`DeltaPQR` and `DeltaTQR` are on the same base QR and on the same side of QR, If PQ=TR and PR=TQ, then which of the following is correct?

Options

A. `DeltaPQRcongDeltaRQT`
B. `DeltaPQRcongDeltaQTR`
C. `DeltaPQRcongDeltaTQR`
D. `DeltaPQRcongDeltaTRQ`

Answer

To solve the problem, we need to analyze the given information about triangles \( \Delta PQR \) and \( \Delta TQR \).

### Step 1: Understand the Given Information
We have two triangles, \( \Delta PQR \) and \( \Delta TQR \), which share a common base \( QR \) and are located on the same side of this base. The information provided states:
- \( PQ = TR \)
- \( PR = TQ \)

### Step 2: Identify the Corresponding Sides
From the information:
- Side \( PQ \) of triangle \( PQR \) is equal to side \( TR \) of triangle \( TQR \).
- Side \( PR \) of triangle \( PQR \) is equal to side \( TQ \) of triangle \( TQR \).
- The base \( QR \) is common to both triangles.

### Step 3: Apply the Side-Side-Side (SSS) Congruence Criterion
Since we have:
- \( PQ = TR \)
- \( PR = TQ \)
- \( QR = QR \) (common side)

We can conclude that both triangles \( \Delta PQR \) and \( \Delta TQR \) are congruent by the Side-Side-Side (SSS) congruence criterion.

### Step 4: Conclusion
Since the two triangles are congruent, we can state that:
- \( \Delta PQR \cong \Delta TQR \)

Now, we can check the options provided in the question to identify which condition is correct based on the congruence of the triangles.

### Final Answer
The correct option is **Option 4: \( \Delta PQR \cong \Delta TQR \)**.

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