Interior Diagonal of Polygons: Proving Existence

[ehrenfest]

Homework Statement

I want to show that every polygon with more than 3 sides has an interior diagonal.

Homework Equations

The Attempt at a Solution

If the polygon is convex, this is obvious. If not, there is an interior angle at some vertex, say V, that is greater than 180 degrees. Then I think a ray emanating from V and sweeping the interior of the polygon must strike another vertex. I cannot find a counterexample to that, but I am not sure how to prove that. It seems like the polygon would be infinite if that is not true, but how can I explain that?

 

[Dick]

If the rays from the vertex don't hit another vertex then the first side that they hit must be the same side no matter what the angle. So if you are viewing a segment from a point can it sweep an angle of over 180 degrees?

 

[NateTG]

You might be able to start by showing that there is an internal line from your vertex to a non-adjacent edge of the polygon.