Angle UTV is given: 68°
Since the triangle is isoceles, the other two angles measure the same.
This way,
[tex]\begin{gathered} 68+x+x=180\rightarrow68+2x=180\rightarrow2x=112 \\ \rightarrow x=112 \end{gathered}[/tex]
Thereby,
[tex]m\angle TUV=56[/tex]
Notice that
[tex]m\angle TUV+m\angle\text{VUW}=90[/tex]
Because UW is tangent to the circle
Therefore,
[tex]\begin{gathered} m\angle TUV+m\angle\text{VUW}=90 \\ \rightarrow m\angle\text{VUW}=90-m\angle TUV \\ \rightarrow m\angle\text{VUW}=90-56 \\ \rightarrow m\angle\text{VUW}=34 \end{gathered}[/tex]
Answer:
[tex]\begin{gathered} m\angle UTV=68 \\ m\angle VUW=34 \end{gathered}[/tex]
