### Let's Think About It

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 \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.

### Guidance

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 \begin{align*}90^\circ\end{align*}. Let’s look at an example.

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

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

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

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

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

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

\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 **\begin{align*}X\end{align*}**. Apply what you have learned about supplementary angles. Supplementary angles add up to ** \begin{align*}180^\circ\end{align*}**, and **\begin{align*}180^\circ\end{align*}** is a straight line. Look at the diagram. The \begin{align*}80^\circ\end{align*} angle and angle \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 \begin{align*}X\end{align*}.

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

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

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

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

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

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

### Guided Practice

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*}

### Examples

Find the complement or supplement in each example.

#### Example 1

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 2

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 3

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*}**

### Follow Up

Remember Marco and his house? If one piece of wood has an angled cut that is \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.

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

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

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

Then calculate the difference.

The difference is \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.

### Video Review

### Explore More

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*}.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 8.3.