## Slopes: Parallel and Perpendicular Lines

If lines are parallel, they slant in exactly the same direction. If they are nonvertical, their steepness is exactly the same.

*Theorem 103:* If two nonvertical lines are parallel, then they have the same slope.

*Theorem 104:* If two lines have the same slope, then the lines are nonvertical parallel lines.

If two lines are perpendicular and neither one is vertical, then one of the lines has a positive slope, and the other has a negative slope. Also, the absolute values of their slopes are reciprocals.

*Theorem 105:* If two nonvertical lines are perpendicular, then their slopes are opposite reciprocals of one another, or the product of their slopes is −1.

*Theorem 106:* If the slopes of two lines are opposite reciprocals of one another, or the product of their slopes is −1, then the lines are nonvertical perpendicular lines.

Horizontal and vertical lines are always perpendicular: therefore, two lines, one of which has a zero slope and the other an undefined slope are perpendicular.

**Example 1:** If line *l* has slope 3/4, then (a) any line parallel to *l* has slope ___, *and (b) any line perpendicular to **l* has slope ___.

a. (a) 3/4 *(Theorem 103)*

b. (b) −4/3 *(Theorem 105)*

**Example 2:** Given points *Q, R, S,* and *T*, tell which sides, if any, of quadrilateral *QRST* in Figure 1 are parallel or perpendicular.

**Figure 1 ***Determining which sides, if any, of a quadrilateral are parallel or perpendicular.*