The interior angle of a regular polygon is :

Question

Accepted Answer

To determine the interior angle of a regular polygon, we need to understand the relationship between the number of sides (n) and the sum of the interior angles. Here's a detailed explanation supporting Option 2 as the correct answer:

### Explanation for Option 2:

1. **Sum of Interior Angles**: The sum of the interior angles of a polygon with n sides is given by the formula $(n - 2) \times 180^\circ$. This formula arises because a polygon can be divided into $(n - 2)$ triangles, and each triangle has an angle sum of $180^\circ$.

2. **Interior Angle of a Regular Polygon**: In a regular polygon, all interior angles are equal. Therefore, to find the measure of each interior angle, we divide the sum of the interior angles by the number of sides (n). This gives us the formula:
   $$
   \text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}
   $$
   This is exactly what Option 2 states, making it the correct choice.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: $(n - 2) \times 180^\circ$**

- This formula correctly calculates the sum of all interior angles of the polygon, not the measure of a single interior angle. It does not account for dividing by the number of sides to find the measure of each angle in a regular polygon.

2. **Option 3: $\frac{n}{2} \times 180^\circ$**

- This formula does not correctly represent any known calculation related to the angles of a polygon. It neither calculates the sum of the interior angles nor the measure of a single interior angle in a regular polygon.

3. **Option 4: $\frac{n(n - 3)}{2}$**

- This formula is unrelated to the calculation of interior angles in polygons. It does not correspond to any standard geometric formula for angles in polygons and is therefore incorrect in this context.

In conclusion, Option 2 is the correct answer because it accurately represents the formula for calculating the measure of each interior angle in a regular polygon. The other options do not provide the correct calculation for this specific geometric property.