Question:

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

Updated On: Jan 6, 2025
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Approach Solution - 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.

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Approach Solution -2

Let exterior angle is xx
Interior angle will be 120+x120+x
interior angle ++ exterior angle =180= 180
so x+120+x=180x+120+x =180
2x+120=1802x+120 = 180
2x=1801202x =180-120
2x=602x = 60
x=30x=30degree. Exterior angle =30=30.
 Let there are nn sides so n×x=360n \times x = 360
n×30=360n\times 30= 360
n=36030n=\frac{360}{30}
n=12n= 12
 so there are 1212 sides of polygon. 

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