1. **State the problem:** We need to find the measure of angle $\angle T$ in trapezoid $TVYZ$ where $TY \cong VZ$ and $TV \parallel YZ$. Given $\angle V = 60^\circ$. 2. **Recall properties:** In trapezoid $TVYZ$, $TV \parallel YZ$ means $\angle T$ and $\angle V$ are consecutive angles between the parallel sides and the legs. 3. **Use the property of trapezoid angles:** Consecutive angles between parallel sides are supplementary, so $$\angle T + \angle V = 180^\circ$$ 4. **Substitute the known angle:** $$\angle T + 60^\circ = 180^\circ$$ 5. **Solve for $\angle T$:** $$\angle T = 180^\circ - 60^\circ = 120^\circ$$ 6. **Check with congruent sides:** Since $TY \cong VZ$, trapezoid $TVYZ$ is isosceles, confirming the angles at $T$ and $Z$ are equal, consistent with $\angle T = 120^\circ$. **Final answer:** $$\boxed{120^\circ}$$