If a regular polygon has 6 sides, by how many degrees is the measure of each interior angle greater than the measure of each exterior angle?

Introduction / Context:
This question is about angle properties in a regular polygon, specifically a regular hexagon. The focus is on the difference between an interior angle and an exterior angle at the same vertex. It tests recall of formulas for interior and exterior angles and understanding of how these angles are related in regular polygons.

Given Data / Assumptions:

• The polygon is regular, meaning all sides and all angles are equal.
• The number of sides n = 6 (a regular hexagon).
• We are interested in the measure of one interior angle and the corresponding exterior angle.
• The difference required is interior angle minus exterior angle.

Concept / Approach:
For a regular polygon, two key formulas are used:

• Interior angle = [(n - 2) * 180] / n degrees.
• Exterior angle = 360 / n degrees.

Additionally, at each vertex, the interior angle and the exterior angle are supplementary, so their sum is 180 degrees. Once we compute both angles for n = 6, we subtract the exterior angle from the interior angle to get the required difference.

Step-by-Step Solution:
Step 1: For n = 6, compute the interior angle using the formula: Interior angle = [(6 - 2) * 180] / 6.Step 2: Simplify: Interior angle = (4 * 180) / 6 = 720 / 6 = 120 degrees.Step 3: Compute the exterior angle: Exterior angle = 360 / 6 = 60 degrees.Step 4: Find the difference: Interior angle minus exterior angle = 120 - 60 = 60 degrees.

Verification / Alternative check:
Since the interior and exterior angles form a linear pair at each vertex, their sum is 180 degrees. If the exterior angle is 60 degrees, then the interior angle is 180 - 60 = 120 degrees.The difference is then 120 - 60 = 60 degrees, which matches the previous calculation.Thus the result is consistent with both the general formula and the supplementary property of the angles.

Why Other Options Are Wrong:
A difference of 90 degrees would correspond to interior and exterior angles like 135 degrees and 45 degrees, which do not occur in a regular hexagon.Values such as 100 degrees or 108 degrees are associated with other polygons and not with a 6 sided regular polygon.A difference of 120 degrees would imply an exterior angle of 30 degrees and interior of 150 degrees, which would correspond to a 12 sided polygon.Only 60 degrees is correct for a regular hexagon.

Common Pitfalls:
A typical mistake is to confuse the formula for the sum of interior angles with the formula for an individual interior angle.Another error is to forget that the exterior angle for a regular polygon is always 360 divided by the number of sides.Some learners may also add the angles instead of subtracting to find the difference requested in the problem.

Final Answer:
The interior angle is greater than the exterior angle by 60°.