Three angles of a hexagon are congruent. The other three angles are also congruent, each with a measure twice that of the first three. What is the measure of each angle?

The sum of the interior angles of a hexagon is 720 degrees. If 3 angles are congruent, each with measure x degrees, and the other 3 angles are also congruent to each other, each with measure 2x degrees, then the total sum of all 6 angles would be 3x + 3(2x) = 9x. If this is equal to 720, then x = 80 degrees, while 2x = 160 degrees.
Therefore, there are 3 80-degree angles and 3 160-degree angles.