Complementary and supplementary are terms used to describe special relationships between two angles. If any two angles are complementary, their measurements add up to a total of 90 degrees or, to put it another way, their measurements add up to a right angle. If two angles are supplementary, their measurements add up to a total of 180 degrees or a straight line/straight angle. If you know the measurement of one of these angles, you can use the complementary or supplementary relationship to find the measurement of the other angle.
TL;DR (Too Long; Didn't Read)
To find the complement of an angle x, subtract x's measurement from 90 degrees: 90 degrees - x degrees = the measure of x's complementary angle.
To find the supplement of an angle y, subtract y's measurement from 180 degrees: 180 degrees - y degrees = the measure of y's supplementary angle.
Finding Complementary Angles
Because the sum of two complementary angles is 90 degrees, if you're given the measurement of one angle x, the measurement of its complementary angle is 90 - x degrees.
So if you're told that an angle measures 30 degrees and asked to find its complement, you could calculate:
90 degrees - 30 degrees = 60 degrees
The measurement of the missing (complementary) angle is 60 degrees.
Check your work by summing the measures of the two complementary angles. In this example, you have 30 degrees + 60 degrees = 90 degrees. If both angles add up to 90 degrees, you did indeed find the complement of the first angle.
The complementary relationship is very useful when it comes to right triangles. This is because the three angles of a triangle always add up to 180 degrees. So if you know one of the angles is a right angle – or in other words, it measures 90 degrees – that leaves exactly 90 degrees to be distributed between the remaining two angles. So they are complementary. (Angles don't have to be right next to each other to be complementary or supplementary.)
Finding Supplementary Angles
Because the sum of any two supplementary angles is 180 degrees, if you're given the measure of an angle y, then you can find its supplement or supplementary angle by calculating 180 - y.
So if you are told that one angle measures 124 degrees and asked to find its supplement, you would calculate:
180 degrees - 124 degrees = 56 degrees
The measure of the missing angle is 56 degrees.
Just as with complementary angles, you can check your work by summing the two angles you just computer. In this case you have 124 degrees + 56 degrees = 180 degrees. If the two angles total to exactly 180 degrees, they are supplementary.