Complementary angles don't sit around saying nice things to each other. If they did, they'd be complimentary angles – get it? Instead, when you add two complementary angles together, they total 90 degrees. This is also the measure of a right angle, so it might help to visualize complementary angles as what you get when you draw a line that separates a right angle into two separate angles. If you're given the measure of one angle, you can use this relationship – adding up to 90 degrees – to find that angle's complement.
TL;DR (Too Long; Didn't Read)
To find the complement of an angle, subtract that angle's measurement from 90 degrees. The result will be the complement.
Subtract the measurement of the first angle from 90 degrees. The result is the measure of the complementary angle. So if the first angle measures 40 degrees, you'd have:
The measure of the complementary angle is 50 degrees.
What About Variables?
What if you're only given the measure of the first angle as a variable? In that case you can still perform the subtraction to find the measure of the complementary angle – you just can't simplify past that step.
So if you're told only that the first angle measures x degrees, the measure of the complementary angle would be:
Complementary Angles Don't Have to Be Adjacent
Although you can visualize complementary angles as the result of splitting a right angle into two separate angles, two complementary angles don't actually have to be positioned right next to each other. In fact, if you're dealing with a right triangle, there will be complementary angles on opposite ends of the triangle's hypotenuse, or diagonal side.
This is because if you total the three angles of a triangle, they always add up to 180 degrees. And because a right triangle has a right or 90-degree angle in it, that only leaves 90 degrees more to be distributed between the other two angles. So, by definition, they must be complementary.
Keep this relationship in mind. If you're ever given a right triangle and the measure of just one of the non-right angles, you'll be able to use the complementary relationship to find the measure of the other angle.
Did you know? Because two complementary angles add up to a total of 90 degrees, they must both, by definition, be acute. (An acute angle measures less than 90 degrees.)
About the Author
Lisa studied mathematics at the University of Alaska, Anchorage, and spent several years tutoring high school and university students through scary -- but fun! -- math subjects like algebra and calculus.