An exterior angle of a n-sided regular polygon = \(\frac{360}{n}\)

For (n - 1) sided regular polygon = \(\frac{360}{n - 1}\)

For (n + 2) sided regular polygon = \(\frac{360}{n + 1}\)

⇒ \(\frac{360}{n - 1} - \frac{360}{n + 2}\) = 6 9Given)

⇒ \(\frac{360(n + 2) - 360(n - 1)}{(n - 1)(n + 2)}\)

⇒ \(\frac{360n + 720 - 360n + 360}{(n - 1)(n + 2)}\)

⇒ \(\frac{1080}{(n - 1)(n + 2)} = \frac{6}{1}\)

⇒ 1080 = 6 (n - 1)(n + 2)

⇒ 180 = (n - 1)(n + 2)

⇒ 180 = n\(^2\)+ 2n - n - 2

⇒ 180 = n\(^2\) + n - 2

⇒ n\(^2\)b+ n - 2 - 180 = 0

⇒ n\(^2\) + n - 182 = 0

⇒ n\(^2\) + 14n - 13n - 182 = 0

⇒ n (n + 14) - 13 (n + 14) = 0

⇒ (n - 13) (n + 14) = 0

⇒ n - 13 = 0 or n + 14 = 0

⇒ n = 13 or n = -14

∴ n = 13 (We can't have a negative number of side)