Let \( n \) denote the number of sides of the regular polygon. The formula for the measure of each exterior angle \( E \) of a regular polygon is given by:
\[ E = \frac{360}{n} \]
The measure of each interior angle \( I \) can be calculated as:
\[ I = 180 - E = 180 - \frac{360}{n} \]
According to the problem statement, each interior angle is greater by 120 degrees than each exterior angle, which gives us the equation:
\[ I = E + 120 \]
Substituting the expressions for \( I \) and \( E \):
\[ 180 - \frac{360}{n} = \frac{360}{n} + 120 \]
To solve for \( n \), we will first simplify the equation:
- Combine like terms:
\[ 180 - 120 = \frac{360}{n} + \frac{360}{n} \]
which simplifies to:
\[ 60 = \frac{720}{n} \]
- Multiply both sides by \( n \):
\[ 60n = 720 \]
- Divide both sides by 60:
\[ n = \frac{720}{60} = 12 \]
Thus, the polygon has \( n = 12 \) sides.
To verify, we can check the measures of the interior and exterior angles.
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Each exterior angle: \[ E = \frac{360}{12} = 30 \text{ degrees} \]
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Each interior angle: \[ I = 180 - E = 180 - 30 = 150 \text{ degrees} \]
Now, we can verify the condition that the interior angle is greater by 120 degrees than the exterior angle:
\[ 150 = 30 + 120 \]
This confirms the condition stated in the problem.
Thus, the number of sides of the polygon is \( \boxed{12} \).
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