Demonstrate by mathematical induction that the sum of the internal angles of a $n$-sided *concave* polygon is $180^\circ(n-2)$.

Question

Demonstrate by mathematical induction that the sum of the internal angles of a $n$-sided concave polygon is $180^\circ(n-2)$.

Have I solved this classic problem for convex polygons, any tips on how to connect the two solutions?

Answer 1

We can restate the claim as this: $\pi$ times the number of sides, minus the sum of internal angles, is $2\pi$.

Suppose a reflex angle occurs at a vertex $B$ from $A$ to $C$. If you replace the polygonal arc $ABC$ with $AC$, you reduce the number of reflex angles by $1$, while reducing the sum of internal angles by $\pi$ radians, because if the reflex angle was $\pi+x$ we replace it with two contributions totalling $\pi-(\pi-x)=x$. We've also reduced the number of sides by $1$, preserving the above quantity we claim to be invariant.

So instead of inducting on the number of sides, induct on the number of reflex angles. Your base case, where there are none of them, is the convex polygons you already understand.