Question

A polygon has $n$sides. Two of its angles are right angles and each of the remaining angles is ${144^{\circ}}$.Find the value of $n$?

Answer 1

Hint:
Find the sum of interior angles of the polygon and equate it to the general formula of sum of interior angles of a polygon i.e. $\left( {n - 2} \right){180^{\circ}}$, and find the value of $n$, which is the number of sides of a polygon.

Complete step by step solution:
Given,
Two of the angles are ${90^{\circ}}$and the remaining angles are ${144^{\circ}}$
Which indicates, sum of angles of a polygon is, $2({90^{\circ}}) + (n - 2){144^{\circ}}$as per the question
So, now equate it to the general formula i.e.$\left( {n - 2} \right){180^{\circ}}$
\[
   \Rightarrow 2({90^{\circ}}) + (n - 2){144^{\circ}} = \left( {n - 2} \right){180^{\circ}} \\
   \Rightarrow {180^{\circ}} + {144^{\circ}}n - {288^{\circ}} = {180^{\circ}}n - {360^{\circ}} \\
   \Rightarrow {180^{\circ}} - {288^{\circ}} + {360^{\circ}} = {180^{\circ}}n - {144^{\circ}}n \\
   \Rightarrow {252^{\circ}} = {36^{\circ}}n \\
   \Rightarrow n = \dfrac{{{{252}^{\circ}}}}{{{{36}^{\circ}}}} \\
   \Rightarrow n = 7 \\
 \]

Which means the polygon has 7 sides, in which two of its angles are ${90^{\circ}}$and 5 of the remaining are ${144^{\circ}}$.

Note:
Sum of interior angles of a regular polygon is $\left( {n - 2} \right){180^{\circ}}$and each interior angle of a regular polygon is $\dfrac{{\left( {n - 2} \right){{180}^{\circ}}}}{n}$where $n$is the number of sides of a polygon.