Let's solve the given problems:

Problem 14:

Given:

• $QRST$ is a parallelogram.
• $TR \cong QS$.
• $\angle QPR = 90^\circ$.

Prove: $QRST$ is a square.

Proof:

1. Properties of a parallelogram:

   • In a parallelogram, opposite sides are congruent, so $QR \cong TS$ and $QT \cong RS$.

2. Given congruence:

   • $TR \cong QS$.

3. Right angle:

   • $\angle QPR = 90^\circ$ (Given). Since parallelograms have opposite angles equal, all angles of $QRST$ are $90^\circ$.

4. Conclusion:

   • Since $QRST$ is a parallelogram with all angles $90^\circ$ and all sides congruent ($QR \cong QT \cong TR \cong TS$), $QRST$ is a square.

Problem 16:

Given:

• $ACDH$ and $BCDF$ are parallelograms.
• $BF \cong AB$.

Prove: $ABHF$ is a rhombus.

Proof:

1. Properties of parallelograms:

   • In parallelogram $ACDH$, $AB \parallel DH$ and $AB \cong DH$.
   • In parallelogram $BCDF$, $BF \parallel CD$ and $BF \cong CD$.

2. Given congruence:

   • $BF \cong AB$.

3. Opposite sides of $ABHF$:

   • Since $AB \cong BF$, all four sides of $ABHF$ are congruent.

4. Conclusion:

   • $ABHF$ is a quadrilateral with all sides congruent, so it is a rhombus.

Do you want a detailed explanation for either problem? Here are some related questions to explore:

1. What additional properties differentiate a rhombus from a square?
2. How do the properties of parallelograms help in proofs?
3. Can a quadrilateral with equal sides not be a rhombus?
4. What other criteria can be used to prove a quadrilateral is a square?
5. How does the concept of congruent triangles assist in such proofs?

Tip: In geometry proofs, always reference known properties (e.g., congruence, parallelism) to build logical arguments.