Q. 184.2( 112 Votes )

# 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.

Answer :

Smallest Angle = 120°

Difference between any two consecutive interior angles of a polygon = 5°

The angles of the polygon will form an A.P. with common difference d as 5° and first term a as 120°.

We know, sum of all angles of a polygon with n sides = 180° (n – 2)

S_{n} = 180° (n – 2)

Equating both we get

⇒

⇒ n (240 + 5n – 5) = 360n – 720

⇒ 5n^{2} + 240n – 5n – 360n + 720 = 0

⇒ 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

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