To determine the sum of the measures of the interior angles of a regular polygon where each exterior angle measures [tex]\(90^\circ\)[/tex], we can follow these steps:
1. Understand the relationship between exterior and interior angles:
- Each exterior angle of a regular polygon can be found by dividing the total sum of exterior angles, which is always [tex]\(360^\circ\)[/tex], by the number of sides (denoted as [tex]\(n\)[/tex]) the polygon has.
- Given that each exterior angle measures [tex]\(90^\circ\)[/tex], we can figure out the number of sides using the formula:
[tex]\[
\text{Exterior Angle} = \frac{360^\circ}{n}
\][/tex]
[tex]\[
90^\circ = \frac{360^\circ}{n}
\][/tex]
- Solving for [tex]\(n\)[/tex]:
[tex]\[
n = \frac{360^\circ}{90^\circ} = 4
\][/tex]
2. Determine the sum of the interior angles of the polygon:
- The sum of the interior angles of a polygon is given by the formula:
[tex]\[
\text{Sum of Interior Angles} = (n-2) \times 180^\circ
\][/tex]
- Substituting [tex]\(n = 4\)[/tex]:
[tex]\[
\text{Sum of Interior Angles} = (4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ
\][/tex]
Therefore, the sum of the measures of the interior angles of the regular polygon is [tex]\(\boxed{360^\circ}\)[/tex].
