Prove that TU is parallel to SR, given that T and U are the midpoints of PS and QR respectively in parallelogram PQRS.

Since PQRS is a parallelogram, its opposite sides are parallel and equal in length.
Therefore, we have:

• $PS \parallel QR$
• $PQ \parallel SR$
• $PS = QR$
• $PQ = SR$

T is the midpoint of PS, so $PT = TS = \frac{1}{2} PS$.
U is the midpoint of QR, so $QU = UR = \frac{1}{2} QR$.

From Step 1, we know $PS = QR$.
From Step 2, $PT = \frac{1}{2} PS$ and $QU = \frac{1}{2} QR$.
Substituting $PS = QR$ into the midpoint equations, we get $PT = \frac{1}{2} QR$ and $QU = \frac{1}{2} QR$.
Thus, $PT = QU$.
Also, since $PS \parallel QR$ (from Step 1) and T lies on PS while U lies on QR, it follows that $PT \parallel QU$.

Consider quadrilateral PTUQ. We have shown that $PT \parallel QU$ and $PT = QU$.
A quadrilateral with one pair of opposite sides that are both parallel and equal in length is a parallelogram.
Therefore, PTUQ is a parallelogram.

Since PTUQ is a parallelogram, its opposite sides are parallel.
Thus, $TU \parallel PQ$.
From Step 1, we know that $PQ \parallel SR$.
Since $TU \parallel PQ$ and $PQ \parallel SR$, by the transitive property of parallel lines, we can conclude that $TU \parallel SR$.