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

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?

• They say "demonstrate by induction". To me, that hints towards cutting off a triangle somewhere. Or just cut anywhere between two vertices that can "see" one another. Oct 8 '19 at 18:24

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.
• I'm not convinced the number of reflex angles always decreases: I think angles at $A$ and $C$ in your example could become reflex angles after replacing $ABC$ with $AC$. Oct 9 '19 at 11:36