1. **State the problem:** We need to find the missing angle $\angle U$ in triangle $UVT$ where $\angle V = 50^\circ$ and the exterior angle at $T$ (formed by extending $TP$) is $115^\circ$. 2. **Recall the exterior angle theorem:** The exterior angle at a vertex of a triangle equals the sum of the two opposite interior angles. Here, the exterior angle at $T$ is $115^\circ$, so $$\angle U + \angle V = 115^\circ$$ 3. **Substitute the known value:** We know $\angle V = 50^\circ$, so $$\angle U + 50^\circ = 115^\circ$$ 4. **Solve for $\angle U$:** $$\angle U = 115^\circ - 50^\circ = 65^\circ$$ 5. **Check with triangle angle sum:** The sum of interior angles in triangle $UVT$ is $180^\circ$: $$\angle U + \angle V + \angle T = 180^\circ$$ We know $\angle T$ is the interior angle at $T$, which is supplementary to the exterior angle $115^\circ$, so $$\angle T = 180^\circ - 115^\circ = 65^\circ$$ Check sum: $$65^\circ + 50^\circ + 65^\circ = 180^\circ$$ This confirms our calculation. **Final answer:** The missing angle $\angle U$ is $65^\circ$.