Everyday words can have a special meaning in math. That's certainly the case for "complementary," which represents the special relationship between any two angles that, when added together, total 90 degrees. This can mean the angles are right next to each other, but they can also be on opposite sides of one edge of a triangle, or not on the same geometric shape at all.
TL;DR (Too Long; Didn't Read)
If two angles are complementary, the sum total of their angle measurements is 90 degrees.
Finding a Missing Complementary Angle
So, what good is it to know that two angles are complementary? To begin with, if you know the value of one angle you can use that to find the value of the other angle, because you know they both total 90 degrees. Or to write it out in mathematical terms,
a + b = 90 degrees, where a is the measure of one angle and b is the measure of the other angle.
Imagine that you know one of the angles in question measures 25 degrees. If you substitute that into the formula, you have:
25 degrees + b = 90 degrees
To find the measure of the other angle, solve for b. This gives you:
b = 65 degrees
So the measure of the other complementary angle is 65 degrees.
Two Complementary Angles Form a Right Angle
Knowing that two angles are complementary opens the door to some other information, too. First, a 90-degree angle is also known as a right angle, which you'll find in many geometric shapes like squares, rectangles and some triangles, and in real-world shapes including boxes and ramps. Two angles don't have to be right next to each other to be complementary, but if they are, you'll automatically know that when taken together, they form that right angle.
Right Triangles Have Complementary Angles
There's a special relationship between all three angles of a triangle, too: If you add their measurements all together, the total will be 180 degrees. If you're dealing with a right triangle, you already know that one of those angles measures 90 degrees. That leaves 90 degrees to be distributed between the other two angles, which – surprise! – means they're complementary. This comes in handy if, for example, you're told that two angles of a triangle are complementary. In that case, you automatically know you're dealing with a right triangle.
The right triangle is also an excellent example of complementary angles not having to be right next to each other; in this case, the complementary angles are at opposite ends of one of the triangle's sides.