Complementary and Supplementary Angles
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Complementary and Supplementary Angles
In this lesson, you will learn
- what adjacent angles are
- what complementary angles are and how to find the complement of an angle
- what supplementary angles are and how to find the supplement of an angle
- why these words describe only pairs of two angles
Adjacent Angles: A Quick Reminder
Before we provide you with a definition of complementary and supplementary angles, here is an important reminder about adjacent angles.
Adjacent angles: Adjacent angles are angles that share a vertex and a common side.
This fact is important to keep in mind because many complementary and supplementary angles that you will be dealing with in geometry are adjacent angles.
Two angles share a vertex, but they do not share a side. Are they adjacent angles?
Show answer
No. Adjacent angles must share both a vertex and a common side.
Complementary Angles Add to 90 Degrees
Complementary angles: Two angles whose measures add to 90 degrees.
It is easy to see when the angles are adjacent like the following:
Angle x and angle y are adjacent angles because they share a ray (line in black) and a vertex (point in black called D).
Now why are angle x and angle y complementary? Notice also that the angle in blue measures 90 degrees.
Since the measure of angle x plus the measure of angle y = 90 degrees, x and y are complementary.
Each angle is called the complement of the other. For example, you could also say that angle x is the complement of angle y.
Angles do not have to be adjacent to be complementary. The following angles are also complementary. As long as the sum of the measures equals 90 degrees, the angles are complementary.
To find the complement of an angle, subtract its measure from 90°.
- Complementary angles add to 90°, so the complement is 90° − 35°.
- 90° − 35° = 55°.
- Check: 35° + 55° = 90°.
The complement of 35° is 55°.
An angle that measures more than 90° has no complement, because no angle could be added to it to make 90°.
Angles x and y are complementary, and angle x measures 62°. What is the measure of angle y?
Show answer
28°. 90° − 62° = 28°.
Supplementary Angles Add to 180 Degrees
Supplementary angles: Two angles whose measures add to 180 degrees.
It is easy too to see this when the angles are adjacent like the following:
Again, angle a and angle b are adjacent angles because they share a ray (line in black) and a vertex (point in blue called D).
Now why are they supplementary?
Notice also that the angle in blue measures 180 degrees because the angle is a straight line and a straight line measures 180 degrees.
Since the measure of angle a plus the measure of angle b = 180 degrees, a and b are supplementary angles.
Each angle is called the supplement of the other. For example, you could also say that angle a is the supplement of angle b.
Again, angles do not have to be adjacent to be supplementary. The following angles are also supplementary since the sum of the measures equals 180 degrees.
To find the supplement of an angle, subtract its measure from 180°.
Mixing up the two words. Complementary angles add to 90°; supplementary angles add to 180°. To keep them apart, notice that “c” comes before “s” in the alphabet, just as 90 comes before 180.
What is the supplement of a 120° angle?
Show answer
60°. 180° − 120° = 60°.
Only Two Angles Can Be Complementary or Supplementary
Can three or more angles be complementary or supplementary? No way!
The definitions of complementary and supplementary angles apply only to two angles. Therefore, even if 3 angles or more add up to 90 or 180 degrees, we do not call them complementary or supplementary angles.
Three angles measure 30°, 30° and 30°. Together they add to 90°. Are they complementary angles?
Show answer
No. Only a pair of two angles can be complementary.
Summary
- Adjacent angles share a vertex and a common side.
- Complementary angles are two angles whose measures add to 90°. The complement of an angle is 90° minus the angle.
- Supplementary angles are two angles whose measures add to 180°. The supplement of an angle is 180° minus the angle.
- Complementary and supplementary angles can be adjacent, but they do not have to be.
- These words describe only pairs of two angles, never three or more.
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