Question

 In a regular polygon, each interior angle is 120 more than each exterior angle. Find the number of diagonals of the polygon.

Answer 1

Assume the exterior angle of the polygon is denoted as x, leading to the interior angle being 120+x. Applying the property that the sum of interior and exterior angles in a polygon is always 180∘180∘, we set up the equation x+(120+x)=180, simplifying to 2+120=1802x+120=180. Solving for x, we find x=30∘.

With the exterior angle established as 30∘, consider a polygon with n sides. The total sum of exterior angles in any polygon is always 360∘, so n×x=360 becomes n×30=360. Solving for n, we find n=12.

Therefore, the polygon in question has 12 sides.

Answer 2

Let exterior angle is \(x\)
Interior angle will be \(120+x\)
interior angle \(+\) exterior angle \(= 180\)
so \(x+120+x =180\)
\(2x+120 = 180\)
\(2x =180-120\)
\(2x = 60\)
\(x=30\)degree. Exterior angle \(=30\).
 Let there are \(n\) sides so \(n \times x = 360\)
\(n\times 30= 360\)
\(n=\frac{360}{30}\)
\(n= 12\)
 so there are \(12\) sides of polygon.