[tex]\huge\boxed{55^\circ}[/tex]
First of all, we know that [tex]\angle Q=95^{\circ}[/tex] because of its vertical angle. [tex]\angle Q[/tex] is also equal to [tex]\dfrac{\widehat{RS}+\widehat{UT}}{2}[/tex] because it's an interior angle.
[tex]\begin{aligned}\angle Q&=\frac{\widehat{RS}+\widehat{UT}}{2}\\95^\circ&=\frac{135^\circ+\widehat{UT}}{2}&&\qquad&&\textsf{Substitute in the known values.}\\190^\circ&=135^\circ+\widehat{UT}&&\qquad&&\textsf{Multiply both sides by $2$.}\\55^\circ&=\widehat{UT}&&\qquad&&\textsf{Subtract $135^\circ$ from both sides.}\\\widehat{UT}&=\boxed{55^\circ}&&\qquad&&\textsf{Switch the sides of the equation to show our answer.}\end{aligned}[/tex]
