Question

What is the sum of the exterior angles of any convex polygon?

Answer 1

There are two exterior angles near each vertex. Let's call them "left" and "right".
The sum of all "left" exterior angels equals to the sum of all "right" ones and equals to `360^o`.

Explanation:

Consider we stand on the vertex of a horizontally positioned regular polygon with `N` sides. Extend one of the sides connected by this vertex and look straight along this extension that we call "left".

Turn from that direction towards the nearest side (clockwise on the illustrative picture above) by exterior angle and move along that side towards the next vertex.

Coming to this next vertex, we extend the side we walked upon and look along this extension. The situation is similar to the one before.

Turn again from that direction towards the nearest side by exterior angle, walk along it to the next vertex, extend this side and look straight ahead.

Continuing this process `N` times, we will turn (clockwise on a picture) by the same external angle `N` times and finally will face the same direction we started from. That is, we have turned `360^o` - that is the sum of all "left" exterior angles.

Similarly, all other exterior angles ("right" ones, when we move counterclockwise) also sum up to `360^o`