To determine the sum of the measures of the interior angles of a regular polygon where each exterior angle measures [tex]\(120^\circ\)[/tex], follow these steps:
1. Determine the number of sides of the polygon:
- Recall that the sum of the exterior angles of any polygon is always [tex]\(360^\circ\)[/tex].
- Since each exterior angle measures [tex]\(120^\circ\)[/tex], we can find the number of sides [tex]\(n\)[/tex] by dividing the sum of the exterior angles by the measure of one exterior angle.
[tex]\[
n = \frac{360^\circ}{120^\circ} = 3
\][/tex]
2. Calculate the sum of the interior angles:
- The formula for the sum of the interior angles of a polygon is [tex]\((n - 2) \times 180^\circ\)[/tex], where [tex]\(n\)[/tex] is the number of sides.
- Substitute [tex]\(n = 3\)[/tex] into the formula:
[tex]\[
\text{Sum of interior angles} = (3 - 2) \times 180^\circ = 1 \times 180^\circ = 180^\circ
\][/tex]
Therefore, the sum of the measures of the interior angles of the polygon is [tex]\(\boxed{180^\circ}\)[/tex].
