8.3: Supplementary and Complementary Angle Pairs
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
"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, 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
We can see that together,
To find the measurement of angle
In order for these two angles to be complementary, as the problem states, they must add up to
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
We know that supplementary angles have a total of
Angle
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
The equation shows what we already know: the sum of supplementary angles is
The measure of the unknown angle in this supplementary pair is
We can check our work by putting this value in for
Now it's time for you to apply what you have learned. Find the complement or supplement in each example.
Example A
Angles
Solution:\begin{align*}57^\circ\end{align*}
Example B
Angles \begin{align*}C\end{align*}
Solution:\begin{align*}121^\circ\end{align*}
Example C
Angles \begin{align*}A\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*}
"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*}
\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
Here are the vocabulary words in this Concept.
 Acute Angle

an angle whose measure is less than \begin{align*}90^\circ\end{align*}
90∘
 Obtuse Angle

an angle whose measure is greater than \begin{align*}90^\circ\end{align*}
90∘
 Right Angle

an angle whose measure is equal to \begin{align*}90^\circ\end{align*}
90∘
 Straight Angle

an angle whose measure is equal to \begin{align*}180^\circ\end{align*}
180∘
 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*}
180∘
 Complementary Angles

angle pairs whose sum is \begin{align*}90^\circ\end{align*}
90∘
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*}
\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*}
\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*}
\begin{align*}68 + 22 = 90^\circ\end{align*}
Video Review
Here is a video for 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*}
2. Angles \begin{align*}A\end{align*}
3. Angles \begin{align*}A\end{align*}
4. Angles \begin{align*}A\end{align*}
5. Angles \begin{align*}A\end{align*}
6. Angles \begin{align*}A\end{align*}
7. Angles \begin{align*}A\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*}.
Acute Angle
An acute angle is an angle with a measure of less than 90 degrees.Complementary angles
Complementary angles are a pair of angles with a sum of .Obtuse angle
An obtuse angle is an angle greater than 90 degrees but less than 180 degrees.Straight angle
A straight angle is a straight line equal to .Supplementary angles
Supplementary angles are two angles whose sum is 180 degrees.Image Attributions
Description
Learning Objectives
Here you'll use reasoning to find angle measures.