Question

Find the number of sides of a regular polygon if each exterior angle is equal to twice its adjacent interior angle

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 equal to twice its adjacent interior angle, therefore, substitute $2i=e$ or $i=\frac{e}{2}$.

$e+\frac{e}{2}={180}^0\Rightarrow \frac{2e+e}{2}={180}^0\Rightarrow \frac{3e}{2}={180}^0\Rightarrow 3e={180}^0\times 2\Rightarrow 3e={360}^0\Rightarrow e=\frac{{360}^0}{3}={120}^0$

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 $e={120}^0$, therefore,

$n=\frac{360}{e}=\frac{360}{120}=3$

Hence, the number of sides is $3$.