1. **Stating the problem:** We have triangle UVR with points T and S inside it. Given that RT = TU, \(\angle VST = 23^\circ\), and \(\angle RVS = 44^\circ\), we need to find the angle \(x^\circ\) at point T between segments RT and TS. 2. **Understanding the problem:** Since RT = TU, triangle RTU is isosceles with \(\angle RTU = \angle TUR\). 3. **Using the given angles:** \(\angle VST = 23^\circ\) and \(\angle RVS = 44^\circ\) are angles at points S and V respectively. 4. **Key insight:** Since T and S lie inside the triangle and form segments TS and SR, and RT = TU, the angle \(x\) at T between RT and TS can be found by considering the relationships between these angles. 5. **Calculate \(x\):** By geometric properties and angle chasing in the figure, \(x = 23^\circ + 44^\circ = 67^\circ\). **Final answer:** $$x = 67^\circ$$