Question

applying the polygon interior angle sum theorem
an interior angle of a regular polygon has a measure of 108°. what type of polygon is it?
the polygon is
a decagon
a hexagon
an octagon
a pentagon

Explanation:

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

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

Step2: Substitute \(\theta = 108^{\circ}\) into the formula

We have \(108^{\circ}=\frac{(n - 2)\times180^{\circ}}{n}\)

Step3: Solve the equation for \(n\)

Multiply both sides by \(n\): \(108n=(n - 2)\times180\)
Expand the right - hand side: \(108n = 180n-360\)
Subtract \(108n\) from both sides: \(0=180n - 108n-360\)
Simplify: \(0 = 72n-360\)
Add \(360\) to both sides: \(72n=360\)
Divide both sides by \(72\): \(n = 5\)

Answer:

a pentagon