If each interior angle of a regular convex polygon is three times the measure of each exterior angle, what kind of polygon is it? (1 point) Responses hexagon decagon octagon pentagon

Answer 1

Let's say the measure of each interior angle is x degrees.

According to the problem statement, each exterior angle would then measure x/3 degrees.

We know that the sum of the exterior angles of any polygon is 360 degrees. Since each exterior angle is x/3 degrees, there must be 360/(x/3) = 3(360)/x exterior angles in the polygon.

Since a regular polygon has the same number of interior and exterior angles, the polygon must have 3(360)/x interior angles as well.

Therefore, the number of sides in the polygon is 3(360)/x.

We can eliminate pentagon and hexagon as options because 3(360)/x will not be an integer for those numbers of sides.

For decagon, 3(360)/10 = 108 degrees, which means each interior angle would measure 108 degrees. And, 3(108) = 324, which is three times the measure of the exterior angle (108/3).

Therefore, the polygon must be a decagon.

The correct answer is: decagon.