If each interior angle of a regular polygon is 175°, then the number of sides are :

Question

Accepted Answer

To determine the number of sides of a regular polygon when each interior angle is 175°, we can use the formula for the interior angle of a regular polygon:

$$
\text{Interior Angle} = \frac{(n-2) \times 180°}{n}
$$

where $ n $ is the number of sides of the polygon.

Given that the interior angle is 175°, we set up the equation:

$$
175° = \frac{(n-2) \times 180°}{n}
$$

Solving for $ n $:

1. Multiply both sides by $ n $ to eliminate the fraction:

$$
   175n = (n-2) \times 180
   $$

2. Expand the right side:

$$
   175n = 180n - 360
   $$

3. Rearrange the equation to isolate terms involving $ n $:

$$
   180n - 175n = 360
   $$

$$
   5n = 360
   $$

4. Divide both sides by 5 to solve for $ n $:

$$
   n = \frac{360}{5} = 72
   $$

Thus, the number of sides of the polygon is 72, which corresponds to Option 2.

**Explanation for why other options are incorrect:**

- **Option 1 (54 sides):** If we substitute $ n = 54 $ into the formula for the interior angle, we get:

$$
  \text{Interior Angle} = \frac{(54-2) \times 180°}{54} = \frac{52 \times 180°}{54} \approx 173.33°
  $$

This is not equal to 175°, so Option 1 is incorrect.

- **Option 3 (60 sides):** If we substitute $ n = 60 $ into the formula for the interior angle, we get:

$$
  \text{Interior Angle} = \frac{(60-2) \times 180°}{60} = \frac{58 \times 180°}{60} = 174°
  $$

This is not equal to 175°, so Option 3 is incorrect.

- **Option 4 (15 sides):** If we substitute $ n = 15 $ into the formula for the interior angle, we get:

$$
  \text{Interior Angle} = \frac{(15-2) \times 180°}{15} = \frac{13 \times 180°}{15} = 156°
  $$

This is not equal to 175°, so Option 4 is incorrect.

Therefore, based on the calculations and logical reasoning, Option 2 (72 sides) is the correct answer.