Question

find the number of sides of a regular polygon with each interior angle measuring 165°

Explanation:

Step1: Recall the formula for the interior angle of a regular polygon

The formula for the interior angle $\theta$ of a regular polygon with $n$ sides is $\theta=\frac{(n - 2)\times180^{\circ}}{n}$.

Step2: Substitute the given interior angle value into the formula

We are given that $\theta = 165^{\circ}$, so $165^{\circ}=\frac{(n - 2)\times180^{\circ}}{n}$.

Step3: Cross - multiply

Cross - multiplying gives us $165n=(n - 2)\times180$.

Step4: Expand the right - hand side

Expanding the right - hand side: $165n = 180n-360$.

Step5: Solve for $n$

Subtract $165n$ from both sides: $0=180n - 165n-360$, which simplifies to $15n=360$. Then $n = 24$.

Answer:

24