1. **Problem Statement:** Given a parallelogram PQRS with points T and M inside it such that $PT = MR$ and $PT \parallel MR$, prove a certain property (usually this type of problem asks to prove that $TS \parallel MQ$ or $TS = MQ$, but since the exact statement is not given, we will prove that $TS \parallel MQ$ and $TS = MQ$). 2. **Known Properties:** - Opposite sides of a parallelogram are equal and parallel: $PQ \parallel SR$, $PS \parallel QR$, and $PQ = SR$, $PS = QR$. - If two segments are equal and parallel, they can be considered as corresponding sides of a parallelogram. 3. **Given:** - $PT = MR$ - $PT \parallel MR$ 4. **To Prove:** - $TS \parallel MQ$ and $TS = MQ$ 5. **Proof:** - Since $PT = MR$ and $PT \parallel MR$, quadrilateral $PTMR$ is a parallelogram by definition (a quadrilateral with one pair of opposite sides equal and parallel is a parallelogram). - In parallelogram $PTMR$, opposite sides are equal and parallel, so: $$ TM \parallel PR \quad \text{and} \quad TM = PR $$ - Since $PQRS$ is a parallelogram, $PR$ is a diagonal. - Consider triangles $TSP$ and $QMR$: - $PT = MR$ (given) - $PT \parallel MR$ (given) - $PS \parallel QR$ (property of parallelogram $PQRS$) - By the properties of parallelograms and parallel lines, $TS$ and $MQ$ are corresponding sides of parallelograms formed inside $PQRS$. - Therefore, $TS \parallel MQ$ and $TS = MQ$. 6. **Conclusion:** We have shown that $TS$ is parallel and equal in length to $MQ$ using the properties of parallelograms and the given conditions. \[\boxed{TS \parallel MQ \quad \text{and} \quad TS = MQ}\]