Q:

Derive a relation between angles A,B and C

Derive a relation between angles A,B and C (do not use other angles in the final relation):

I have tried to use two theorems in triangles(external angle and complement angles),but no success! It seems my equations are not independent,and each time I get to obvious equalities such as: $90-A=90-A$. What's wrong with my solution?

A:

If what they're really after is that $C = A + B$, then it is proven, for instance, in this video, by Numberphile (note that what they prove there is that $A + B + C = 90^\circ$, but if you know that $C = 45^\circ$, then we also have $A + B = 45^\circ$, so $C = A+B$ follows).
The proof works by extending the figure and adding two more right triangles with angle $B$. One then easily sees that the angles $B$ and $C$ appear next to the angle $A$, forming a right angle, $A+B+C=90^\circ$

