To find the sum of the measures of the interior angles of a regular polygon where each exterior angle measures [tex]\(72^\circ\)[/tex], we can follow these steps:
1. Determine the number of sides ([tex]\(n\)[/tex]) of the polygon:
Since each exterior angle of a regular polygon is equal, we can use the fact that the sum of all exterior angles is always [tex]\(360^\circ\)[/tex]. Therefore, we can find the number of sides by dividing [tex]\(360^\circ\)[/tex] by the measure of one exterior angle:
[tex]\[
n = \frac{360^\circ}{72^\circ} = 5
\][/tex]
So, the polygon has 5 sides, meaning it is a pentagon.
2. Calculate the sum of the interior angles of the polygon:
The sum of the interior angles of a polygon with [tex]\(n\)[/tex] sides is given by the formula:
[tex]\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\][/tex]
For our 5-sided polygon, we substitute [tex]\(n = 5\)[/tex]:
[tex]\[
\text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\][/tex]
Therefore, the sum of the measures of the interior angles of the regular polygon is [tex]\(540^\circ\)[/tex].
The correct answer is:
[tex]\[
\boxed{540^\circ}
\][/tex]
