Supplementary and
Complementary Angle Pairs
Jen Kershaw
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Printed: August 19, 2014
AUTHOR
Jen Kershaw
www.ck12.org
C HAPTER
Chapter 1. Supplementary and Complementary Angle Pairs
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Supplementary and
Complementary Angle Pairs
Here you’ll use reasoning to find angle measures.
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 30◦ ," 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 90◦ , 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, C and D form a right angle. Therefore they are complementary, and they add up to 90◦ .
We know that C has a measure of 44◦ . How can we find the measure of angle D?
To find the measurement of angle D, we simply subtract the measure of angle C from 90.
6
C + 6 D = 90◦
44◦ + 6 D = 90◦
D = 90 − 44
6
6
D = 46◦
In order for these two angles to be complementary, as the problem states, they must add up to 90◦ . Angle D therefore
measures 46◦ . We can check our calculation by adding angles C and D. Their sum must be equal to 90◦ .
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44◦ + 46◦ = 90◦
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 P and Q are supplementary angles. If angle P measures 112◦ , what is the measure of angle Q?
We know that supplementary angles have a total of 180◦ Therefore we can subtract the measurement of the
angle we know, angle P, from 180◦ to find the measure of angle Q.
6
P + 6 Q = 180◦
112◦ + 6 Q = 180◦
Q = 180 − 112
6
6
Q = 68◦
Angle Q is 68◦ . We can check our calculation by adding angles P and Q. Remember, in order to be supplementary
angles, their sum must be equal to 180◦ .
68◦ + 112◦ = 180◦
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 X? We can, if we apply what we have learned about supplementary angles.
We know that supplementary angles add up to 180◦ , and that 180◦ is a straight line. Look at the diagram.
The 80◦ angle and angle X 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 X.
80 + x = 180
The equation shows what we already know: the sum of supplementary angles is 180◦ . We can find the measure of
the unknown angle by solving for X.
80 + x = 180
x = 180 − 80
x = 100◦
The measure of the unknown angle in this supplementary pair is 100◦ .
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Chapter 1. Supplementary and Complementary Angle Pairs
We can check our work by putting this value in for X in the equation.
80 + 100 = 180
Now it’s time for you to apply what you have learned. Find the complement or supplement in each example.
Example A
Angles A and B are complementary. Angle A is 33◦ . Find the measure of angle B.
Solution: 57◦
Example B
Angles C and D are supplementary. Angle C is 59◦ . Find the measure of angle D.
Solution: 121◦
Example C
Angles A and B are supplementary. Angle A is 169◦ . Find the measure of angle B.
Solution: 11◦
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 30◦ ," 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 180c irc. Given this, we can write an equation.
x + 30 = 180
The 30 is the measure of the angle that Justin and Susan figure out.
Now we can solve for the unknown variable.
x = 150◦
This is our answer.
Vocabulary
Acute Angle
an angle whose measure is less than 90◦ .
Obtuse Angle
an angle whose measure is greater than 90◦ .
Right Angle
an angle whose measure is equal to 90◦ .
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Straight Angle
an angle whose measure is equal to 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 180◦ .
Complementary Angles
angle pairs whose sum is 90◦ .
Guided Practice
Here is one for you to try on your own.
What is the measure of angle R?
Answer
How can we use what we have learned to find the measure of angle R? 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 90◦ . Again, we can set up an equation to solve for R, the unknown angle.
R + 22 = 90
This equation represents what we know, that the sum of these two complementary angles is 90◦ . Now we solve for
R.
R + 22 = 90
R = 90 − 22
R = 68◦
The measure of the unknown angle is 68◦ . We can check our answer by putting this value in for R in the equation.
68 + 22 = 90◦
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Chapter 1. Supplementary and Complementary Angle Pairs
Video Review
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/1274
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 A and B are complementary. Angle A is 63◦ . Find the measure of angle B.
2. Angles A and B are complementary. Angle A is 83◦ . Find the measure of angle B.
3. Angles A and B are complementary. Angle A is 3◦ . Find the measure of angle B.
4. Angles A and B are complementary. Angle A is 23◦ . Find the measure of angle B.
5. Angles A and B are complementary. Angle A is 70◦ . Find the measure of angle B.
6. Angles A and B are complementary. Angle A is 29◦ . Find the measure of angle B.
7. Angles A and B are complementary. Angle A is 66◦ . Find the measure of angle B.
8. Angles A and B are complementary. Angle A is 87◦ . Find the measure of angle B.
9. Angles A and B are supplementary. Angle A is 33◦ . Find the measure of angle B.
10. Angles A and B are supplementary. Angle A is 103◦ . Find the measure of angle B.
11. Angles A and B are supplementary. Angle A is 73◦ . Find the measure of angle B.
12. Angles A and B are supplementary. Angle A is 78◦ . Find the measure of angle B.
13. Angles A and B are supplementary. Angle A is 99◦ . Find the measure of angle B.
14. Angles A and B are supplementary. Angle A is 110◦ . Find the measure of angle B.
15. Angles A and B are supplementary. Angle A is 127◦ . Find the measure of angle B.
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