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Complementary Angles

Two angles that add to 90 degrees.

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Complementary Angles

What if you were given two angles of unknown size and were told they are complementary? How would you determine their angle measures? After completing this Concept, you'll be able to use the definition of complementary angles to solve problems like this one.

Watch This

CK-12 Complementary Angles

Watch this video beginning at around the 3:20 mark.

James Sousa: Angle Basics--Complementary Angles

Then watch the first part of this video.

James Sousa: Complementary Angles

Guidance

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

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

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

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

\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

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

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

CK-12 Complementary Angles

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*}12^\circ\end{align*}

Answers:

1. Because complementary angles have to add up to \begin{align*}90^\circ\end{align*}, the other angle must be \begin{align*}90^\circ-45^\circ=45^\circ\end{align*}.

2. Because complementary angles have to add up to \begin{align*}90^\circ\end{align*}, the other angle must be \begin{align*}90^\circ-82^\circ=8^\circ\end{align*}.

3. Because complementary angles have to add up to \begin{align*}90^\circ\end{align*}, the other angle must be \begin{align*}90^\circ-19^\circ=71^\circ\end{align*}.

4. Because complementary angles have to add up to \begin{align*}90^\circ\end{align*}, the other angle must be \begin{align*}90^\circ-12^\circ=78^\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*}4^\circ\end{align*}
  2. \begin{align*}89^\circ\end{align*}
  3. \begin{align*}54^\circ\end{align*}
  4. \begin{align*}32^\circ\end{align*}
  5. \begin{align*}27^\circ\end{align*}
  6. \begin{align*}(x+y)^\circ\end{align*}
  7. \begin{align*}z^\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 complementary angles.
  1. If \begin{align*}m\angle INJ = 63^\circ\end{align*}, find \begin{align*}m\angle KNJ\end{align*}.

For 10-11, 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*}.

Vocabulary

complementary angles

complementary angles

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

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