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. Understanding Exterior Angles of a Polygon:
The exterior angles of any polygon always add up to [tex]\( 360^\circ \)[/tex].
2. Find the Number of Sides:
The measure of each exterior angle is given as [tex]\( 120^\circ \)[/tex]. The number of sides of the polygon can be found by dividing [tex]\( 360^\circ \)[/tex] by the measure of each exterior angle:
[tex]\[
\text{Number of sides} = \frac{360^\circ}{120^\circ} = 3
\][/tex]
So, the polygon has 3 sides and is a triangle.
3. Calculate the Sum of the Interior Angles:
The formula to find the sum of the interior angles of a polygon with [tex]\( n \)[/tex] sides is:
[tex]\[
\text{Sum of the interior angles} = (n - 2) \times 180^\circ
\][/tex]
For a triangle ([tex]\( n = 3 \)[/tex]):
[tex]\[
\text{Sum of the interior angles} = (3 - 2) \times 180^\circ = 1 \times 180^\circ = 180^\circ
\][/tex]
Therefore, the sum of the measures of the interior angles of this polygon is:
[tex]\[ 180^\circ \][/tex]
So, the correct answer is [tex]\( \boxed{180^\circ} \)[/tex].
