Algebra 1 parallel and perpendicular lines worksheet answers

How to use Algebra to find parallel and perpendicular lines.

Table of Contents Show

Parallel Lines
Vertical Lines
Not The Same Line
Example: is y = 3x + 2 parallel to y − 2 = 3x ?
Perpendicular Lines
Quick Check of Perpendicular
Vertical Lines
Parallel and Perpendicular Lines Worksheets
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Parallel Lines

How do we know when two lines are parallel?

Their slopes are the same!

Example:

Find the equation of the line that is:

parallel to y = 2x + 1
and passes though the point (5,4)

The slope of y=2x+1 is: 2

The parallel line needs to have the same slope of 2.

We can solve it using the "point-slope" equation of a line:

y − y1 = 2(x − x1)

And then put in the point (5,4):

y − 4 = 2(x − 5)

And that answer is OK, but let's also put it in y = mx + b form:

y − 4 = 2x − 10

y = 2x − 6

Vertical Lines

But this does not work for vertical lines ... I explain why at the end.

Not The Same Line

Be careful! They may be the same line (but with a different equation), and so are not parallel.

How do we know if they are really the same line? Check their y-intercepts (where they cross the y-axis) as well as their slope:

Example: is y = 3x + 2 parallel to y − 2 = 3x ?

For y = 3x + 2: the slope is 3, and y-intercept is 2

For y − 2 = 3x: the slope is 3, and y-intercept is 2

In fact they are the same line and so are not parallel

Perpendicular Lines

Two lines are Perpendicular when they meet at a right angle (90°).

To find a perpendicular slope:

When one line has a slope of m, a perpendicular line has a slope of −1m

In other words the negative reciprocal

Example:

Find the equation of the line that is

perpendicular to y = −4x + 10
and passes though the point (7,2)

The slope of y=−4x+10 is: −4

The negative reciprocal of that slope is:

m = −1−4 = 14

So the perpendicular line will have a slope of 1/4:

y − y1 = (1/4)(x − x1)

And now put in the point (7,2):

y − 2 = (1/4)(x − 7)

And that answer is OK, but let's also put it in "y=mx+b" form:

y − 2 = x/4 − 7/4

y = x/4 + 1/4

Quick Check of Perpendicular

When we multiply a slope m by its perpendicular slope −1m we get simply −1.

So to quickly check if two lines are perpendicular:

When we multiply their slopes, we get −1

Like this:

Are these two lines perpendicular?

Line	Slope
y = 2x + 1	2
y = −0.5x + 4	−0.5

When we multiply the two slopes we get:

2 × (−0.5) = −1

Yes, we got −1, so they are perpendicular.

Vertical Lines

The previous methods work nicely except for a vertical line:

In this case the gradient is undefined (as we cannot divide by 0):

m = yA − yBxA − xB = 4 − 12 − 2 = 30 = undefined

So just rely on the fact that:

a vertical line is parallel to another vertical line.
a vertical line is perpendicular to a horizontal line (and vice versa).

Summary

parallel lines: same slope
perpendicular lines: negative reciprocal slope (−1/m)

Parallel and Perpendicular Lines Worksheets

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Identifying Parallel Lines Worksheets
These Parallel and Perpendicular Lines Worksheets are great for practicing identifying parallel lines from pictures. These worksheets will produce 6 problems per page.

Identifying Perpendicular Lines Worksheets
These Parallel and Perpendicular Lines Worksheets are great for practicing identifying perpendicular lines from pictures. These worksheets will produce 6 problems per page.

Identifying Parallel, Perpendicular, and Intersecting Lines Worksheets
These Parallel and Perpendicular Lines Worksheets are great for practicing identifying parallel, perpendicular, and intersecting lines from pictures. These worksheets will produce 6 problems per page.

Identifying Parallel, Perpendicular, and Intersecting Lines from a Graph
These Parallel and Perpendicular Lines Worksheets will show a graph of a series of parallel, perpendicular, and intersecting lines and ask a series of questions about the graph. These worksheets will produce 10 problems per page.

Given Slopes of Two Lines Determine if the Lines are Parallel, Perpendicular, or Neither
These Parallel and Perpendicular Lines Worksheets will give the slopes of two lines and ask the student if the lines are parallel, perpendicular, or neither. These worksheets will produce 10 problems per page.

Given Slope of a Line Find Slopes for Parallel and Perpendicular Lines Worksheets
These Parallel and Perpendicular Lines Worksheets will give the slope of a line and ask the student to determine the slope for any line that is parallel and the slope that is perpendicular to the given line. These worksheets will produce 10 problems per page.

Find the Equation of a Parallel Line Passing Through a Given Equation and Point
These Parallel and Perpendicular Lines Worksheets will ask the student to find the equation of a parallel line passing through a given equation and point. These worksheets will produce 6 problems per page.

Find the Equation of a Perpendicular Line Passing Through a Given Equation and Point
These Parallel and Perpendicular Lines Worksheets will ask the student to find the equation of a perpendicular line passing through a given equation and point. These worksheets will produce 6 problems per page.

Given a Pair of Lines Determine if the Lines are Parallel, Perpendicular, or Intersecting
These Parallel and Perpendicular Lines Worksheets will give the student a pair of equations for lines and ask them to determine if the lines are parallel, perpendicular, or intersecting. These worksheets will produce 6 problems per page.