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# 8.3: Supplementary and Complementary Angle Pairs

Difficulty Level: At Grade Created by: CK-12
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Marco is building a house. He bought lots of wood to make the frame of the house. He wants right angles for his corners. If he uses a piece of wood that is cut at a 55o\begin{align*}55^o\end{align*} angle, what must be the angle measure of the other piece of wood that he uses to complete the corner?

In this concept, you will learn how reasoning can help you figure out the measures of missing angles.

### Supplementary and Complementary Angle Pairs

Some special angle pairs are identified by their sum. If you know the measure of one angle, you can calculate the measure of the second angle. For instance, complementary angles always add up to 90\begin{align*}90^\circ\end{align*}. Let’s look at an example.

Together, C\begin{align*}C\end{align*} and D\begin{align*}D\end{align*} form a right angle. Therefore they are complementary, and they add up to 90\begin{align*}90^\circ\end{align*}. C\begin{align*}C\end{align*} has a measure of 44\begin{align*}44^\circ\end{align*}.

To find the measurement of angle D\begin{align*}D\end{align*}, simply subtract the measure of angle C\begin{align*}C\end{align*} from 90o\begin{align*}90^o\end{align*}.

C+D44+DDD=90=90=9044=46\begin{align*}\angle{C}+ \angle{D} & = 90^\circ\\ 44^\circ + \angle{D} & = 90^\circ\\ \angle {D} & = 90-44\\ \angle {D} & = 46^\circ\end{align*}

Angle D\begin{align*}D\end{align*} therefore measures 46\begin{align*}46^\circ\end{align*}. You can check the calculation by adding angles C\begin{align*}C\end{align*} and D\begin{align*}D\end{align*}. The sum must be equal to 90\begin{align*}90^\circ\end{align*}.

44+46=90\begin{align*}44^\circ + 46^\circ = 90^\circ\end{align*}

The same process can be used to find the unknown angle in a pair of supplementary angles. Let's look at another example.

Angles P\begin{align*}P\end{align*} and Q\begin{align*}Q\end{align*} are supplementary angles. If angle P\begin{align*}P\end{align*} measures 112\begin{align*}112^\circ\end{align*}, what is the measure of angle Q\begin{align*}Q\end{align*}?

Supplementary angles have a total of 180\begin{align*}180^\circ\end{align*}. Subtract the measurement of P\begin{align*}P\end{align*}, from 180\begin{align*}180^\circ\end{align*} to find the measure of angle Q\begin{align*}Q\end{align*}.

P+Q112+QQQ=180=180=180112=68\begin{align*}\angle{P} + \angle{Q} & = 180^\circ\\ 112^\circ + \angle {Q} & = 180^\circ\\ \angle{Q} & = 180-112\\ \angle{Q} & = 68^\circ\end{align*}

Angle Q\begin{align*}Q\end{align*} is 68\begin{align*}68^\circ\end{align*}. You can check the calculation by adding angles P\begin{align*}P\end{align*} and Q\begin{align*}Q\end{align*}. Remember, in order to be supplementary angles, their sum must equal 180\begin{align*}180^\circ\end{align*}.

68+112=180\begin{align*}68^\circ + 112^\circ = 180^\circ\end{align*}

This process can often be used to find the measure of unknown angles. Use logical reasoning to interpret the information in order to find the unknown measure.

Take a look at the diagram below.

Let's find the value of angle X\begin{align*}X\end{align*}. Apply what you have learned about supplementary angles. Supplementary angles add up to   180\begin{align*}180^\circ\end{align*}, and 180\begin{align*}180^\circ\end{align*} is a straight line. Look at the diagram. The 80\begin{align*}80^\circ\end{align*} angle and angle X\begin{align*}X\end{align*} together form a straight line, so they are supplementary angles. That means you can set up an equation to solve for X\begin{align*}X\end{align*}.

80+x=180\begin{align*}80 + x = 180\end{align*}

The equation shows the sum of supplementary angles is 180\begin{align*}180^\circ\end{align*}. Find the measure of the unknown angle by solving for X\begin{align*}X\end{align*}.

80+xxx=180=18080=100\begin{align*}80 + x & = 180\\ x & = 180 - 80\\ x & = 100^\circ\end{align*}

The measure of the unknown angle in this supplementary pair is 100\begin{align*}100^\circ\end{align*}.

You can check your work by putting this value in for X\begin{align*}X\end{align*} in the equation.

80+100=180\begin{align*}80 + 100 = 180\end{align*}

### Examples

Example 1

Earlier, you were given a problem about Marco and his house.

