Question

how many sides does a regular polygon have if each interior angle measures $140^\circ$?

Explanation:

Step1: Relate interior to exterior angle

The sum of an interior angle and its corresponding exterior angle of a polygon is $180^\circ$.
$\text{Exterior angle} = 180^\circ - 140^\circ = 40^\circ$

Step2: Use total exterior angle sum

The sum of all exterior angles of any regular polygon is $360^\circ$. Let $n$ be the number of sides.
$n = \frac{360^\circ}{\text{Measure of one exterior angle}}$

Step3: Calculate number of sides

Substitute the exterior angle value into the formula.
$n = \frac{360}{40} = 9$

Answer:

9