I am not 100% certain of this solution, but I think it is probably correct.
You need to make a drawing. Call the left end of the longer base of the trapezoid A and the right end B.
Call the left end of the smaller base C and the right end D.
Extend the non-parallel sides of the trapezoid until they meet at a point E.
Draw the horizontal through A.
Extend line through points EDB until it meets the horizontal at F.
∠AEF = 130° (180° - 30° - 20°)
ΔABE is similar to ΔCDE.
Let y be the length of segment DE.
x/55 = (y+11)/57
Use the value for y and the law of sines to compute ∠ECD = ∠EAB (Remember ∠CDE = 130°)
The angle you need for this problem is ∠BAF and it = 30° - ∠EAB
There may be another way to solve this but I don't see it.
If you get another solution, especially one easier and/or better, I would certainly like to know about it.