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In this problem, we are given a quadrilateral \( MQNP \) with point \( R \) at its center, and we know that \( MP \) is the perpendicular bisector of \( QN \), \( MN = MQ \), and \( PN = PQ \). The task is to identify which pair of triangles is congruent by the Side-Side-Side (SSS) criterion.

### Solution Explanation

1. **Identify the Congruent Segments**:
   - Since \( MP \) is the perpendicular bisector of \( QN \), it implies that \( QN \) is bisected at \( R \), making \( QR = RN \).
   - \( MN = MQ \) and \( PN = PQ \) are given as congruent sides.

2. **Using SSS Criterion**:
   - To use the SSS (Side-Side-Side) criterion for triangle congruence, we need three pairs of corresponding sides that are equal.
   - In this configuration:
     - Triangle \( MNR \) and \( PNR \) share the side \( NR \).
     - \( MN = PN \) (by the given information).
     - \( MR = PR \) because \( MP \) is the perpendicular bisector of \( QN \).

Thus, triangles \( MNR \) and \( PNR \) have three pairs of corresponding sides that are equal, making them congruent by the SSS criterion.

### Answer
The correct answer is:
**A. \( \triangle MNR \) and \( \triangle PNR \)**

Would you like more details, or do you have any questions?

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Here are 5 related questions to deepen understanding:

1. Why does the perpendicular bisector property of \( MP \) help establish congruence between triangles?
2. How does the SSS criterion for triangle congruence differ from other congruence criteria like SAS or ASA?
3. What role does symmetry play in identifying congruent triangles in geometric figures?
4. If we changed one of the equalities, such as \( MN \neq MQ \), how would it affect the congruence of triangles?
5. How could we prove triangle congruence using other criteria if SSS wasn't applicable?

### Tip:
Always start with known properties (like perpendicular bisectors or equal segments) to simplify and guide your geometric proofs.

Consider the quadrilateral shown in the figure where MP is the perpendicular bisector of QN, MN = MQ, and PN = PQ. Complete the statement about a pair of triangles. Select the letter before the correct pair of triangles to make the statement true: [Options: A. ΔMNR and ΔPNR, B. ΔMQR and ΔPNR, C. ΔMNR and ΔMQR, D. ΔMNQ and ΔPNQ] are congruent by SSS.

Solve a geometry problem involving a quadrilateral with a perpendicular bisector. Given sides MN = MQ and PN = PQ, identify which triangles are congruent by the SSS criterion. Suitable for high school students in Grades 9-10.

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