The interior angles of a polygon are in AP.The smallest angle is 120° and common sense difference is is 5° find the no.ofsides of polygon

The smallest interior angle is 120 deg. and the corresponding exterior angle is 60 deg. The sum of the exterior angles of the polygon is 360, and the exterior angles are in an AP.

Sn = (n/2)[2a +(n-1)d]

360 = (n/2)[(2x60) +(n-1)x(-5)], or

720 = n[120 -5n + 5], or

720 = n[-5n + 125], or

144 = n[-n +25], or

n^2 - 25n + 144 = 0, or

(n-9)(n-16) = 0

or n = 9. The figure is an irregular nonagon. The sum of the nine exterior angles which are 60,55,50,45,40,35,30,25,20 = 360.

Check: Sn = (9/2)[2*20 + (9–1)*5] = (9/2)[40 +8*5]] = (9/2)[40+40] = 9*80/2 = 360. Correct.

n = 9. The figure is an irregular nonagon.