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°
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Right Angled Triangle
In 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? Well, alphabetically they are:
 Complementary add to 90°
 Supplementary add to 180°
You can also think:
 "C" of Complementary is for "Corner" (a Right Angle), and
 "S" of Supplementary is for "Straight" (180° is a straight line)
Or you can think:
 when you are right you get a compliment (sounds like complement)
 "supplement" (like a vitamin supplement) is something extra, so is bigger