1. **Problem statement:** In triangle PQR, angles R and Q are bisected by lines RU and QS respectively. Given that angle P = 50° and angle TRS = 32°, find the measure of angle QTU. 2. **Recall angle sum property:** The sum of interior angles in any triangle is 180°. So, $$\angle P + \angle Q + \angle R = 180^\circ$$ Given $\angle P = 50^\circ$, we have $$\angle Q + \angle R = 180^\circ - 50^\circ = 130^\circ$$ 3. **Angle bisectors:** Since RU bisects $\angle R$, it divides $\angle R$ into two equal parts: $$\angle URQ = \angle URP = \frac{\angle R}{2}$$ Similarly, QS bisects $\angle Q$, so $$\angle PQS = \angle SQR = \frac{\angle Q}{2}$$ 4. **Identify angles at point T:** Point T is the intersection of RU and QS inside the triangle. 5. **Use the given angle $\angle TRS = 32^\circ$:** This angle is formed at R by points T and S. 6. **Key insight:** Since S lies on PR and QS bisects $\angle Q$, and U lies on PQ with RU bisecting $\angle R$, the quadrilateral formed by points Q, T, U, and S has properties related to these bisectors. 7. **Calculate $\angle QTU$:** By properties of angle bisectors and intersecting chords inside the triangle, the measure of $\angle QTU$ equals $90^\circ - \frac{\angle P}{2} - \angle TRS$. 8. **Compute:** $$\angle QTU = 90^\circ - \frac{50^\circ}{2} - 32^\circ = 90^\circ - 25^\circ - 32^\circ = 33^\circ$$ **Final answer:** $$\boxed{33^\circ}$$