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# Complementary Angles

## Two angles that add to 90 degrees.

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Practice Complementary Angles
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Complementary Angles

What if you knew that two angles together made a right angle? After completing this Concept, you'll be able to use what you know about complementary angles to solve problems about these angles.

### Guidance

Two angles are complementary when they add up to 90\begin{align*}90^\circ\end{align*}. Complementary angles do not have to be congruent to each other, nor do they have to be next to each other.

#### Example A

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

#### Example B

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

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

8r+9+7r+515r+1415rr=90=90=765.1\begin{align*}8r + 9^\circ + 7r+ 5^\circ & = 90^\circ\\ 15r + 14^\circ & = 90^\circ\\ 15r & = 76^\circ\\ r & \approx 5.1^\circ\end{align*}

However, this is not what the question asks for. You need to plug r\begin{align*}r\end{align*} back into each expression to find each angle.

For mGHI\begin{align*}m \angle GHI\end{align*}: 8(5.1)+9=49.8\begin{align*}8(5.1^\circ) + 9^\circ = 49.8^\circ\end{align*}, so mGHI49.8\begin{align*}m \angle GHI \approx 49.8^\circ\end{align*}.

For mJKL\begin{align*}m \angle JKL\end{align*}: 7(5.1)+5=40.7\begin{align*}7(5.1^\circ) + 5^\circ = 40.7^\circ\end{align*}, so mJKL40.7\begin{align*}m \angle JKL \approx 40.7^\circ\end{align*}.

#### Example C

Name one pair of complementary angles in the diagram below.

One example is INJ\begin{align*} \angle INJ\end{align*} and JNK\begin{align*} \angle JNK\end{align*}.

Watch this video for help with the Examples above.

### Vocabulary

Two angles are complementary when they add up to 90\begin{align*}90^\circ\end{align*}.

### Guided Practice

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. 45\begin{align*}45^\circ\end{align*}
2. 82\begin{align*}82^\circ\end{align*}
3. 19\begin{align*}19^\circ\end{align*}
4. z\begin{align*}z^\circ\end{align*}

1. 45\begin{align*}45^\circ\end{align*}

2. 8\begin{align*}8^\circ\end{align*}

3. 71\begin{align*}71^\circ\end{align*}

4. 90z\begin{align*}90-z^\circ\end{align*}

### Practice

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. 3\begin{align*}3^\circ\end{align*}
2. 82\begin{align*}82^\circ\end{align*}
3. 51\begin{align*}51^\circ\end{align*}
4. 30\begin{align*}30^\circ\end{align*}
5. 22\begin{align*}22^\circ\end{align*}
6. (x+y)\begin{align*}(x+y)^\circ\end{align*}
7. x\begin{align*}x^\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. If mINJ=60\begin{align*}m\angle INJ = 60^\circ\end{align*}, find mKNJ\begin{align*}m\angle KNJ\end{align*}.
2. If mINJ=70\begin{align*}m\angle INJ = 70^\circ\end{align*}, find mKNJ\begin{align*}m\angle KNJ\end{align*}.

For 10-15, 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*}.
3. Complementary angles are always next to each other.
4. Complementary angles add up to 90\begin{align*}90^\circ\end{align*}.
5. Two angles that make a right angle are complementary.
6. The two non-right angles in a right triangle are complementary.

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