The difference between any two consecutive interior angles of a polygon is `5o`. If the smallest angle is `120o`, find the number of the sides of the polygon.

The difference between any two consecutive interior angles of a polygo

1.	The difference between any two consecutive interior angles of a polygon is `5o`. If the smallest angle is `120o`, find the number of the sides of the polygon.
Answer» Let the number of sides of the polygon be n . Then, the sum of its interior angle = ( 2n-4) right angles = ` {( n-2)xx 180}^(@)` Since the difference between any two consecutive interior angles of the polygon is constant, its angles are in AP. In this AP, we have a =12-0 and d=5 Now, ` S_(n) = ( n-2) xx 180` ` Rightarrow n/2. { 2 x 120 + ( n-1) xx 5} = (n -2) xx 180` ` Rightarrow ( 235n)/2 + (5n^(2))/2 = 180 n - 360 Rightarrow 5n^(2) =- 125n + 720=0` ` Rightarrow n^(2) -25n + 144=0 Rightarrow ( n-90 (n-16) =0` but , when n=16 , we have last angle =` { 120 + ( 16 -10 xx 5} ^(@) = 195 ^(@)` , which is not possible n=9 Hence, the number of sides of the given polygon =9