Parallel and Perpendicular Lines

Two lines are parallel if they never intersect; they are perpendicular if they meet at a right angle. Both conditions can be stated precisely using slopes: parallel lines have equal slopes, and perpendicular lines have slopes whose product is .

Quick Reference

Relationship	Slope Condition	Example
Parallel		Slopes and
Perpendicular		Slopes and

Parallel Lines

Two lines in the same plane are called parallel if they never intersect, no matter how far they are extended. The figure below shows several lines of the form , which are all parallel to each other:

Several parallel blue lines of the form y = 2x + b plotted on a coordinate plane, all having the same slope. — Parallel lines of the form y = 2x + b all share the same slope

Parallel Lines Theorem: Two nonvertical lines with slopes and are parallel if and only if .

Proof (click to expand)

() Parallel lines have equal slopes.

Let the two lines be and . To find an intersection, set them equal:

Parallel lines have no intersection, so this equation must have no solution. That happens exactly when , i.e., .

() Lines with equal slopes are parallel.

Suppose . Setting gives .

If , there is no solution: the lines do not intersect and are parallel.
If , the equations are identical: the lines coincide, which can also be considered parallel.

Example 1. Write the equation of the line parallel to that passes through .

Solution. First, rewrite the given line in slope-intercept form:

The slope is . The parallel line has the same slope.

Method (a): Point-Slope Form

Method (b): Slope-Intercept Form

Write and substitute :

Both methods give .

Perpendicular Lines

Two lines are perpendicular if they intersect at a right angle (). The figure below shows lines of slope and lines of slope , which are perpendicular to each other:

A coordinate plane showing blue parallel lines with slope 2 and red parallel lines with slope -1/2 that intersect the blue lines at right angles. — Lines of slope 2 and lines of slope −1/2 are perpendicular

Perpendicular Lines Theorem: Two nonvertical lines with slopes and are perpendicular if and only if:

Equivalently, the slopes of perpendicular lines are negative reciprocals of each other: .

A horizontal line (slope ) is perpendicular to every vertical line (undefined slope).

Proof using the Pythagorean Theorem (click to expand)

Assume lines and with slopes and intersect perpendicularly at :

Geometric diagram showing two perpendicular lines intersecting at P, with points A on L₁ and B on L₂ one unit to the right of P, forming a triangle used to apply the Pythagorean theorem. — Proof of the perpendicularity condition using the Pythagorean Theorem

Move one unit to the right from to reach point on and point on . By the Distance Formula:

The lines are perpendicular if and only if (Pythagorean Theorem):

Example 2. Write the equation of the line perpendicular to that passes through .

Solution. Rewrite the given line in slope-intercept form:

The slope of the given line is . The perpendicular slope is:

Method (a): Point-Slope Form

Method (b): Slope-Intercept Form

Write and substitute :

The equation is .

Graph showing the blue line 2x − 5y + 10 = 0 and the perpendicular red line y = -5/2x - 13/2 passing through (-3, 1). — The original line (blue) and its perpendicular through (−3, 1) (red)

Example 3. Show that , , and are vertices of a right triangle.

Solution. Compute the slopes of all three sides:

Coordinate plane showing triangle ABC with vertices at A(1,4), B(5,1), and C(2,−3). The right angle is at vertex B. — Triangle ABC with a right angle at B

Check whether any pair of slopes multiplies to :

Since , sides and are perpendicular. Therefore triangle is a right triangle with the right angle at vertex .

Frequently Asked Questions

How do I know if two lines are parallel?

Two nonvertical lines are parallel if and only if they have the same slope. Rewrite both equations in slope-intercept form and compare the slopes. If and , the lines are distinct and parallel. If both the slopes and intercepts are equal, the lines are the same.

How do I find the slope of a perpendicular line?

If a line has slope

, any line perpendicular to it has slope

(the negative reciprocal). For example, if the slope is

, the perpendicular slope is

. Note: horizontal lines (slope 0) are perpendicular to vertical lines (undefined slope), which is the one case the formula does not apply to.

What is a negative reciprocal?

The negative reciprocal of a number

: flip the fraction and change the sign. For example:

Negative reciprocal of is .
Negative reciprocal of is .

Two nonzero numbers

and

are negative reciprocals of each other exactly when

Can two lines be both parallel and perpendicular?

No. Parallel lines have equal slopes, while perpendicular lines have slopes with product

. If

, then

, which equals

only if

(imaginary). For real lines, no two distinct lines can be simultaneously parallel and perpendicular.

How do I write the equation of a line parallel or perpendicular to a given line?

Follow these steps:

Find the slope of the given line by rewriting it in slope-intercept form.
For a parallel line, use the same slope. For a perpendicular line, take the negative reciprocal.
Use the point-slope form with the new slope and the given point to write the equation.

Parallel and Perpendicular Lines

Parallel and Perpendicular Lines

Quick Reference

Parallel Lines

Perpendicular Lines

Frequently Asked Questions

Quick Links

Social Media