Slopes of Parallel and Perpendicular Lines

Objective:To discover the relationships between the slopes of

parallel lines and perpendicular lines.

What type of lines do lines “p” and “q” look like?

Answer

p and q appear to be parallel lines!

How can we use mathematics to be sure they are indeed parallel?

Calculate the slope of each.

slope of p = slope of q = = 4

14

1

12

3

What do you notice about the slopes of these two lines?• The slopes are congruent.

parallel

congruent

Parallel Conjucture:

• Two lines are parallel if their slopes are equal.

• Recall the definition of parallel lines then write a few sentences describing why it makes sense that parallel lines would have equal slopes.

Are these lines parallel?

slope a = -

slope b = -6

1NO –The slopes are not

5

1

Are these lines parallel?

slope c =

slope d = =

5

4

5

410

8YES – Slopes are

You will be shown the slopes of two lines. Write down whether each pair of lines are parallel or not.

1. slope of line r =

slope of line s =

4

3

4

3

These lines are parallel.

3. slope of line t =

slope of line u =

These lines are parallel.

8

6

4

3

4. slope of line t =

slope of line u =

Not parallel because the slopes are not congruent to

each other.

8

9

9

8

Ex. Determine whether the graphs of y = -3x + 4 and 6x + 2y = -10 are parallel lines.

Step 1: make both equations in the y-intercept form to compare slopes

6x + 2y = -10 can be changed to y-intercept form by solving for y

6x + 2y = -10

-6x -6x

2y = -6x -10

2 2 2

y = -3x - 5

Subtract 6x

Divide by 2

Step 2: compare the slopes of both equations.

The first equation y = -3x + 4 has a slope of -3 and the second equation has the same slope of -3

Therefore, the graph of the lines will be parallel.

Practice: Determine whether the graphs are parallel lines (without graphing)

1) 3x - y = -5 and 5y -15x = 10

2) 4y = -12x + 16 and y = 3x + 4