1. **Problem statement:** We are given a right triangle PQR with right angle at P, and segment PQ is parallel to segment TS. We need to find which ratio is equivalent to $\tan Q$. 2. **Recall the definition of tangent:** For an angle $Q$ in a right triangle, $\tan Q = \frac{\text{opposite side}}{\text{adjacent side}}$. 3. **Identify sides relative to angle Q:** - Opposite side to angle $Q$ is $PR$. - Adjacent side to angle $Q$ is $PQ$. 4. **Using the parallel lines PQ and TS:** Since $PQ \parallel TS$, triangles $PQR$ and $TSR$ are similar by the AA criterion (corresponding angles are equal). 5. **From similarity, corresponding sides are proportional:** $$\frac{PQ}{TS} = \frac{PR}{TR} = \frac{QR}{SR}$$ 6. **Express $\tan Q$ in terms of segments involving T, S, R:** Since $\tan Q = \frac{PR}{PQ}$, from the similarity ratio: $$\frac{PR}{PQ} = \frac{TR}{TS}$$ 7. **Therefore,** $$\tan Q = \frac{TR}{TS}$$ **Final answer:** $\boxed{\frac{TR}{TS}}$