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# Complementary Angles

## Two angles that add to 90 degrees.

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Complementary Angles

### Complementary Angles

Two angles are complementary if they add up to 90\begin{align*}90^\circ\end{align*}. Complementary angles do not have to be congruent or adjacent.

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

### Examples

#### Example 1

Find the measure of an angle that is complementary to ABC\begin{align*}\angle ABC\end{align*} if mABC\begin{align*}m\angle ABC\end{align*} is 82\begin{align*}82^\circ\end{align*}.

9082=8\begin{align*}90^\circ-82^\circ=8^\circ\end{align*}.

#### Example 2

Find the measure of an angle that is complementary to ABC\begin{align*}\angle ABC\end{align*}if mABC\begin{align*}m\angle ABC\end{align*} is 12\begin{align*}12^\circ\end{align*}

9012=78\begin{align*}90^\circ-12^\circ=78^\circ\end{align*}.

#### Example 3

The two angles below are complementary. mGHI=x\begin{align*}m\angle GHI = x\end{align*}. What is x\begin{align*}x\end{align*}?

Because the two angles are complementary, they add up to 90\begin{align*}90^\circ\end{align*}. Make an equation.

x+34x=90=56\begin{align*}x + 34^\circ & = 90^\circ\\ x & = 56^\circ\end{align*}

#### Example 4

The two angles below are complementary. Find the measure of each angle.

The two angles add up to 90\begin{align*}90^\circ\end{align*}. Make an equation.

(8r+9)+(7r+6)=90(15r+15)=9015r=75r=5\begin{align*}(8r+9)+(7r+6) = 90^\circ \\ (15r+15)=90^\circ \\ 15r = 75^\circ \\ r = 5^\circ\end{align*}

However, you need to find each angle. Plug r\begin{align*}r\end{align*} back into each expression.

mGHImJKL=8(5)+9=49=7(5)+6=41\begin{align*}m \angle GHI & = 8(5^\circ) + 9^\circ = 49^\circ\\ m\angle JKL & = 7(5^\circ) + 6^\circ = 41^\circ\end{align*}

#### Example 5

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

Because complementary angles have to add up to 90\begin{align*}90^\circ\end{align*}, the other angle must be 9070=20\begin{align*}90^\circ-70^\circ=20^\circ\end{align*}.

### Review

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

1. 4\begin{align*}4^\circ\end{align*}
2. 89\begin{align*}89^\circ\end{align*}
3. 54\begin{align*}54^\circ\end{align*}
4. 32\begin{align*}32^\circ\end{align*}
5. 27\begin{align*}27^\circ\end{align*}
6. (x+y)\begin{align*}(x+y)^\circ\end{align*}
7. z\begin{align*}z^\circ\end{align*}

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

1. Name two complementary angles.
1. If mINJ=63\begin{align*}m\angle INJ = 63^\circ\end{align*}, find mKNJ\begin{align*}m\angle KNJ\end{align*}.

For 10-11, determine if the statement is true or false.

1. Complementary angles add up to 180\begin{align*}180^\circ\end{align*}.
2. Complementary angles are always 45\begin{align*}45^\circ\end{align*}.

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### Vocabulary Language: English Spanish

complementary angles

Two angles are complementary if they add up to $90^\circ$.