Students will see two proofs for the sum of exterior angles of a convex polygon. In the first proof, students will see a geometric proof that the sum of exterior angles of a triangle is \(360^\circ.\) From the method of proof, it's easy to infer that the sum of exterior angles of any convex polygon is also \(360^\circ.\) That's because an n-sided polygon will make \(n\) linear pairs, thus \(180n.\) We subtract from this the sum of interior angles, which is \(180^\circ(n - 2)\), which gives us \(360^\circ:\)
$$\begin{align} & 180n - 180(n - 2) \\ =\ & 180(n - (n - 2)) \\ =\ & 180(n - n + 2) \\ =\ & 180(2) \\ =\ & 360 \end{align}$$The second proof is entirely visual, and can be seen here. In addition to problems involving convex polygons in general, students will consider the special case of regular polygons, and solve two types of problems involving them. The first type is to find the number of sides given one interior angle. The second is to find one interior angle given the number of sides. Next, give your students this interesting problem. After that, give them this puzzle. An additional solution can be found here. This problem is difficult. I doubt many students will solve it, but the solutions within the first link are clever and easy to understand, so I think they're worth seeing.
Next, give your students this challenge:
A Chain of Eight Polyhedra by NRICH
Conclude by leading this investigation:
Daedalus and Icarus try to Escape
by MathPickle