In a regular polygon, the measure of each interior angle is 90 degrees greater than the measure of each exterior angle. How many sides does this regular polygon have?

Introduction / Context:
This question checks your understanding of the relationship between interior and exterior angles of a regular polygon. Regular polygons frequently appear in school geometry and competitive exams because their angles follow simple, elegant formulas. Every regular polygon has equal sides and equal angles, and there is a fixed link between each interior angle and its corresponding exterior angle. Knowing how to use this relationship allows you to move quickly from one type of angle to the other and then deduce the number of sides of the polygon, which is what is required in this problem.

Given Data / Assumptions:
• The polygon is regular, which means all sides and all interior angles are equal.
• Let the measure of each interior angle be I degrees.
• Let the measure of each exterior angle be E degrees.
• It is given that I is 90 degrees greater than E, so I = E + 90.
• For any polygon, each interior angle and its corresponding exterior angle form a linear pair, so I + E = 180 degrees.

Concept / Approach:
The key concepts are: (1) an interior angle and its adjacent exterior angle are supplementary and sum to 180 degrees, and (2) in a regular polygon, all exterior angles are equal and their sum is always 360 degrees. Once we find the measure of one exterior angle, we can use the formula number of sides n = 360 / E. The given condition that the interior angle is 90 degrees greater than the exterior angle provides a simple equation in terms of E, which we can solve to determine the angle values, and then the number of sides.

Step-by-Step Solution:
Step 1: Use the relationship between interior and exterior angles: I + E = 180. Step 2: Use the condition that I = E + 90. Step 3: Substitute I = E + 90 into I + E = 180 to get (E + 90) + E = 180. Step 4: Simplify the equation: 2E + 90 = 180, so 2E = 90 and E = 45 degrees. Step 5: The measure of each exterior angle of the regular polygon is therefore 45 degrees. Step 6: Use the formula for a regular polygon: number of sides n = 360 / E, so n = 360 / 45 = 8.

Verification / Alternative check:
We can verify the result by checking the interior angle. If E = 45 degrees, then I = 180 - 45 = 135 degrees from the linear pair relationship. The condition in the question says that the interior angle is 90 degrees more than the exterior angle. Indeed, 135 - 45 = 90, so the condition is satisfied. Also, the sum of all exterior angles is 360 degrees, and 8 * 45 = 360, which confirms that a regular polygon with 8 sides (an octagon) fits perfectly with the standard polygon angle properties.

Why Other Options Are Wrong:
If the polygon had 9 sides, the exterior angle would be 360 / 9 = 40 degrees, which would make the interior angle 140 degrees, and the difference would be 100 degrees, not 90. For 10 sides, the exterior angle would be 36 degrees and the interior angle 144 degrees, giving a difference of 108 degrees. For 12 sides, the exterior angle is 30 degrees and the interior angle 150 degrees, giving a difference of 120 degrees. None of these match the required 90 degree difference, so only 8 sides is correct.

Common Pitfalls:
Many learners forget that interior and exterior angles are supplementary in a simple convex polygon, and they may mistakenly try to use incorrect formulas. Others confuse the sum of interior angles with the sum of exterior angles, but remember that the sum of all exterior angles of any convex polygon is always 360 degrees. Another common error is to treat the 90 degree difference as I = 90 or E = 90 directly, which is not what the problem states. Carefully translating the statement into equations I = E + 90 and I + E = 180 avoids these mistakes.

Final Answer:
The regular polygon has 8 sides.