In a regular polygon, any interior angle exceeds the exterior angle by 120 degrees. Then, the number of diagonals of this polygon is:

Question

Accepted Answer

1. **Relation Between Interior and Exterior Angles**:
   - The sum of the interior and exterior angles of a polygon is always 180°.
   - Given that the interior angle exceeds the exterior angle by 120°:
     $$
     	ext{Interior Angle} = 	ext{Exterior Angle} + 120
     $$
     Substituting into the sum relation:
     $$
     	ext{Exterior Angle} + (	ext{Exterior Angle} + 120) = 180
     $$
     Simplify:
     $$
     2 	imes 	ext{Exterior Angle} = 60
     $$
     $$
     	ext{Exterior Angle} = 30^\circ
     $$

2. **Number of Sides of the Polygon**:
   - The exterior angle of a regular polygon is given by:
     $$
     	ext{Exterior Angle} = \frac{360}{n}
     $$
     Substituting $$ 	ext{Exterior Angle} = 30^\circ $$:
     $$
     \frac{360}{n} = 30
     $$
     Solve for $$ n $$:
     $$
     n = 12
     $$

3. **Number of Diagonals**:
   - The formula for the number of diagonals in a polygon with $$ n $$ sides is:
     $$
     	ext{Diagonals} = \binom{n}{2} - n
     $$
   - For $$ n = 12 $$:
     $$
     \binom{12}{2} = \frac{12 	imes 11}{2} = 66
     $$
   - Subtract the number of sides (12):
     $$
     	ext{Diagonals} = 66 - 12 = 54
     $$

Thus, the number of diagonals in the polygon is **54**.