convex polygon considering three angles

Question

If I choose three vertexes A,B,C in a convex polygon, it so happens that the sum of angleA,angleB,angleC appears to be 180 or larger. Why is this true? I tried drawing the polygon and making triangles, but I can't say that every vertex and every convex polygon works for it.

Answer 1

Let $a,b,c$ be the interior angles at $A,B,C$ respectively. $$a\ge\angle BCA\\b\ge\angle ABC\\c\ge\angle BCA$$ \begin{align}a+b+c&\ge\angle BCA+\angle ABC+\angle BCA\\a+b+c&\ge 180\end{align}