let x + y + z + v = 360 (sum of all angles in a quad)

1) Insufficient. Two right angles will give you 360-180 = 180. if x and y were right angles, z + v = 180. one of the angles does not have to be 60 degrees.

2) Insufficient. x = 2y. that still leaves us with 2y + y + z + v = 360

Together, there are still 2 possibilities:

First possibility:

x + y = 180

z = 2v = 180 + 3v = 360

v = 180/3 = 60

Second possibility:

x = 90; y = 90; x = 2z (z =45) v = 135

one of the angles is not 60 degrees here.

Insufficient

Therefore IMO (E)

Yes, you may have thought "since you can't have an angles 2x90 = 180, then the only option is 60 and 120 for the remaining 180".

But either of right angles can serve as the doubled piece, leaving 90 90 45 and 135