# Prove that every internal angles of a convex polygon is less than $180$ degrees

It's not clear to me why a geometrical proof is so hard to find. The definition says a polygon is convex if we can connect any pair of two points of the polygon with a line that's contained in that polygon. In most sources, for unknown reason, it is treated as an obvious fact.

This is the proof I've found - page 2. Could you verify whether it is correct? I have doubts with regard to this proof - there's a contradiction if we allow angles $> 180$, but how do we know if we can connect two points inside the polygon with a line inside the polygon, if it has angles $<180$? Maybe we would arrive at contradiction if we assumed angles $>170$. We should prove that all angles $< 180$ are okay here.

• Is there a way to prove the converse of this statement? Apr 2, 2021 at 23:15

Let $A$ be any vertex of the polygon. Let $B$, $C$ be the vertices adjacent to $A$.

Angle $BAC$ (the internal angle) is inside the non-degenerate triangle $BAC$, and hence is less than $180^{\circ}$.

• Where are you using convexity? Feb 20, 2015 at 7:34
• Convexity ensures that angle BAC is inside the triangle. Feb 20, 2015 at 9:32
• Convexity ensures that angle BAC is inside the triangle. - this is what we are trying to prove here... you cannot assume it. Feb 20, 2015 at 10:33
• I'm not assuming it (as far as I can see). I'm saying that because the polygon is convex, line BC is inside the polygon, and so the angle opposite it (angle BAC) must also be inside the triangle. Feb 20, 2015 at 13:52

Definitions:

$1$. A convex polygon is defined as a polygon with all its interior angles less than $180°$.

$2$. A simple polygon is concave iff at least one of its internal angles is greater than $180°$ .

Let's say an internal angle in a convex polygon is more than $180°$, but it is a contradiction to the definition of a convex polygon.