To determine the sum of the measures of the interior angles of a regular polygon where each exterior angle measures [tex]\(90^\circ\)[/tex], let's follow these steps:
### Step 1: Understand the Relationship Between Exterior Angles and Number of Sides
For any polygon, the sum of the exterior angles is always [tex]\(360^\circ\)[/tex]. To find the number of sides ([tex]\(n\)[/tex]) of the polygon, we can use the fact that:
[tex]\[ \text{Number of sides} = \frac{\text{Sum of exterior angles}}{\text{Measure of one exterior angle}} \][/tex]
Given that each exterior angle measures [tex]\(90^\circ\)[/tex], we can calculate:
[tex]\[ n = \frac{360^\circ}{90^\circ} = 4 \][/tex]
So, the polygon is a quadrilateral with 4 sides.
### Step 2: Calculate the Sum of Interior Angles
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]
For our quadrilateral (where [tex]\(n = 4\)[/tex]):
[tex]\[ \text{Sum of interior angles} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ \][/tex]
### Conclusion
The sum of the measures of the interior angles of a regular polygon where each exterior angle measures [tex]\(90^\circ\)[/tex] is [tex]\(360^\circ\)[/tex].
Thus, the correct answer is:
D. [tex]\(360^\circ\)[/tex]
