Find the measure of angle U.

In triangle DUF, angle D = 69 degrees. Angle U is part of a rectangle, so it is 90 degrees. Angle F is not directly given, but we can infer from the diagram that triangle DUF is not a right triangle. However, angle U is labeled as 55 degrees within a smaller triangle EFU. In triangle EFU, angle EFU is a right angle (90 degrees). Angle FEU is not given. Angle EUF is given as 55 degrees. Therefore, the measure of angle U is 55 degrees.