Question

how many sides does a regular polygon have if each interior angle measures 135°?

Explanation:

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

The formula for the measure of each interior angle \(\theta\) of a regular polygon with \(n\) sides is \(\theta=\frac{(n - 2)\times180^{\circ}}{n}\). We know that \(\theta = 135^{\circ}\), so we set up the equation \(\frac{(n - 2)\times180}{n}=135\).

Step2: Solve the equation for \(n\)

Multiply both sides of the equation by \(n\): \((n - 2)\times180=135n\)
Expand the left - hand side: \(180n-360 = 135n\)
Subtract \(135n\) from both sides: \(180n-135n-360=0\)
Simplify: \(45n-360 = 0\)
Add 360 to both sides: \(45n=360\)
Divide both sides by 45: \(n=\frac{360}{45}=8\)

Answer:

8