1. **Problem statement:** Show by induction that the angle sum of a convex $n$-gon is $$(n-2)\pi$$ for integer $n \geq 3$. 2. **Base case:** For $n=3$, a triangle, the angle sum is known to be $$\pi + \pi + \pi = (3-2)\pi = \pi$$ which is true. 3. **Inductive hypothesis:** Assume for some $k \geq 3$, the angle sum of a convex $k$-gon is $$(k-2)\pi$$. 4. **Inductive step:** Consider a convex $(k+1)$-gon. By drawing a diagonal from one vertex to a non-adjacent vertex, we split it into a triangle and a convex $k$-gon. 5. The angle sum of the $(k+1)$-gon is the sum of the angle sums of the triangle and the $k$-gon: $$\text{Angle sum}_{k+1} = \pi + (k-2)\pi = (k-1)\pi = ((k+1)-2)\pi$$ 6. This completes the induction proof that the angle sum of a convex $n$-gon is $$(n-2)\pi$$ for all integers $n \geq 3$. --- 7. **Part (ii):** Find the angle at each vertex of a regular $n$-gon. 8. Since the polygon is regular, all interior angles are equal. The sum of all interior angles is $$(n-2)\pi$$. 9. Therefore, each interior angle is: $$\frac{(n-2)\pi}{n} = \pi - \frac{2\pi}{n}$$ --- 10. **Part (iii):** Deduce for which $n$ copies of a regular $n$-gon can tile the plane. 11. For tiling, the interior angles must fit around a point exactly, so some integer $m$ copies of the angle must sum to $2\pi$: $$m \times \left(\pi - \frac{2\pi}{n}\right) = 2\pi$$ 12. Simplify: $$m\pi - \frac{2m\pi}{n} = 2\pi$$ 13. Divide both sides by $\pi$: $$m - \frac{2m}{n} = 2$$ 14. Multiply both sides by $n$: $$mn - 2m = 2n$$ 15. Factor $m$: $$m(n - 2) = 2n$$ 16. So: $$m = \frac{2n}{n-2}$$ 17. For $m$ to be a positive integer, $\frac{2n}{n-2}$ must be an integer. 18. Check integer values $n \geq 3$: - $n=3$: $m=\frac{6}{1}=6$ (integer) - $n=4$: $m=\frac{8}{2}=4$ (integer) - $n=5$: $m=\frac{10}{3}\notin \mathbb{Z}$ - $n=6$: $m=\frac{12}{4}=3$ (integer) - For $n>6$, $m$ is not integer. 19. Therefore, only $n=3,4,6$ allow regular $n$-gons to tile the plane. **Final answers:** - Angle sum of convex $n$-gon: $$(n-2)\pi$$ - Each interior angle of regular $n$-gon: $$\pi - \frac{2\pi}{n}$$ - Regular $n$-gons tile the plane only for $n=3,4,6$.