The difference between the exterior and interior angles at a vertex of a regular polygon is `150^@`. The number of sides of the polygon is

Options

A. 10
B. 15
C. 24
D. 30

Answer

To solve the problem, we need to find the number of sides of a regular polygon given that the difference between its exterior and interior angles at a vertex is \(150^\circ\).

### Step-by-step Solution:

1. **Define the Angles**:
Let the interior angle be \(i\) and the exterior angle be \(e\).

2. **Set Up the Equations**:
From the problem, we know:
\[ i - e = 150^\circ \quad \text{(Equation A)} \]
Also, we know that the interior and exterior angles at a vertex sum up to \(180^\circ\):
\[ i + e = 180^\circ \quad \text{(Equation B)} \]

3. **Add the Two Equations**:
Now, we will add Equation A and Equation B:
\[ (i - e) + (i + e) = 150^\circ + 180^\circ \]
This simplifies to:
\[ 2i = 330^\circ \]

4. **Solve for the Interior Angle**:
Divide both sides by 2 to find \(i\):
\[ i = \frac{330^\circ}{2} = 165^\circ \]

5. **Find the Exterior Angle**:
Now, substitute \(i\) back into Equation B to find \(e\):
\[ e = 180^\circ - i = 180^\circ - 165^\circ = 15^\circ \]

6. **Calculate the Number of Sides**:
The number of sides \(n\) of a regular polygon can be found using the formula:
\[ n = \frac{360^\circ}{e} \]
Substituting the value of \(e\):
\[ n = \frac{360^\circ}{15^\circ} = 24 \]

### Final Answer:
The number of sides of the polygon is \(24\).

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