Lesson Complementary, Supplementary angles

Throughout the past weeks or so I have seen dozens and dozens of questions on complementary and supplementary angles, which has compelled me to write a lesson on them. Truth is, this is one of the easiest things to learn in a standard geometry course. In addition, they are simply definitions, not concepts or theorems, making them even simpler.


By definition, two angles are "complementary" if the measures of the angles add up to 90 degrees. Similarly, two angles are "supplementary" if their sum is 180 degrees.

Let us look at some examples:

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Example 1: Find the complement of a 27 degree angle.
Solution: Letting the complement be "x", we can apply the definition of complementary angles and say that

27 + x = 90
x = 90 - 27 = 63 (degrees) ∎


Sometimes you will have to find the complement or supplement of an angle that has variables in it. This shouldn't make the problem any harder; you just use the same procedure.

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Example 2: Find the supplement of an angle that measures (2y + 8) degrees.
Solution: By definition, the supplement is
180 - (2y + 8) = 180 - 2y - 8 = 172 - 2y. ∎


Oftentimes you will get a question regarding complementary/supplementary angles that will try to trick you. You will have to read carefully and fully, and be able to translate the question into an equation that can be solved. Let us look at this example:

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Example 3: The supplement of an angle is 12 less than three times the complement of the angle. Find the complement of the angle.
Solution: If we let the angle be x, then by definition, the supplement of the angle is 180-x and the complement is 90-x. We translate into an algebra equation:

180-x = 3(90-x) - 12 (supplement is 3*complement, minus 12)

180 - x = 270 - 3x - 12 = 258 - 3x

Adding 3x to both sides,

180 + 2x = 258

Subtracting 180 from both sides,

2x = 78 --> x = 39

However 39 is *not* the correct answer. The question asks for the complement of the angle. Here, we find 90-x, which is 90-39, or 51. ∎


Even though you will rarely see the phrases "complementary angles" or "supplementary angles" in advanced geometry problems, the concepts themselves are crucial. Oftentimes, in advanced geometry problems, you might see something like "Prove that ABC + CBD = 90" instead of "Prove that ABC and CBD are complementary" because foreign students might not be familiar with the terminology. In either case, you will still need this concept.