Have you ever tried to figure out an angle measure? Look at what happened at the art museum.

Justin was looking at a painting with two intersecting lines on it. One of the lines formed a straight line and the other intersected with the first line.

"What do you think the measure is of the smaller angle?" he asked Susan who was standing nearby.

"I think it is about \begin{align*}30^\circ\end{align*}," Susan said.

"That's exactly what I was thinking," Justin added.

If Susan and Justin are correct, can you figure out the other missing angle?

**This Concept will show you how reasoning can help you figure out the measures of missing angles.**

### Guidance

**As we have seen in the Angle Pairs Concept, we identify complementary and supplementary angles by their sum. This means that we can also find the measure of one angle in a pair if we know the measure of the other angle.** For instance, because we know that complementary angles always add up to \begin{align*}90^\circ\end{align*}, we can calculate the measurement of one angle in a pair of complementary angles. Let’s see how this works.

We can see that 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*}. We know that \begin{align*}C\end{align*} has a measure of \begin{align*}44^\circ\end{align*}. How can we find the measure of angle \begin{align*}D\end{align*}?

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

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

In order for these two angles to be complementary, as the problem states, they must add up to \begin{align*}90^\circ\end{align*}. Angle \begin{align*}D\end{align*} therefore measures \begin{align*}46^\circ\end{align*}. We can check our calculation by adding angles \begin{align*}C\end{align*} and \begin{align*}D\end{align*}. Their sum must be equal to \begin{align*}90^\circ\end{align*}.

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

**We can follow the same process to find the unknown angle in a pair of supplementary angles. As with complementary angles, if we know the measure of one angle in the pair, we can find the measure of the other.**

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*}?

**We know that supplementary angles have a total of \begin{align*}180^\circ\end{align*} Therefore we can subtract the measurement of the angle we know, angle \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*}. We can check our 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 be equal to \begin{align*}180^\circ\end{align*}.

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

**We can call this finding the complement or the supplement.**

**Armed with our knowledge of complementary and supplementary angles, we can often find the measure of unknown angles. We can use logical reasoning to interpret the information we have been given in order to find the unknown measure.** Take a look at the diagram below.

**Can we find the measure of angle \begin{align*}X\end{align*}? We can, if we apply what we have learned about supplementary angles. We know that supplementary angles add up to \begin{align*}180^\circ\end{align*}, and that \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 we can deduce that they are supplementary angles. That means we can set up an equation to solve for \begin{align*}X\end{align*}.

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

The equation shows what we already know: the sum of supplementary angles is \begin{align*}180^\circ\end{align*}. We can 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*}.**

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

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

Now it's time for you to apply what you have learned. Find the complement or supplement in each example.

#### Example A

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*}.

**Solution: \begin{align*}57^\circ\end{align*}**

#### Example B

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*}.

**Solution: \begin{align*}121^\circ\end{align*}**

#### Example C

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*}.

**Solution: \begin{align*}11^\circ\end{align*}**

Here is the original problem once again.

Justin was looking at a painting with two intersecting lines on it. One of the lines formed a straight line and the other intersected with the first line.

"What do you think the measure is of the smaller angle?" he asked Susan who was standing nearby.

"I think it is about \begin{align*}30^\circ\end{align*}," Susan said.

"That's exactly what I was thinking," Justin added.

If Susan and Justin are correct, can you figure out the other missing angle?

To figure this out, we can use reasoning and the dilemma to hunt for clues. First, notice that the painting had one straight line. We know that the measure of a straight line is \begin{align*}180^circ\end{align*}. Given this, we can write an equation.

\begin{align*}x + 30 = 180\end{align*}

The 30 is the measure of the angle that Justin and Susan figure out.

Now we can solve for the unknown variable.

\begin{align*}x = 150^\circ\end{align*}

**This is our answer.**

### Vocabulary

- Acute Angle
- an angle whose measure is less than \begin{align*}90^\circ\end{align*}.

- Obtuse Angle
- an angle whose measure is greater than \begin{align*}90^\circ\end{align*}.

- Right Angle
- an angle whose measure is equal to \begin{align*}90^\circ\end{align*}.

- Straight Angle
- an angle whose measure is equal to \begin{align*}180^\circ\end{align*}.

- Degrees
- how an angle is measured.

- Angle Pairs
- when the measures of two angles are added together to form a special relationship.

- Supplementary Angles
- angle pairs whose sum is \begin{align*}180^\circ\end{align*}.

- Complementary Angles
- angle pairs whose sum is \begin{align*}90^\circ\end{align*}.

### Guided Practice

Here is one for you to try on your own.

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

**Answer**

How can we use what we have learned to find the measure of angle \begin{align*}R\end{align*}? Can we determine whether the two angles have a relationship with each other? Together, they form a right angle. They must be a pair of complementary angles, so we know their sum is \begin{align*}90^\circ\end{align*}. Again, we can set up an equation to solve for \begin{align*}R\end{align*}, the unknown angle.

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

This equation represents what we know, that the sum of these two complementary angles is \begin{align*}90^\circ\end{align*}. Now we solve for \begin{align*}R\end{align*}.

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

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

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

### Video Review

This is a James Sousa video on complementary and supplementary angles.

### Practice

Directions: 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*}.