Question

he measure of an interior angle of a polygon with n sides is given. Find n.

160° n =

156° n =

Answer 1

To find the number of sides \( n \) of a polygon given the measure of an interior angle, we can use the formula for the measure of an interior angle of a regular polygon:

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

We can rearrange this formula to solve for \( n \):

\[
\text{Interior Angle} \times n = (n - 2) \times 180°
\]

\[
\text{Interior Angle} \times n = 180n - 360
\]

\[
180n - \text{Interior Angle} \times n = 360
\]

\[
n(180 - \text{Interior Angle}) = 360
\]

\[
n = \frac{360}{180 - \text{Interior Angle}}
\]

Now we'll apply this formula for both angles:

1. **For an interior angle of 160°:**

\[
n = \frac{360}{180 - 160} = \frac{360}{20} = 18
\]

2. **For an interior angle of 156°:**

\[
n = \frac{360}{180 - 156} = \frac{360}{24} = 15
\]

Therefore:
- For an interior angle of 160°, \( n = 18 \).
- For an interior angle of 156°, \( n = 15 \).