1. Perpendicular and Parallel Lines
Chapter 3
Perpendicular and Parallel Lines

2. 3.1 – Lines and Angles
Two lines are PARALLEL LINES if they are coplanar and they do not intersect.
Two lines are SKEW LINES if they are not coplanar and they do not intersect.
Two planes that do not intersect are called PARALLEL PLANES.

3. Parallel Postulates
If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. P

4. Transversals and Angles
A TRANSVERSAL is a line that intersects two or more coplanar lines at different points.

5. Corresponding Angles
Two angles are corresponding angles if they occupy corresponding positions. 1 2 3 4 5 6 8 7

6. Alternate Interior Angles
Two angles are alternate interior angles if they lie between the two lines on opposite sides of the transversal. 1 2 3 4 5 6 8 7

7. Alternate Exterior Angles
Two angles are alternate exterior angles if they lie outside the two lines on opposite sides of the transversal. 1 2 3 4 5 6 8 7

8. Consecutive Interior Angles
Two angles are consecutive int. angles if they lie between the two lines on the same side of the transversal. (aka: same side interior angles) 1 2 3 4 5 6 8 7

9. Parallel, skew, or perpendicular
UT and WT are: _________
RS and VW are: ___________
TW and WX are: ___________ R S U T X V W

10. Lines and Planes Name a line parallel to HJ. Name a line skew to GH.
Name a line perpendicular to JH. M L K J N G H

11. Angles 6 and 10 are __________ angles. 12 and 6 are __________ angles.5 7 8 10 9 11 12

12. 3.2 Perpendicular Lines
If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. g h
g h

13. Perpendicular Lines
If two sides of two adjacent angles are perpendicular, then the angles are complementary.

14. Perpendicular Lines
If two lines are perpendicular, then they intersect to form four right angles.

15. 3.3 Parallel Lines and Transversals
Corresponding Angles Postulate
If 2 parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2

16. Alternate Interior Angles
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 5 3 4 6

17. Consecutive Interior Angles
If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. 5 3 6 4

18. Alternate Exterior Angles
If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 5 3 4 6

19. Perpendicular Transversal
If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

20. Examples
Find m 1 and m 2. 1 2
105

21. Examples
Find m 1 and m 2. 1 2
135

22. Examples
Find x. 64 2x

23. Examples
Find x.
(3x – 30)

24. 3.4 Parallel Lines
3.4 is basically the converse of 3.3.

25. Corresponding Angles Converse
If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. 1 2

26. Alternate Interior Angles Converse
If two parallel lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. 3 4

27. Consecutive Interior Angles Converse
If two parallel lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel. 5 6

28. Alternate Exterior Angles Converse
If two parallel lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. 5 6

29. 3.5 Properties of Parallel Lines
If two lines are parallel to the same line, then they are parallel to each other. p q r

30. Properties of Parallel Lines
In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. p m n

31. Proving Lines are Parallel
State which pair of lines are parallel and why.
m A = m 6 B D F 5 6 A C E

32. Proving Lines are Parallel
m 6 = m 8 B D 8 7 5 4 6 A C

33. Proving Lines are Parallel
1 = 90 , 5 = 90 Y Z 1 2 4 5 X W

34. Which lines are parallel if given:~
1 = 4
6 = 4
= 5
1 = 7
1 = 8
= 180 ~ l m t ~ j ~ 8 6 ~ 2 1 4 5 3 k 7

35. Find x and y. Given: AB II EF , AE II BF A B C (2x-7) (3x-34) (4y+1) D

36. 3.6-3.7 Coordinate Geometry Slope of a line: The equation of a line:
Slope-intercept form: y = mx + b
Point-slope form: y – y1 = m (x – x1)

37. Parallel and Perpendicular Slopes
In a coordinate plane, two nonvertical lines are parallel iff they have the same slope. (Any two vertical lines are parallel.)
In a coordinate plane, two nonvertical lines are perpendicular iff the product of their slopes is -1. (Vertical and horizontal lines are perpendicular.)

38. Parallel and Perpendicular Slopes
Parallel lines have the same slopes.
Perpendicular lines have slopes that are OPPOSITE RECIPROCALS. (Line 1 has a slope of -2. Line 2 is perpendicular to Line 1 and has a slope of ½ .

39. Horizontal Lines Horizontal lines have a slope = 0 (zero)
The equation of a horizontal line is y = b

40. Vertical Lines Vertical lines have undefined slope (und)
The equation of a vertical line is x = a

41. Finding the equation of a line
Write the equation of the line that goes through (1,1) with a slope of 2.

42. Finding the equation of a line
Find the slope of the line between the points (0,6) and (5,2). Then write the equation of the line.

43. Parallel? Line 1 passes through (0,6) and (2,0).

44. Perpendicular? Line 1 passes through (4,2) and (1,-4).

45. Writing equations
Write the equation of the line which passes through point (4,9) and has a slope of -2.

46. Equations of lines
Write the equation of the line parallel to the line y = -2/5 x + 3 and passing through the point (-5,0).

47. Equations of lines
Write the equation of the line parallel to the line 3x + 2y = 4 and passing through the point (-4, 5).

48. Equations of lines
Write the equation of the line parallel to the x-axis and passing through the point (3,-6).

49. Equations of lines
Write the equation of the line parallel to the y-axis and passing through the point (4,8).

50. Parallel Lines?
Line p1 passes through (0,-3) & (1,-2). Line p2 passes through (5,4) & (-4,-4). Line p3 passes through (-6,-1) & (3,7). Find the slope of each line. Which lines are parallel?

51. Slope
Find the slope of the line passing through the given points.

52. Parallel?
Determine if the two lines are parallel.

53. Slope
What is the slope of the line?

54. Slope
Find the slope of the line that passes through the given points.

55. Parallel?
Determine if the two lines are parallel.

56. Perpendicular? With the given slopes, are the lines perpendicular?
m1 = 2 ; m2 = - ½
m1 = -1 ; m2 = 1
m1 = 5/7 ; m2 = - 7/5

57. Perpendicular?
Find the slopes of the two lines. Determine if they are perpendicular.

58. Perpendicular? Line 1: y = 3x ; Line 2: y = (-1/3)x – 2

59. Perpendicular? Line 1: 3y + 2x = -36 ; Line 2: 4y – 3x = 16

60. Writing equations
Write the equation of the line perpendicular to the line y = (-3/4)x + 6 that goes through the point (8,0)

61. Writing Equations
Write the equation of the line that goes through point (-3,-4) and that is perpendicular to y = 3x + 5.

62. Perpendicular, Parallel or Neither
y = -2x – 1
y = -2x – 3
y = 4x + 10
y = -2x + 5