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Warm Up Find the reciprocal. 1. 2 2. 3.

Warm Up Find the reciprocal. 1. 2 2. 3. Find the slope of the line that passes through each pair of points. 4. (2, 2) and (–1, 3) 5. (3, 4) and (4, 6) 6. (5, 1) and (0, 0) . 3. 2. Objectives.

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Warm Up Find the reciprocal. 1. 2 2. 3.

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  1. Warm Up Find the reciprocal. 1. 2 2. 3. Find the slope of the line that passes through each pair of points. 4. (2, 2) and (–1, 3) 5.(3, 4) and (4, 6) 6. (5, 1) and (0, 0) 3 2

  2. Objectives Identify and graph parallel and perpendicular lines. Write equations to describe lines parallel or perpendicular to a given line.

  3. Vocabulary parallel lines perpendicular lines

  4. These two lines are parallel. Parallel linesare lines in the same plane that have no points in common. In other words, they do not intersect.

  5. Directions: Identify which lines are parallel. Hint: Look at the slope (if the slopes are the same, then the lines are parallel). Remember: an equation must be written in slope intercept form to compare slopes!!!

  6. The lines described by and both have slope . These lines are parallel. The lines described by y = x and y = x + 1 both have slope 1. These lines are parallel. Example 1

  7. –2x – 2x – 1 – 1 Example 2 Write all equations in slope-intercept form to determine the slope. 2x + 3y =8 y + 1 = 3(x – 3) y + 1 = 3x – 9 3y = –2x + 8 y = 3x – 10

  8. The lines described by and represent parallel lines. They each have the slope . y = 2x – 3 y + 1 = 3(x – 3) Example 2 Continued The lines described by y = 2x – 3 and y + 1 = 3(x – 3) are not parallel with any of the lines.

  9. Use the ordered pairs and the slope formula to find the slopes of MJ and KL. MJ is parallel to KL because they have the same slope. JK is parallel to ML because they are both horizontal. Example 3 Show that JKLM is a parallelogram. Since opposite sides are parallel, JKLM is a parallelogram.

  10. Perpendicular lines are lines that intersect to form right angles (90°). Perpendicular lines have slopes with opposite signs (one positive and one negative) AND are reciprocals of each other.

  11. Directions: Identify which lines are perpendicular Hint: Look at the slope (if the slopes have opposite signs (one positive and one negative) AND are reciprocals of each other, then the lines are perpendicular). Remember: an equation must be written in slope intercept form to compare slopes!!!

  12. y = 3; x = –2; y = 3x; y = 3 x = –2 The slope of the line described by y = 3x is 3. The slope of the line described by y =3x is . Example 4 The graph described by y = 3 is a horizontal line, and the graph described by x = –2 is a vertical line. These lines are perpendicular.

  13. y = 3 x = –2 y =3x Example 4 Continued These lines are perpendicular because the product of their slopes is –1.

  14. y = –4; y – 6 = 5(x + 4); x = 3; y = x = 3 The slope of the line described by y – 6 = 5(x + 4) is 5. The slope of the line described by y = is y = –4 y – 6 = 5(x + 4) Example 5 The graph described by x = 3 is a vertical line, and the graph described by y = –4 is a horizontal line. These lines are perpendicular.

  15. x = 3 y = –4 y – 6 = 5(x + 4) Example 5 Continued These lines are perpendicular because the product of their slopes is –1.

  16. If ABC is a right triangle, AB will be perpendicular to AC. slope of slope of AB is perpendicular to AC because Example 6 Show that ABC is a right triangle. Therefore, ABC is a right triangle because it contains a right angle.

  17. slope of = XY slope of YZ = 4 slope of = WZ slope of XW = 4 The product of the slopes of adjacent sides is –1. Therefore, all angles are right angles, and WXYZ is a rectangle. Lesson Summary 3. Show that WXYZ is a rectangle.

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