Complementary angles always come in pairs, That is there has to be two of them to complement each other. When they are together they will always make up a right angle (that’s 90 degrees to you).

Why are they called complementary? Although I like to think it’s because they get on so well, the truth is a bit more obscure and its all the fault of the Romans, who were completely preoccupied with conjugating their verbs, one of which, 'completum' meant ‘having been filled up’. Once it has been thoroughly conjugated 'completum' spits out 'complere' and eventually 'complementum', which is where we get all of our complements from and at least some of our compliments. "What a gorgeous angle you have there, I bet it would fit wonderfully next to mine", being a fairly uncommon example.

Anyway, if you’ve got yourself an angle and it doesn’t quite cut the mustard in filling up the corner of your box, whatever angle that you need to do the job of ‘filling up’ the rest of the corner is the complementary.

Or to put the same thing in terms that a geometer would feel at home with: The angle subtended between the non-adjacent sides of two adjacent complementary angles and their common vertex equals a right angle.

What is so great about a right angle anyway?

Euclid has left the building in distress!