Complementary Angles
Two angles are Complementary when they
add up to 90 degrees (a Right Angle ).
These two angles (40° and 50°) are Complementary Angles, because they add up to 90°:
Notice that together they make a right angle .
But the angles don't have to be together.
These two are complementary because 27° + 63° = 90°
Play With It ...
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Right Angled TriangleIn a right angled triangle, the two nonright angles are complementary, because in a triangle the three angles add to 180°, and 90° has already been taken by the right angle. 
When two angles add to 90°, we say they "Complement" each other. Complementary comes from Latin completum meaning "completed" ... because the right angle is thought of as being a complete angle. 

Spelling: be careful, it is not "Complimentary Angle" (with an "i") ... that would be an angle you get for free! 
Complementary vs Supplementary
A related idea is Supplementary Angles  those add up to 180°
How to remember which is which? Easy! Think:
 "C" of Complementary stands for "Corner" (a Right Angle), and
 "S" of Supplementary stands for "Straight" (180 degrees is a straight line)
Or you can think: when you are right you get a compliment (sounds like complement).
Or you could also think: "Supplement" (like a Vitamin Supplement) is something extra, so it it bigger.