Question

given the measure of an interior angle of a regular polygon, how many sides does the polygon have?
177.5

the regular polygon has \\(\square\\) sides.
(type an integer or a decimal.)

Explanation:

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

The formula for the measure of an interior angle \( I \) of a regular polygon with \( n \) sides is \( I=\frac{(n - 2)\times180^{\circ}}{n} \). We are given \( I = 177.5^{\circ} \), so we set up the equation:
\( 177.5=\frac{(n - 2)\times180}{n} \)

Step2: Solve the equation for \( n \)

Multiply both sides by \( n \):
\( 177.5n=(n - 2)\times180 \)
Expand the right side:
\( 177.5n = 180n-360 \)
Subtract \( 177.5n \) from both sides:
\( 0 = 180n-177.5n - 360 \)
Simplify the left side:
\( 0 = 2.5n-360 \)
Add 360 to both sides:
\( 2.5n=360 \)
Divide both sides by 2.5:
\( n=\frac{360}{2.5}=144 \)

Answer:

144