In a regular decagon (a polygon with 10 equal sides), what is the measure of each interior angle in degrees?

Introduction:
This geometry problem checks your knowledge of interior angles of regular polygons. A regular decagon has 10 equal sides and 10 equal interior angles. You must apply the standard formula for interior angles of a polygon and then find the measure of each angle.

Given Data / Assumptions:

• The polygon is a regular decagon.
• Number of sides n = 10.
• All interior angles are equal in a regular polygon.
• We work in degrees.

Concept / Approach:
The sum of interior angles of an n sided polygon is given by: Sum of interior angles = (n − 2) * 180 degrees. For a regular polygon, each interior angle is this sum divided by n. We substitute n = 10 for a decagon and simplify to get the measure of one interior angle.

Step-by-Step Solution:
Step 1: Number of sides n = 10. Step 2: Use the formula for sum of interior angles: Sum = (n − 2) * 180 degrees. Step 3: Substitute n = 10: Sum = (10 − 2) * 180 = 8 * 180 = 1440 degrees. Step 4: In a regular decagon, each interior angle is equal, so: Each interior angle = Sum / n = 1440 / 10. Step 5: Compute 1440 / 10 = 144 degrees.

Verification / Alternative check:
As a rough check, note that for large n, interior angles approach 180 degrees. For n = 10, we expect a value somewhat less than 180 degrees but greater than 120 degrees (which would correspond to a hexagon). The result of 144 degrees fits neatly between these limits and matches typical values known for regular polygons.

Why Other Options Are Wrong:

• 120° and 140°: These are too small for a regular decagon, which has more sides than a hexagon or octagon and therefore larger interior angles.
• 150° and 160°: These are too large. They would correspond to polygons with more than 10 sides.

Common Pitfalls:
Students sometimes confuse the formula for exterior angles with that for interior angles, or forget to divide the total sum by the number of sides. Another frequent mistake is using (n * 180) instead of (n − 2) * 180. Always remember that a triangle (n = 3) has 180 degrees in total, which leads to the general formula (n − 2) * 180.

Final Answer:
Each interior angle of a regular decagon measures 144°.