Trigonometry - finding a length

There are 6 points (U,V,W,X,Y,Z) in a 2D space.

* Point Y lies on line WX

* Point Z lies on line UX

* VW and YZ are perpendicular to WX

* UY and VY are equal length

VW = 400, XY = 200, YZ = 100 and angle UYV is 20 degrees. Find the length of WY?

http://xs221.xs.to/xs221/07452/mode7diag.jpg

I haven't had much luck tackling this question. Tried right-angle trig, Pythagoras and angle theorems but I cannot get angle UYV into a formula with any of the provided lengths. Could someone point me to the right direction?

Or at least, can someone prove whether UX and WX are equal or not? (that's the only part I'm missing)