Let's have a look...

Parallel and perpendicular lines each have their own unique characteristics.

Parallel lines extend forever in each direction, and they never touch or intersect each other. When parallel lines are represented using an equation, their slopes have identical values, but their $y$-intercepts have different values.

In contrast, perpendicular lines intersect each other at a $90{^\circ}$ angle. When non-vertical perpendicular lines are represented using an equation, their slopes have values that are opposite reciprocals.

It is also possible for a set of lines to be neither parallel nor perpendicular. These lines intersect but do not intersect at a $90{^\circ}$ angle.

Examples of parallel lines, perpendicular lines, and lines that are neither parallel nor perpendicular are shown below.

Parallel Lines

Graph of two parallel lines.

Perpendicular Lines

Graph of two perpendicular lines with a right angle mark showing that the two lines intersect each other at a $90{^\circ}$angle.

Neither

Graph of two intersecting lines that are not perpendicular to each other.

Let's Practice

It is important to be able to identify parallel lines, perpendicular lines, and lines that are neither parallel nor perpendicular using images, graphs, and equations. Use the activity below to practice your identification skills. Look at the image on each slide. Decide if the lines are parallel, perpendicular, or neither. Then, check your answer.

Slide: 1 / 4

Do the vertical fence boards in this image depict lines that are parallel, perpendicular, or neither? Explain.

Fence made up of several sets of fence boards that do not touch or intersect.

Answer

These are parallel lines since the vertical fence boards do not touch or intersect.

Are the lines in this graph parallel, perpendicular, or neither? Explain.

Graph of two lines with a right angle mark showing that the two lines intersect each other at a $90{^\circ}$ angle.

Answer

These are perpendicular lines because the lines intersect at a $90{^\circ}$ angle.

Do these equations describe lines that are parallel, perpendicular, or neither?

Equation 1: $y = 4x + 1$

Equation 2: $y - 1 = - 4(x + 1)$

Answer

These lines are neither parallel nor perpendicular. The slopes are not identical, and they are not opposite reciprocals.

Equation 1: $y = \require{color}\colorbox{yellow}{$ 4 $}x + 1$

Equation 2: $y - 1 = \require{color}\colorbox{yellow}{$ - 4 $}(x + 1)$

Do these equations describe lines that are parallel, perpendicular, or neither?

Equation 1: $y - 11 = - \frac{1}{2}(x - 3)$

Equation 2: $y - 3 = - \frac{1}{2}(x + 2)$

Answer

These lines are parallel because their slope values are identical, $m = - \frac{1}{2}$ and their $y$-intercepts are different.

Equation 1:
$y - 11 = \ \require{color}\colorbox{yellow}{$ - \frac{1}{2} $}(x - 3)$
$y - 11 = \ \require{color}\colorbox{yellow}{$ - \frac{1}{2} $}x - \frac{3}{2}$
$y = \ \require{color}\colorbox{yellow}{$ - \frac{1}{2} $}x + \ \require{color}\colorbox{lime}{$ \frac{25}{2} $}$

Equation 2:
$y - 3 = \ \require{color}\colorbox{yellow}{$ - \frac{1}{2} $}(x + 2)$
$y - 3 = \ \require{color}\colorbox{yellow}{$ - \frac{1}{2} $}x - 1$
$y = \ \require{color}\colorbox{yellow}{$ - \frac{1}{2} $}x + \ \require{color}\colorbox{aqua}{$ 2 $}$