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Complementary Angles

Two angles that add to 90 degrees.

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Practice Complementary Angles
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Estimated6 minsto complete
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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 \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. \begin{align*}m \angle GHI = x\end{align*}. What is \begin{align*}x\end{align*}?

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

\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 \begin{align*}90^\circ\end{align*}. Make an equation.

\begin{align*}8r + 9^\circ + 7r+ 5^\circ & = 90^\circ\\ 15r + 14^\circ & = 90^\circ\\ 15r & = 76^\circ\\ r & = 5.067^\circ\end{align*}

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

\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 C

Name one pair of complementary angles in the diagram below.

One example is \begin{align*} \angle INJ\end{align*} and \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 \begin{align*}90^\circ\end{align*}.

Guided Practice

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

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

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

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

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

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

Practice

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

1. \begin{align*}3^\circ\end{align*}
2. \begin{align*}82^\circ\end{align*}
3. \begin{align*}51^\circ\end{align*}
4. \begin{align*}30^\circ\end{align*}
5. \begin{align*}22^\circ\end{align*}
6. \begin{align*}(x+y)^\circ\end{align*}
7. \begin{align*}x^\circ\end{align*}

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

1. If \begin{align*}m\angle INJ = 60^\circ\end{align*}, find \begin{align*}m\angle KNJ\end{align*}.
2. If \begin{align*}m\angle INJ = 70^\circ\end{align*}, find \begin{align*}m\angle KNJ\end{align*}.

For 10-15, determine if the statement is true or false.

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