Final answer:
The measure of angle U in triangle WFI, congruent to triangle EAU, is 66°.
Step-by-step explanation:
To find the measure of angle U in triangle WFI which is congruent to triangle EAU, we must recognize that the sum of the angles in a triangle always equals 180°. Given that the measure of angle F is 77° and the measure of angle W is 37°, we can add these two angles and subtract their sum from 180° to find the measure of the third angle.
First, let's add the given angles: m∠F + m∠W = 77° + 37° = 114°. Now we subtract this sum from the total sum of angles in a triangle: 180° - 114° = 66°. Therefore, the measure of angle U is 66°.