What is the measure, in degrees, of each exterior angle of a regular polygon that has 9 equal sides (a regular nonagon)?

Introduction: This geometry question tests your understanding of exterior angles of a regular polygon. For a regular polygon, all exterior angles are equal, and there is a simple relationship between the number of sides and each exterior angle.

Given Data / Assumptions:

• The polygon is regular with 9 sides (a regular nonagon).
• Each exterior angle is equal in measure.
• We work with degrees.

Concept / Approach: The sum of the measures of the exterior angles of any convex polygon, taking one exterior angle at each vertex, is always 360 degrees. For a regular polygon with n sides, each exterior angle is: Exterior angle = 360 / n degrees. We substitute n = 9 and compute this value.

Step-by-Step Solution: Step 1: Identify the number of sides: n = 9. Step 2: Use the formula for one exterior angle of a regular polygon: Exterior angle = 360 / n. Step 3: Substitute n = 9: Exterior angle = 360 / 9 degrees. Step 4: Compute 360 / 9 = 40 degrees.

Verification / Alternative check: As a check, you can find the interior angle first. Interior angle of a regular n sided polygon is: ( (n − 2) * 180 ) / n. For n = 9: Interior angle = (7 * 180) / 9 = 1260 / 9 = 140 degrees. Since each interior and exterior angle are supplementary for a regular polygon, interior + exterior = 180, so exterior = 180 − 140 = 40 degrees, which matches our earlier result.

Why Other Options Are Wrong:

• 36, 30, 24, and 45 degrees do not equal 360 / 9.
• Choosing any of these would give an incorrect sum of exterior angles, not equal to 360 degrees.

Common Pitfalls: Students sometimes confuse formulas for interior and exterior angles and use (n − 2) * 180 / n incorrectly. Others forget that the sum of all exterior angles is always 360 degrees, regardless of the number of sides. Remember that for a regular polygon, finding each exterior angle is as simple as dividing 360 degrees by the number of sides.

Final Answer: Each exterior angle of the regular 9 sided polygon measures 40 degrees.