A regular polygon P has 135 diagonals. Find the interior angle of the polygon.

We know, Number of diagonals = n(n - 3)/2, where n is the number of the sides in the polygon.

⇒ 135 = n(n - 3)/2

⇒ n(n - 3) = 270

⇒ n2 - 3n = 270

⇒ n2 - 3n - 270 = 0

⇒ n2 - 18n + 15n - 270 = 0

⇒ n(n - 18) + 15(n - 18) = 0

⇒ (n - 18)(n + 15) = 0

⇒ n = 18

∴ Number of sides in polygon = 18

Exterior angle = 360°/n = 360°/18 = 20°

And Interior angle = 180° - Exterior angle = 180° - 20° = 160°