Question

The exterior angle of a regular polygon is our fourth of its interior angle. How many sides does the polygon have?

Answer 1

sum of all interior $\angle s$ of a regular polygon = $180-x{180}^{\circ }(x-2)$

each interior $\angle =\frac{{180}^{\circ }(x-2)}{x}$

sum of all exterior $\angle ={360}^{\circ }$

Each exterior $\angle ={360}^{\circ }/x$

Given $\frac{{360}^{\circ }}{x}=\frac{1}{4}\left(\frac{{180}^{\circ }(x-2)}{x}\right)$

$\Rightarrow x=10$