Each interior angle of a regular polygon exceeds its exterior angle by `132^(@)`. How many sides does the polygon have ?

Options

A. 9
B. 15
C. 12
D. none of these

Answer

To solve the problem, we need to find the number of sides of a regular polygon given that each interior angle exceeds its exterior angle by \(132^\circ\).

### Step-by-Step Solution:

1. **Understanding the Relationship Between Interior and Exterior Angles**:
- Let the interior angle be \(x\) and the exterior angle be \(y\).
- According to the problem, we have:
\[ x - y = 132^\circ \]

2. **Using the Sum of Angles**:
- We know that the sum of the interior and exterior angles in a polygon is:
\[ x + y = 180^\circ \]

3. **Setting Up the Equations**:
- Now we have two equations:
1. \(x - y = 132^\circ\) (Equation 1)
2. \(x + y = 180^\circ\) (Equation 2)

4. **Adding the Two Equations**:
- Adding Equation 1 and Equation 2:
\[ (x - y) + (x + y) = 132^\circ + 180^\circ \]
\[ 2x = 312^\circ \]
\[ x = \frac{312^\circ}{2} = 156^\circ \]

5. **Finding the Exterior Angle**:
- Now substitute \(x\) back into Equation 2 to find \(y\):
\[ 156^\circ + y = 180^\circ \]
\[ y = 180^\circ - 156^\circ = 24^\circ \]

6. **Finding the Number of Sides**:
- The exterior angle \(y\) of a regular polygon is related to the number of sides \(n\) by the formula:
\[ y = \frac{360^\circ}{n} \]
- Substituting the value of \(y\):
\[ 24^\circ = \frac{360^\circ}{n} \]
- Rearranging gives:
\[ n = \frac{360^\circ}{24^\circ} = 15 \]

### Conclusion:
The polygon has **15 sides**.