Question

1. the measure of an interior angle of a regular polygon is $165\\ ^\circ$. find the number of sides in the polygon.

Explanation:

Step1: Recall interior angle formula

The formula for the measure of an interior angle of a regular \(n\)-sided polygon is:
$$\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}$$

Step2: Set up equation with given value

Substitute the given interior angle \(165^\circ\) into the formula:
$$165 = \frac{(n-2) \times 180}{n}$$

Step3: Solve for \(n\)

Multiply both sides by \(n\):
$$165n = 180(n-2)$$
Expand the right-hand side:
$$165n = 180n - 360$$
Rearrange terms to isolate \(n\):
$$180n - 165n = 360$$
$$15n = 360$$
$$n = \frac{360}{15}$$

Answer:

24