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Complementary Angles

Two angles that add to 90 degrees.

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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.

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CK-12 Foundation: Chapter1ComplementaryAnglesA

James Sousa: Complementary 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 & \approx 5.1^\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.

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

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

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter1ComplementaryAnglesB

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*}

Answers:

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*}

Interactive Practice

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.

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