1. **Problem statement:** Given a circle with center $S$, a tangent line $TU$ at point $T$ on the circle, and the angle $\angle STU = 64^\circ$, find the measure of $\angle U$. 2. **Key fact:** The tangent to a circle is perpendicular to the radius drawn to the point of tangency. This means $\angle STU = 90^\circ$. 3. **Given:** $\angle TSU = 64^\circ$ (angle at center $S$ between $ST$ and $SU$). We want to find $m\angle U = \angle UT S$. 4. **Triangle $STU$:** It has angles $\angle STU = 90^\circ$ (tangent-radius), $\angle TSU = 64^\circ$, and $\angle UT S = m\angle U$ (unknown). 5. **Sum of angles in triangle:** $$\angle STU + \angle TSU + \angle UT S = 180^\circ$$ 6. Substitute known values: $$90^\circ + 64^\circ + m\angle U = 180^\circ$$ 7. Solve for $m\angle U$: $$m\angle U = 180^\circ - 90^\circ - 64^\circ = 26^\circ$$ **Final answer:** $$m\angle U = 26^\circ$$