# The difference between any two consecutive interior angles of a polygon is 50. If the smallest angle is 1200, find the number of the sides of the polygon.

We know that if the number of sides is n, then the sum of interior angles of a polygon with

n sides = (2n - 4) right angles

= (180n- 360) degrees

Now the sequence of angles

120°, 125°, 130°,... form an A.P. with a = 120°, d = 5°

Sn = ⇒ 180n – 360 = or 2(180n – 360) = n[240 + 5n - 5)]

or 360n - 720 = 240n + 5n2 + 5n

or 5n2 – 125n + 720 = 0

or n2 - 25n + 144 =0

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

or n = 9, 16

But n = 16 is not possible as it gives the last term of A.P.

= a + (n - 1)d

= 120 + (16-9) x 5

= 195°

Thus the number of sides = 9.

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