# data[].attributes.headline
Equations for parallel and perpendicular lines (3.6)

# data[].attributes.subject
Equations for parallel and perpendicular lines (3.6)

# data[].attributes.attachment_link.readable_text
Definition of Parallel and Perpendicular

Parallel lines Lines in the same plane that do not intersect; their slopes are the same. are lines in the same plane that never intersect. Two nonvertical lines in the same plane, with slopes and , are parallel if their slopes are the same, . Consider the following two lines:

Consider their corresponding graphs:

then the slope of a perpendicular line is the opposite reciprocal:

The mathematical notation reads " m perpendicular." We can verify that two slopes produce perpendicular lines if their product is −1.

Geometrically, we note that if a line has a positive slope, then any perpendicular line will have a negative slope. Furthermore, the rise and run between two perpendicular lines are interchanged.

Perpendicular lines have slopes that are opposite reciprocals, so remember to find the reciprocal and change the sign. In other words,

Determining the slope of a perpendicular line can be performed mentally. Some examples follow.

Example 2: Determine the slope of a line perpendicular to .

Solution: First, solve for y and express the line in slope-intercept form.

Try this! Find the slope of the line perpendicular to .

Finding Equations of Parallel and Perpendicular Lines

We have seen that the graph of a line is completely determined by two points or one point and its slope. Often you will be asked to find the equation of a line given some geometric relationship-for instance, whether the line is parallel or perpendicular to another line.

It is important to have a geometric understanding of this question. We were asked to find the equation of a line parallel to another line passing through a certain point.

Through the point (6, −1) we found a parallel line, , shown dashed. Notice that the slope is the same as the given line, but the y-intercept is different. If we keep in mind the geometric interpretation, then it will be easier to remember the process needed to solve the problem.

Geometrically, we see that the line , shown dashed below, passes through (−1, −5) and is perpendicular to the given line.

It is not always the case that the given line is in slope-intercept form. Often you have to perform additional steps to determine the slope. The general steps for finding the equation of a line are outlined in the following example.

Example 5: Find the equation of the line passing through (8, −2) and perpendicular to .
Solution:
Step 2: Substitute the slope you found and the given point into the point-slope form of an equation for a line. In this case, the slope is and the given point is (8, −2).

Solution: Find the slope m by solving for y.

When finding an equation of a line perpendicular to a horizontal or vertical line, it is best to consider the geometric interpretation.

Solution: We recognize that is a horizontal line and we want to find a perpendicular line passing through (−3, −2).

If we draw the line perpendicular to the given horizontal line, the result is a vertical line.

We can rewrite the equation of any horizontal line, , in slope-intercept form as follows:

Written in this form, we see that the slope is . If we try to find the slope of a perpendicular line by finding the opposite reciprocal, we run into a problem: , which is undefined. This is why we took care to restrict the definition to two nonvertical lines. Remember that horizontal lines are perpendicular to vertical lines.

Key Takeaways

Parallel lines have the same slope.

To find an equation of a line, first use the given information to determine the slope. Then use the slope and a point on the line to find the equation using point-slope form.

Horizontal and vertical lines are perpendicular to each other.

Topic Exercises

Part A: Parallel and Perpendicular Lines
Determine the slope of parallel lines and perpendicular lines. Determine if the lines are parallel, perpendicular, or neither.
Part B: Equations in Point-Slope Form
Find the equation of the line.
Answers

1: and

5: and

13: and

15: and

17: and

21: Parallel

23: Perpendicular

25: Perpendicular

27: Perpendicular

29: Parallel

31: Neither

33: Parallel

35: Parallel

# data[].attributes.attachment_link.description
Parallel lines Lines in the same plane that do not intersect; their slopes are the same. are lines in the same plane that never intersect. Two nonvertical lines in the same plane, with slopes and , are parallel if their slopes are the same, .

# data[].attributes.headline
Standard way to covert ay = mx + b (3.5) notes 

# data[].attributes.subject
Standard way to covert ay = mx + b (3.5) notes 

# data[].attributes.headline
Interaction form of slope y=mx +b (how to write equations from a graph from an equation)

# data[].attributes.subject
Interaction form of slope y=mx +b (how to write equations from a graph from an equation)

# data[].attributes.headline
"How to compare slopes of lines to know if it is parallel, perpendicular or none"

# data[].attributes.subject
"How to compare slopes of lines to know if it is parallel, perpendicular or none"

# data[].attributes.attachment_link.title
ALGEBRA: How to Identify  Parallel and Perpendicular Lines Using the Slope of a Line

# data[].attributes.headline
Slope  ( definition of how to find the slope using a graph.) ( how to find slope using slope formula.)

# data[].attributes.subject
Slope  ( definition of how to find the slope using a graph.) ( how to find slope using slope formula.)

# data[].attributes.attachment_link.readable_text
Example 1

Find the slope of the line in the graph below

Plot and label 2 points on the line, anywhere on the line. Remember that the slope of a line never changes, so you can choose whatever 2 points you want and you will always get the same slope.

Calculate the rise and run (You can draw it on the graph if it helps).

Use the slope formula.

The slope is 2/4, which , of course, you can simplify to ½.

# data[].attributes.attachment_link.title
How to Find Slope from Graph. Tutorial, Examples , Practice Problems

# data[].attributes.attachment_link.description
Example 1 Find the slope of the line in the graph below Plot and label 2 points on the line, anywhere on the line. Remember that the slope of a line never changes, so you can choose whatever 2 points you want and you will always get the same slope.

# data[].attributes.headline
"Angles formed by parallel lines cut by transverse"

# data[].attributes.subject
"Angles formed by parallel lines cut by transverse"

# data[].attributes.attachment_link.title
Parallel and Perpendicular Lines, Transversals, Alternate Interior Angles, Alternate Exterior Angles

# data[].attributes.headline
4 Types of lines  perpendicular, skew , intersectng and parallel.

# data[].attributes.subject
4 Types of lines  perpendicular, skew , intersectng and parallel.

# data[].attributes.headline
"Relationships of angles as adjacent vertical supplementary complement and for linear"

# data[].attributes.subject
"Relationships of angles as adjacent vertical supplementary complement and for linear"

# data[].attributes.headline
'Angles identification classification types of angles and measurment "

# data[].attributes.subject
'Angles identification classification types of angles and measurment "

# data[].attributes.headline
"Line Segments (definition, midpoint, midpoint formula, distance formula)"

# data[].attributes.subject
"Line Segments (definition, midpoint, midpoint formula, distance formula)"

# data[].attributes.headline
" Resolver ecuaciones de varios pasos (3 tipos: con términos distributivos y combinados, con variables en ambos lados y con ambos)"

# data[].attributes.subject
" Resolver ecuaciones de varios pasos (3 tipos: con términos distributivos y combinados, con variables en ambos lados y con ambos)"

# data[].attributes.attachment_link.title
Solving Multi Step Equations with Variables on Both Sides of Equal Sign