# Supplementary Angles

## Two angles that add to 180 degrees.

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Supplementary Angles

### Supplementary Angles

Two angles are supplementary if they add up to \begin{align*}180^\circ\end{align*}. Supplementary angles do not have to be congruent or adjacent.

What if you were given two angles of unknown size and were told they are supplementary? How would you determine their angle measures?

### Examples

#### Example 1

Find the measure of an angle that is supplementary to \begin{align*}\angle ABC\end{align*} if \begin{align*}m\angle ABC\end{align*} is \begin{align*}118^\circ\end{align*}.

\begin{align*}180^\circ-118^\circ=62^\circ\end{align*}.

#### Example 2

Find the measure of an angle that is supplementary to \begin{align*}\angle ABC\end{align*} if \begin{align*}m\angle ABC\end{align*} is \begin{align*}32^\circ\end{align*}.

\begin{align*}180^\circ-32^\circ=148^\circ\end{align*}.

#### Example 3

The two angles below are supplementary. If \begin{align*}m\angle MNO = 78^\circ\end{align*} what is \begin{align*}m\angle PQR\end{align*}?

Set up an equation. However, instead of equaling \begin{align*}90^\circ\end{align*}, now the sum is \begin{align*}180^\circ\end{align*}.

\begin{align*}78^\circ + m\angle PQR & = 180^\circ\\ m\angle PQR & = 102^\circ\end{align*}

#### Example 4

What are the measures of two congruent, supplementary angles?

Supplementary angles add up to \begin{align*}180^\circ\end{align*}. Congruent angles have the same measure. So, \begin{align*}180^\circ \div 2 = 90^\circ\end{align*}, which means two congruent, supplementary angles are right angles, or \begin{align*}90^\circ\end{align*}.

#### Example 5

Find the measure of an angle that is a supplementary to \begin{align*}\angle MRS\end{align*} if \begin{align*} m\angle MRS\end{align*} is \begin{align*} 70^\circ\end{align*}.

Because supplementary angles have to add up to \begin{align*}180^\circ\end{align*}, the other angle must be \begin{align*}180^\circ-70^\circ=110^\circ\end{align*}.

### Review

Find the measure of an angle that is supplementary to \begin{align*}\angle ABC\end{align*} if \begin{align*}m\angle ABC\end{align*} is:

1. \begin{align*}114^\circ\end{align*}
2. \begin{align*}11^\circ\end{align*}
3. \begin{align*}91^\circ\end{align*}
4. \begin{align*}84^\circ\end{align*}
5. \begin{align*}57^\circ\end{align*}
6. \begin{align*}x^\circ\end{align*}
7. \begin{align*}(x+y)^\circ\end{align*}

Use the diagram below for exercises 8-9. Note that \begin{align*}\overline{NK} \perp \overleftrightarrow{IL}\end{align*}.

1. Name two supplementary angles.
1. If \begin{align*}m\angle INJ = 63^\circ\end{align*}, find \begin{align*}m\angle JNL\end{align*}.

For exercise 10, determine if the statement is true or false.

1. Supplementary angles add up to \begin{align*}180^\circ\end{align*}.

For 11-12, find the value of \begin{align*}x\end{align*}.

To see the Review answers, open this PDF file and look for section 1.8.

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