# Determine the Complement and Supplement of a Given Angle

In this lesson we will determine the complement of a given angle as well as a supplement of a given angle. Let’s begin by reviewing the definitions of complementary angles and supplementary angles. Complementary angles are two angles that have a sum of ninety degrees. And here’s an example of two angles that are complementary. These two angles are complementary because the sum of the measures is equal to ninety degrees. Notice thirty degrees plus sixty degrees is equal to ninety degrees. In general though, if we call this angle one and this angle two, when two angles are complementary the measure of angle one plus the measure of angle two equals ninety degrees. Another way to think of this is if we were to combine two complementary angles they would form a ninety degree angle as shown here. So if we break up this right angle into two angles using this ray here, and call this angle one and this angle two. The sum of the measures must equal ninety degrees and therefore together they form a right angle. Two angles are supplementary when the angles have a sum of one hundred eighty degrees. Here’s an example of two supplementary angles because one hundred thirty-nine degrees plus forty-one degrees equals one hundred eighty degrees. Or in general, if we call this angle one and this angle two, the measure of angle one plus the measure of angle two equals one hundred eighty degrees. Another way to think of this is, if we were able to combine two supplementary angles they must form a straight angle. Beginning with this straight angle here, if we break this up into two separate angles, let’s say here
using this ray. And we call this angle one and this angle two, the sum of the measures must equal one hundred eighty degrees and therefore, together they form a straight angle. Now let’s take a look at our two questions. The first question is, find the complement of a fifty-one degree angle. Well we know complementary angles must have a sum of ninety degrees. So to model this, let’s begin with a right angle. Let’s break this up into two separate angles using this ray here. Let’s label this angle fifty-one degrees. The compliment of this angle would be the measure of an angle. So the sum of the measures is equal to ninety degrees, which would have to be this missing angle here. Notice to find the complement of a fifty-one degree angle we begin with ninety degrees and subtract fifty-one degrees, which gives us thirty-nine degrees. The complement of a fifty-one degree angle is an angle that measures thirty-nine degrees because thirty-nine degrees plus fifty-one degrees is equal to ninety degrees . So we’ll say the complement of a fifty-one degree angle is a thirty-nine degree angle. Next we’re asked to find the supplement of a seventeen degree angle, which would be an angle that if we add to the measure to seventeen degrees would be equal to one hundred eighty degrees. To model this let’s form a straight angle. Let’s break this up into separate angles let’s say using this ray here. Let’s measure this small angle seventeen degrees. So the measure of this angle here would be the supplement of the seventeen degree angle. We know that sum of the measures must equel one hundred eighty degrees and therefore, the measure of this angle must equal one hundred eighty degrees, minus seventeen degrees, which equals one hundred sixty-three degrees. Which means the supplement of a seventeen degree angle is a one hundred sixty-three degree angle. I hope you found this helpful.