In a regular polygon, each interior angle is 150 degrees more than its corresponding exterior angle. Knowing that at each vertex the interior and exterior angles are supplementary (sum to 180 degrees), how many sides does the regular polygon have?

Introduction / Context:
This question links the geometry of regular polygons with angle relationships. For any convex polygon, the interior and exterior angle at a vertex are supplementary, and for a regular polygon, all such angles are equal. The problem states a specific relationship between an interior angle and its exterior angle and asks for the number of sides of the polygon.

Given Data / Assumptions:
The polygon is regular, so all interior angles and all exterior angles are equal.Interior angle at a vertex is 150 degrees more than the exterior angle at that vertex.Interior angle and exterior angle at a vertex are supplementary, so their sum is 180 degrees.We must determine the number of sides of the polygon.

Concept / Approach:
Let the exterior angle be E degrees and the interior angle be I degrees. Two equations are available: I + E = 180 from supplementary angles, and I = E + 150 from the given condition. Solving these gives the measure of the exterior angle. For a regular n sided polygon, each exterior angle equals 360 / n, so once E is known, we can solve for n.

Step-by-Step Solution:
Step 1: Let the exterior angle be E and interior angle be I.Step 2: From the condition, I = E + 150.Step 3: From supplementary angles, I + E = 180.Step 4: Substitute I from Step 2 into Step 3: (E + 150) + E = 180.Step 5: Simplify: 2E + 150 = 180, so 2E = 30 and E = 15 degrees.Step 6: For a regular polygon with n sides, each exterior angle is 360 / n, so 360 / n = 15.Step 7: Solve for n: n = 360 / 15 = 24.Step 8: Therefore, the polygon has 24 sides.

Verification / Alternative check:
Check the interior angle corresponding to E = 15 degrees. Using I + E = 180, we get I = 180 − 15 = 165 degrees. Compare with the given condition: I should be 150 degrees more than E, and indeed 15 + 150 = 165. This confirms the angle relationship. With n = 24, each exterior angle 360 / 24 equals 15 degrees, which is consistent with our calculations.

Why Other Options Are Wrong:
If n = 12, each exterior angle would be 30 degrees and interior angle 150 degrees, giving a difference of 120 degrees, not 150. For n = 36, each exterior angle would be 10 degrees, and interior angle 170 degrees, giving a difference of 160 degrees. For n = 6, each exterior is 60 degrees and interior is 120 degrees, with a difference of 60 degrees. None of these match the stated 150 degree difference.

Common Pitfalls:
Some learners confuse the exterior angle at one vertex (360 / n) with the entire exterior angle sum of 360 degrees. Others mistakenly think that interior plus exterior equals 360 at a vertex, which is incorrect in this context. Remembering that the interior and its corresponding exterior angle form a straight line helps keep the relationship I + E = 180 clear.

Final Answer:
The regular polygon has 24 sides.