Slopes of Parallel and Perpendicular Lines

Recall that two lines are parallel if they are in the same plane and do not intersect.

Two lines are perpendicular if they intersect at a right angle

Example 1: Consider the following graph with two lines that are parallel.

Since both lines have the same “steepness”, their slopes must be the same value.

1m

2m

1 2m m

Example 2: Consider the following graph with two lines that are perpendicular.

Notice that the line in red has a positive slope (rising), while the line in blue has a negative slope (falling).

1m2m

1m2m

1

rise

runm

n

1

ru

3

1

1

1

3m

1m2m

2

rise

runm

n

-3

ru

1

-3

1

1

3m

2

3

1m

1m2m

1

1

3m

2

3

1m

This is a quite interesting result. Not only are the slopes opposite in sign, but they are reciprocals!

As it turns out, this is always the case, assuming that neither line is vertical, which has undefined slope.

Example 3: Special case #1

When both lines are vertical, they are parallel.

1m2m

1 2, are undefinedm m

Recall that vertical lines have undefined slope. Thus, …

Example 4: Special case #2

When one line is vertical, and the other is horizontal, then the lines are perpendicular.

1m

2m

The slope of the vertical line is undefined …

… and the slope of the horizontal line is zero.

SUMMARY

Parallel lines Perpendicular lines

1 2m m

Slopes are the sameSlopes are opposite reciprocals

(assume no vertical lines)

21

1m

m

a) Determine the slope of Line 2 that is parallel to Line 1

Example 5: Line 1 has a slope of – 3.

2 1m m

2 3m

b) Determine the slope of Line 3 if it is perpendicular to Line 1

31

1m

m 1

3

1

3

3

1

3m

3

a) Determine the slope of Line 2 that is parallel to Line 1

Example 6: Line 1 has a slope of 3/4.

2 1m m

2

3

4m

3

4

Line 1 has a slope of 3/4.

b) Determine the slope of Line 3 if it is perpendicular to Line 1

3

4

3m

31

1m

m

3 4

1 4

31

4

3

While we have written out all the details, these problems are usually done in your head. For part b again …

Use two steps:

1

3

4m

1) The reciprocal is:4

3

2) Take the opposite:4

3

3

4

3m