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
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.