# Supplementary angles and examples

Supplementary angles are angles whose measures sum to 180°. In the lesson below, we will review this idea along with taking a look at some example problems.

In the image below, you see one of the common ways in which supplementary angles come up. The angles with measures $$a$$° and $$b$$° lie along a straight line. Since straight angles have measures of 180°, the angles are supplementary.

## Example problems with supplementary angles

Let’s look at a few examples of how you would work with the concept of supplementary angles.

### Example

In the figure, the angles lie along line $$m$$. What is the value of $$x$$?

### Solution

The two angles lie along a straight line, so they are supplementary. Therefore: $$x + 118 = 180$$. Solving this equation:
\begin{align}x + 118 &= 180\\ x &= \boxed{62} \end{align}

### Example

The angles $$A$$ and $$B$$ are supplementary. If $$m\angle A = (2x)^{\circ}$$ and $$m\angle B = (2x-2)^{\circ}$$, what is the value of $$x$$?

### Solution

Since the angles are supplementary, their measures add to 180°. In other words: $$2x + (2x – 2) = 180$$. Solving this equation gives the value of $$x$$.

\begin{align}2x + (2x – 2) &= 180\\ 4x – 2 &= 180\\ 4x &= 182\\ x &= \boxed{45.5} \end{align}

The previous example could have asked for some different information. Let’s look at a similar example that asks a slightly different question.

### Example

The angles $$A$$ and $$B$$ are supplementary. If $$m\angle A = (2x+5)^{\circ}$$ and $$m\angle B = (x-20)^{\circ}$$, what is $$m \angle A$$?

### Solution

This time you are being asked for the measure of the angle and not just $$x$$. But, the value of $$x$$ is needed to find the measure of the angle. So, first set up an equation and find $$x$$.

\begin{align}2x+5 + x – 20 &= 180\\ 3x-15 &= 180 \\ 3x &= 195\\ x&= 65\end{align}

The measure of angle $$A$$ is then:
$$m\angle A = (2x+5)^{\circ}$$ and $$x = 65$$

$$m\angle A = (2(65)+5)^{\circ} = \boxed{135^{\circ}}$$

## Summary

There isn’t much to working with supplementary angles. You just have to remember that their sum is 180° and that any set of angles lying along a straight line will also be supplementary.