Let C be the center of the circle. The measure of arc VSU is [tex]2+114x[/tex], so the measure of the minor arc VU is [tex]360-(2+114x)=358-114x[/tex]. The central angle VCU also has measure [tex]358-114x[/tex].
Triangle CUV is isosceles, so the angles CVU and CUV are congruent. The interior angles of any triangle are supplementary (they add to 180 degrees) so
[tex]m\angle VCU+2m\angle CUV=180[/tex]
[tex]\implies m\angle CUV=\dfrac{180-(358-114x)}2=57x-89[/tex]
UT is tangent to the circle, so CU is perpendicular to UT. Angles CUV and VUT are complementary, so
[tex](57x-89)+(31x+3)=90[/tex]
[tex]\implies88x=176[/tex]
[tex]\implies x=2[/tex]
So finally,
[tex]m\widehat{VSU}=2+114\cdot2=230[/tex]
degrees.
