Complementary Angles: Definition, Theorem & Examples

Complementary Angles: Definition, Theorem & Examples
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  • 0:00 What Are Complementary Angles?
  • 1:30 Let's Look at Some Examples
  • 2:37 Theorems Using…
  • 3:50 Lesson Summary
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Lesson Transcript
Instructor: Miriam Snare

Miriam has taught middle- and high-school math for over 10 years and has a master's degree in Curriculum and Instruction.

In this lesson, we will define complementary angles and see how they may appear in diagrams. We will discuss two common theorems that involve complementary angles. Then you can test your knowledge with a quiz.

What Are Complementary Angles?

Complementary angles are two angles whose measures add up to 90 degrees. You can think of them as two puzzle pieces that form one 90 degree angle when they are put together. When talking about complementary angles, it's important to remember that they're always in a pair. When discussing one of the angles in a complementary pair, we can say that one angle is the complement of the other or one angle is complementary to the other.

One angle measuring 90 degrees by itself is called a right angle. Since a right angle doesn't need another puzzle piece to complete the 90 degrees, a right angle doesn't have a complement and can't be called a complement by itself.

Three or more angles are also not called complementary, even if their measures happen to add up to 90 degrees.

Complementary angles always have positive measures. Since their measures add up to 90 degrees, each of the complements must be acute, measuring less than 90 degrees. Two angles measuring 45 degrees are complementary, but that's not the only pair of possible measures. The two puzzle pieces that together form a right angle can be any combination of two positive numbers that add up to 90 degrees. Angles measuring 30 and 60 degrees are a complementary pair. An angle measuring 1 degree would the complement to an angle measuring 89 degrees.

Let's Look at Some Examples

This first diagram shows complementary angles that are adjacent, meaning that the angles share a side and a vertex, or the corner point of the angle. Since 65 + 25 = 90, angles STA and ATR are complementary.

The next diagram shows two complementary angles that are not adjacent, but are in the same figure. Angles GDO and DGO are complementary because 70 + 20 = 90. Angle DOG is a right angle as indicated by the little box we see at the vertex of DOG.

In this final diagram, we have two complementary angles that aren't attached in any way, unlike the ones above. Any two angles that you can find that add up to 90 degrees are complementary. They don't have to be in the same figure. So, angles MIC and KEY are complementary because 47 + 43 = 90. If you imagine moving the angles next to each other so that ray IC lines up with ray EK, then they would fit together to create a 90 degree angle.

Theorems Using Complementary Angles

There are two frequently used theorems that involve complementary angles. The first theorem is:

If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.

Before we start with the theorem, let's remember that congruent angles are angles that have the same measure.

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