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Supplementary Angles

Two angles that add to 180 degrees.

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Supplementary Angles

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

Watch This

CK-12 Basic Geometric Definitions

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

James Sousa: Angle Basics--Supplementary Angles

Then watch the second part of this video.

James Sousa: Supplementary Angles

Guidance

Two angles are supplementary if they add up to \begin{align*}180^\circ\end{align*}. Supplementary angles do not have to be congruent or adjacent.

Example A

The two angles below are supplementary. If @$\begin{align*}m\angle MNO = 78^\circ\end{align*}@$ what is @$\begin{align*}m\angle PQR\end{align*}@$?

Set up an equation. However, instead of equaling @$\begin{align*}90^\circ\end{align*}@$, now the sum is @$\begin{align*}180^\circ\end{align*}@$.

@$$\begin{align*}78^\circ + m\angle PQR & = 180^\circ\\ m\angle PQR & = 102^\circ\end{align*}@$$

Example B

What are the measures of two congruent, supplementary angles?

Supplementary angles add up to @$\begin{align*}180^\circ\end{align*}@$. Congruent angles have the same measure. So, @$\begin{align*}180^\circ \div 2 = 90^\circ\end{align*}@$, which means two congruent, supplementary angles are right angles, or @$\begin{align*}90^\circ\end{align*}@$.

Example C

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

Because supplementary angles have to add up to @$\begin{align*}180^\circ\end{align*}@$, the other angle must be @$\begin{align*}180^\circ-70^\circ=110^\circ\end{align*}@$.

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Guided Practice

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

  1. @$\begin{align*}45^\circ\end{align*}@$
  2. @$\begin{align*}118^\circ\end{align*}@$
  3. @$\begin{align*}32^\circ\end{align*}@$
  4. @$\begin{align*}2^\circ\end{align*}@$

Answers:

  1. @$\begin{align*}180^\circ-45^\circ=135^\circ\end{align*}@$.
  2. @$\begin{align*}180^\circ-118^\circ=62^\circ\end{align*}@$.
  3. @$\begin{align*}180^\circ-32^\circ=148^\circ\end{align*}@$.
  4. @$\begin{align*}180^\circ-2^\circ=178^\circ\end{align*}@$.

Explore More

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

  1. @$\begin{align*}114^\circ\end{align*}@$
  2. @$\begin{align*}11^\circ\end{align*}@$
  3. @$\begin{align*}91^\circ\end{align*}@$
  4. @$\begin{align*}84^\circ\end{align*}@$
  5. @$\begin{align*}57^\circ\end{align*}@$
  6. @$\begin{align*}x^\circ\end{align*}@$
  7. @$\begin{align*}(x+y)^\circ\end{align*}@$

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

  1. Name two supplementary angles.
  1. If @$\begin{align*}m\angle INJ = 63^\circ\end{align*}@$, find @$\begin{align*}m\angle JNL\end{align*}@$.

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

  1. Supplementary angles add up to @$\begin{align*}180^\circ\end{align*}@$.

For 11-12, find the value of @$\begin{align*}x\end{align*}@$.

Vocabulary

Supplementary angles

Supplementary angles

Supplementary angles are two angles whose sum is 180 degrees.

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