1. **Problem statement:** Given that line segment $ \overleftrightarrow{ST} $ is tangent to circle $ \odot Q $ at point $ S $, and $ m\angle U = 65^\circ $, find $ m\angle T $. 2. **Key fact:** The tangent to a circle is perpendicular to the radius at the point of tangency. Therefore, $ \angle QST = 90^\circ $. 3. **Triangle setup:** Triangle $ TSU $ has angles $ \angle T $, $ \angle U = 65^\circ $, and $ \angle S $. 4. **Find $ \angle S $:** Since $ \angle QST = 90^\circ $ and $ S $ lies on the circle, $ \angle S = 90^\circ $. 5. **Sum of angles in triangle:** The sum of interior angles in triangle $ TSU $ is $ 180^\circ $. So, $$ \angle T + \angle U + \angle S = 180^\circ $$ 6. **Substitute known values:** $$ \angle T + 65^\circ + 90^\circ = 180^\circ $$ 7. **Simplify:** $$ \angle T + 155^\circ = 180^\circ $$ 8. **Solve for $ \angle T $:** $$ \angle T = 180^\circ - 155^\circ = 25^\circ $$ **Final answer:** $ m\angle T = 25^\circ $.