Question

The interior angles of a convex polygon are in A.P. the smallest angle is ${120}^{\circ }$ and the common difference is $5^{\circ }$ the number of its sides are

• A. 9
• B. 10
• C. 52
• D. 60

Answer 1

The correct option is A

9

If n is the number of its sides, the sum of its interior angles is

$\frac{n}{2}[2\times {120}^{\circ }+(n-1\left)5^{\circ }\right]=\frac{5n}{2}[n+47]$ ....(1)

But the sum of the interior angles is ${180}^{\circ }n-{360}^{\circ }=180(n-2)$ . . . .(2)

$\left(1\right)and\left(2\right)\Rightarrow \frac{5n}{2}(n+47)=180(n-2)\Rightarrow n^2-25n+144=0\Rightarrow n=9,16$

In a convex polygong no angle exceeds ${180}^{\circ }.\therefore n\ne 16andn=9$