In a regular polygon, any interior angle exceeds the exterior angle by 120 degrees. Then, the number of diagonals of this polygon is
Explanation:
Let the exterior angle be x°, hence interior angle is (x + 120)°.
Sum of interior and exterior angles is 180°. ⇒ x + (x + 120) = 180 ⇒ x = 30°
∴ Interior angle of this regular polygon = 30 + 120 = 150°
Each interior angle of a regular polygon of n sides = (n - 2)/n × 180° = 150° ⇒ (n - 2)/n = 5/6 ⇒ n = 12
Now, number of diagonals in a n-sided polygon = n × (n - 3)/2 = 12 × 9/2 = 54
Hence, 54.
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