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Chapter 3

Chapter 3. Perpendicular and Parallel Lines. 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.

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Chapter 3

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  1. Chapter 3 Perpendicular and Parallel Lines

  2. 3.1 – Lines and Angles • Two lines are PARALLEL LINESif 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. • 7 and 9 are __________ angles. 6 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. 1 = 4 6 = 4 2 + 3 = 5 1 = 7 1 = 8 2 + 3 + 8 = 180 Which lines are parallel if given: ~ ~ 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 F E

  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). • Line 2 passes through (-2,6) and (0,1). • Line 3 passes through (-6,5) and (-4,0)

  44. Perpendicular? • Line 1 passes through (4,2) and (1,-4). • Line 2 passes through (-1,2) and (5,-1).

  45. 3.6-3.7 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?

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