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 $&gt; 180$, but how do we know if we can connect two points inside the polygon with a line inside the polygon, if it has angles $&lt;180$? Maybe we would arrive at contradiction if we assumed angles $&gt;170$. We should prove that all angles $&lt; 180$ are okay here.
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 $180^{\circ}$.
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.