What is the measure of each interior angle of a regular decagon (a regular polygon with 10 equal sides)?

Introduction / Context:
This question checks knowledge of the relationship between the number of sides of a regular polygon and its interior angles. A regular decagon is a polygon with 10 equal sides and 10 equal interior angles. Using the standard formula for the interior angle of a regular n sided polygon, we can find the required measure in degrees.

Given Data / Assumptions:

• The polygon is a regular decagon, so n = 10 sides.
• All interior angles of a regular polygon are equal.
• Sum of interior angles of an n sided polygon = (n - 2) * 180 degrees.
• Each interior angle of a regular polygon = sum of interior angles / n.

Concept / Approach:
First we compute the total sum of interior angles using the formula (n - 2) * 180. Then we divide this sum by n to get each interior angle, since in a regular polygon all interior angles are equal. Substituting n = 10 for a decagon leads to the required value. The key is to apply the formula carefully and simplify the fraction correctly.

Step-by-Step Solution:
Number of sides n = 10.Sum of interior angles S = (n - 2) * 180 = (10 - 2) * 180 = 8 * 180.Compute 8 * 180 = 1440 degrees.Each interior angle of a regular decagon = S / n = 1440 / 10.1440 / 10 = 144 degrees.Therefore each interior angle is 144°.

Verification / Alternative check:
Another way is to recall the formula for each interior angle directly: interior angle = ((n - 2) * 180) / n. Substituting n = 10 gives interior angle = (8 * 180) / 10 = 1440 / 10 = 144 degrees. Both the stepwise method and the direct formula produce the same result, confirming that 144° is correct.

Why Other Options Are Wrong:
120° is the interior angle of a regular hexagon (n = 6), not a decagon. 140° and 150° do not correspond to any standard regular polygon with 10 sides. 160° would give a sum of 160 * 10 = 1600 degrees, which would imply (n - 2) * 180 = 1600 and does not yield an integer n. Only 144° matches the decagon formula exactly.

Common Pitfalls:
Students often confuse interior and exterior angles. For a regular polygon, each exterior angle is 360 / n, and sometimes learners mistakenly compute this instead. Others forget to divide the total sum by the number of sides. It is helpful to remember that interior angle plus exterior angle at each vertex equals 180 degrees, which for n = 10 gives 144 + 36 = 180, a useful consistency check.

Final Answer:
144°