Need help with this geometry question

UT = TW + WU = 27 + 27 = 54

WT = 27

ST = UT = 54

arc(XT) =  angle XZT = 52 degrees

arc(ST) = angle SZT = 104

arc(US) = arc(ST) + arc(UT) = 104 + 104 = 208

I know it was super drawn out and there were probably quicker ways to do it, but I was justusing methods that came to mind rather than going for efficiency.  let me know if any part is confusing.  

Step-by-step explanation:

Let's start by just building on what we know

Since we know sv and vz can be made into the side of two triangles, SZV and TZV, we can use some similar triangle rules.

In SZV and TZV we know SZ = TZ since they are both radii, ZV = ZV because they are the same line and angle ZVS = angle ZVT since they are supplementary and one is 90 degrees.  This allows us to solve and see if they are congruent or not.  To do so we use the normal trig functions

sin(ZTV) = ZV/ZT = ZV/12

sin(ZSV) = ZV/SV = ZV/12

So the angles could be equal, but just to be sure lets look at all possible answers.  with sine an angle x gives the same result as 180-x.  Since both angles are acute thugh we know this can't be true so both angles are equal.    Now we can say both triangles are congruent by AAS.  Since they are congruent we can then confirm SV = VT.

In general you can say if a radii intersects a chord at a right angle then the intersection is a bisection.  So now we know VT = 27

We can use similar methods to show UW = WT and if you made two triangles ZWT and ZWU you will see they are congruent.  and since arc UT is 104 degrees that means angle UZT is 104 degrees.  Since ZW bisects UT and the two triangles are congruent ZW also bisects angle UZT so UZW = TZW = 52 degrees.

Now we have 4 triangles, SZV, TZV, ZWT and ZWU and we know SZV and TZV are congruent and ZWT and ZWU are congruent.  Now, if we can make one from each pair congruent we would knwo all four are congruent.  Since TZV and ZWT share a side  that's a good place to start.  Looking at these two  triangles we know those two sides are equal, and they both have a right angle.  Also we know VZ = ZW fromt he instructions, so this is again a chance to check with trig.

Let's look at angle ZTW and ZTV.

sin(ZTW) = ZW/ZT = 27/ZT

sin(ZTV) = VZ/ZT = 27/ZT

Using the same reasoning as before we can say that the two angles are equal, so the two triangles are congruent.  Now we can say a lot.

angle UZT = angle SZT = 104 degrees

angle SZV = angle VZT = angle TZW = angle WZU = 52

SV = VT = TW = WU = 27

Let's start solving.

UT = TW + WU = 27 + 27 = 54

WT = 27

ST = UT = 54

arc(XT) =  angle XZT = 52 degrees

arc(ST) = angle SZT = 104

arc(US) = arc(ST) + arc(UT) = 104 + 104 = 208

the answer is 40.

proof: