Question

In an equiangular polygon, the measure of each exterior angle is $25\%$ of the measure of each interior angle. How many sides it has?

Answer 1

The sum of exterior angle and its adjacent interior is ${180}^0$, that is,

$e+i={180}^0$

Since each exterior angle is $25$% of the measure of each interior angle, therefore, substitute $\frac{25}{100}i=e$ or $i=4e$.

$e+4e={180}^0\Rightarrow 5e={180}^0\Rightarrow e=\frac{{180}^0}{5}={36}^0$

We know that the measure of exterior angle is $e={\left(\frac{360}{n}\right)}^0$ where $n$ is the number of sides.

Here, it is given that the exterior angle is $e={36}^0$, therefore,

$n=\frac{360}{e}=\frac{360}{36}=10$

Hence, the number of sides is $10$.