4.How many sides does a regular polygon have if each of its interior angles is \(165^{\circ}\)?

Let the number of sides of a regular polygon be n.
We have, Sum of all interior angles = \((n – 2) \times 180^{\circ}\)

and, measure of its each angle=\(\frac{(n-2) \times 180^{\circ}}{n}\)

here we have each angle as \(165^{\circ}\)

\(\Rightarrow \frac{(n-2) \times 180^{\circ}}{n}=165\)

\(\Rightarrow 180n - 360 =165 n \quad \)[Cross multiplication and opening the brackets]

\(\Rightarrow 180n -165 n=360\)

\(\Rightarrow 15n = 360\)

\(\Rightarrow n = \frac{360}{15}=24\)

So, we have the number of sides of regular polygon as 24.