G10.2 The sides of an $N$-sided regular polygon are produced to form a "star". If the angle at each point of that "star" is $108^{\circ}$, find $N$. (For example, the "star" of a six-sided polygon is given as shown in the diagram.)
Consider an isosceles triangle formed by each point. The vertical angle is $108^{\circ}$.
Each of the base angle $=\frac{180^{\circ}-108^{\circ}}{2}=36^{\circ}$
$36 N=360$ (sum of ext. $\angle$ s of polygon) $\Rightarrow N=10$