b Alternate exterior angles Subtract 7x from each side. Substitution Add 25 to each side. Divide each side by 4. Solve Plan For line PQ to be parallel to MN, the alternate exterior angles must be congruent. Substitute the given angle measures into this equation and solve for x. Once you know the value of x, use substitution to find

79. Example 5-2c Answer: Original equation Simplify. Examine Verify the angle measure by using the value of x to find Since

80. Example 5-2d ALGEBRA Find x and m  GBA so that Answer:

81. Example 5-3e Given: Prove:

82. Example 5-3f Proof: 1. Given 1. 5. Substitution 5. ReasonsStatements 2. Consecutive Interior Thm. 2.. 3. Def. of suppl. s 3. 4. Def. of congruent s 4. 6. Def. of suppl. s6.. 7. If cons. int. s are suppl., then lines are. 7.

83. Example 5-3g Given: Prove:

84. Example 5-3h Proof: 1. Given1. 2. Alternate Interior Angles2. 3. Substitution3. 4. Definition of suppl.  s 4.. 5. Definition of suppl.  s 5. 6. Substitution6. 7. If cons. int.  s are suppl., then lines are. 7. ReasonsStatements

85. Example 5-4a Answer:

86. Example 5-4b Answer: Since the slopes are not equal, r is not parallel to s.

87. End of Lesson 5

88. Lesson 6 Contents Example 1Distance from a Point to a Line Example 2Construct a Perpendicular Segment Example 3Distance Between Lines

89. Example 6-1a Draw the segment that represents the distance from Since the distance from a line to a point not on the line is the length of the segment perpendicular to the line from the point, Answer:

90. Example 6-1b Answer: Draw the segment that represents the distance from

91. Example 6-2a Construct a line perpendicular to line s through V(1, 5) not on s. Then find the distance from V to s.

92. Example 6-2b Graph line s and point V. Place the compass point at point V. Make the setting wide enough so that when an arc is drawn, it intersects s in two places. Label these points of intersection A and B.

93. Example 6-2c Put the compass at point A and draw an arc below line s. (Hint: Any compass setting greater than will work.)

94. Example 6-2d Using the same compass setting, put the compass at point B and draw an arc to intersect the one drawn in step 2. Label the point of intersection Q.

95. Example 6-2e Draw. and s. Use the slopes of and s to verify that the lines are perpendicular.

96. Example 6-2f Answer: The distance between V and s is about 4.24 units. The segment constructed from point V(1, 5) perpendicular to the line s, appears to intersect line s at R(–2, 2). Use the Distance Formula to find the distance between point V and line s.

97. Example 6-2g Construct a line perpendicular to line m through Q(–4, –1) not on m. Then find the distance from Q to m. Answer:

98. Example 6-3a You will need to solve a system of equations to find the endpoints of a segment that is perpendicular to both a and b. The slope of lines a and b is 2. Find the distance between the parallel lines a and b whose equations are and respectively.

99. Example 6-3b Point-slope form Add 3 to each side. First, write an equation of a line p perpendicular to a and b. The slope of p is the opposite reciprocal of 2, Simplify. Use the y-intercept of line a, (0, 3), as one of the endpoints of the perpendicular segment.

100. Example 6-3c Next, use a system of equations to determine the point of intersection of line b and p. Substitute 2x–3 for y in the second equation.

101. Example 6-3d Substitute 2.4 for x in the equation for p. Simplify on each side. The point of intersection is (2.4, 1.8). Group like terms on each side.

102. Example 6-3e Distance Formula Then, use the Distance Formula to determine the distance between (0, 3) and (2.4, 1.8). Answer: The distance between the lines is or about 2.7 units.

103. Example 6-3f Answer: Find the distance between the parallel lines a and b whose equations are and respectively.

104. End of Lesson 6

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