Question

question
how many sides does a regular polygon have if each interior angle measures 140°?

Explanation:

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

The formula for each interior angle \(\theta\) of a regular polygon with \(n\) sides is \(\theta=\frac{(n - 2)\times180^{\circ}}{n}\), where \(n\) is the number of sides. We know that \(\theta = 140^{\circ}\), so we set up the equation \(\frac{(n - 2)\times180}{n}=140\).

Step2: Solve the equation for \(n\)

First, multiply both sides of the equation by \(n\) to get rid of the denominator: \((n - 2)\times180=140n\).
Expand the left - hand side: \(180n-360 = 140n\).
Subtract \(140n\) from both sides: \(180n-140n-360=140n - 140n\), which simplifies to \(40n-360 = 0\).
Add 360 to both sides: \(40n-360 + 360=0 + 360\), so \(40n=360\).
Divide both sides by 40: \(n=\frac{360}{40}=9\).

Answer:

The regular polygon has 9 sides.