A geometry problem by Nahom Assefa

The polygon formed will be a regular polygon, with angle $80 + 80 = 160$

$\begin{aligned} \frac{(x-2)\cdot 180}{x} &= 160 \\ 180x - 360 &= 160x \\ 20x &= 360 \\ x &= \boxed{18} \end{aligned}$

As we trace a path along the inner (or outer) sides, we rotate $20$ degrees at each corner. We'll end up moving in the original direction when we reach $360$ degrees. Thus the resulting $n$ -gon has $(360/20) = 18$ sides.

Alternatively, the interior angles of this $n$ -gon would each be $160 (= 80+80)$ degrees. The interior angle of a regular $n$ -gon has $(n-2)*180/$ n degrees. Set $160 = (n-2)*180/n$ and solve for $n$ to get $n=18$ .