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

**Solution:**

The angles of the polygon will form an A.P with d = 5° and a = 120°.

We know that the sum of all angles of a polygon with n sides is 180° (n - 2)

Therefore,

⇒ S_{n} = 180° (n - 2)

⇒ n/2 [2a + (n - 1) d] = 180° (n - 2)

⇒ n/2 [240° + (n - 1) 5] = 180° (n - 2)

⇒ n [240° + (n - 1)5] = 360 (n - 2)

⇒ 240n + 5n^{2} - 5n = 360n - 720

⇒ 5n^{2} - 125n + 720 = 0

⇒ n^{2} - 25n + 144 = 0

⇒ n^{2} -16n - 9n + 144 = 0

⇒ n (n - 16) - 9 (n - 16) = 0

⇒ (n - 9)(n - 16) = 0

⇒ n = 9 or 16

NCERT Solutions Class 11 Maths Chapter 9 Exercise 9.2 Question 18

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

**Summary:**

The number of sides of the polygon where the smallest angle is 120 degrees and any two consecutive interior angles differ by 5 degrees is either 9 or 16