If one piece of wood has an angled cut that is 55o\begin{align*}55^o\end{align*}, what is the measure of the angled cut for the second piece of wood?

First, set up an equation that represents the relationship between the two angles.

55o+M=90o\begin{align*}55^o + M = 90^o\end{align*}

Next, subtract the given angle from the sum of the two angles.

M=90o55o\begin{align*}M = 90^o - 55^o\end{align*}

Then calculate the difference.

The difference is 35o\begin{align*}35^o\end{align*}.

The answer is that the second piece of wood is cut at a \begin{align*}35^o\end{align*} angle.

#### Example 2

Solve the following problem.

What is the measure of angle \begin{align*}R\end{align*}?



First, set up an equation that represents the relationship between the two angles.

\begin{align*}R + 22 = 90\end{align*}

Next, subtract the given angle from the sum of the two angles.

\begin{align*}R = 90^o - 22^o\end{align*}

Then, calculate the difference.

The difference is \begin{align*}68^o\end{align*}.

The answer is angle \begin{align*}R = 68^o\end{align*} .

The measure of the unknown angle is \begin{align*}68^\circ\end{align*}. You can check your answer by putting this value in for \begin{align*}R\end{align*} in the equation.

\begin{align*}68 + 22 = 90^\circ\end{align*}

Find the complement or supplement in each example.

#### Example 3

Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are complementary. Angle \begin{align*}A\end{align*} is \begin{align*}33^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.

First, set up an equation that represents the relationship between the angles.

\begin{align*}33^o + B = 90^o\end{align*}

Next, subtract the given angle from the sum of the two angles.

\begin{align*}B = 90^o - 33^o\end{align*}

Then, calculate the difference.

The difference is \begin{align*}57^o\end{align*}.

The answer is angle \begin{align*}B =\end{align*} \begin{align*}57^\circ\end{align*}.

#### Example 4

Angles \begin{align*}C\end{align*} and \begin{align*}D\end{align*} are supplementary. Angle \begin{align*}C\end{align*} is \begin{align*}59^\circ\end{align*}. Find the measure of angle \begin{align*}D\end{align*}.

First, set up an equation that represents the relationship between the angles.

\begin{align*}59^o + D = 180^o\end{align*}

Next, subtract the given angle from the sum of the two angles.

\begin{align*}D = 180^o - 59^o\end{align*}

Then, calculate the difference.

The difference is \begin{align*}121^o\end{align*}.

The answer is angle \begin{align*}D =\end{align*} \begin{align*}121^\circ\end{align*}

#### Example 5

Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are supplementary. Angle \begin{align*}A\end{align*} is \begin{align*}169^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.

First, set up an equation that represents the relationship between the angles.

\begin{align*}169^o + B = 180^o\end{align*}

Next, subtract the given angle from the sum of the angles.

\begin{align*}B = 180^o - 169^o\end{align*}

Then, calculate the difference.

The difference is \begin{align*}11^o\end{align*}.

The answer is angle \begin{align*}B = \end{align*} \begin{align*}11^\circ\end{align*}

### Review

Find the measure of missing angle for each pair of complementary or supplementary angles.

1. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are complementary. Angle \begin{align*}A\end{align*} is \begin{align*}63^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.
2. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are complementary. Angle \begin{align*}A\end{align*} is \begin{align*}83^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.
3. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are complementary. Angle \begin{align*}A\end{align*} is \begin{align*}3^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.
4. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are complementary. Angle \begin{align*}A\end{align*} is \begin{align*}23^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.
5. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are complementary. Angle \begin{align*}A\end{align*} is \begin{align*}70^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.
6. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are complementary. Angle \begin{align*}A\end{align*} is \begin{align*}29^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.
7. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are complementary. Angle \begin{align*}A\end{align*} is \begin{align*}66^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.
8. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are complementary. Angle \begin{align*}A\end{align*} is \begin{align*}87^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.
9. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are supplementary. Angle \begin{align*}A\end{align*} is \begin{align*}33^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.
10. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are supplementary. Angle \begin{align*}A\end{align*} is \begin{align*}103^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.
11. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are supplementary. Angle \begin{align*}A\end{align*} is \begin{align*}73^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.
12. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are supplementary. Angle \begin{align*}A\end{align*} is \begin{align*}78^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.
13. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are supplementary. Angle \begin{align*}A\end{align*} is \begin{align*}99^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.
14. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are supplementary. Angle \begin{align*}A\end{align*} is \begin{align*}110^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.
15. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are supplementary. Angle \begin{align*}A\end{align*} is \begin{align*}127^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.

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