`DeltaPRQandDeltaLMN` are congruent in any correspondence. PQ = 5 cm, PR = 4 cm and `angleP=30^(@)`. If LM = 5 cm and QR = MN, then LN equals

Options

A. 3 cm
B. 5 cm
C. 4 cm
D. can't be determined

Answer

To solve the problem, we will use the properties of congruent triangles. Given that triangles \( \Delta PRQ \) and \( \Delta LMN \) are congruent, we can establish the relationships between their corresponding sides and angles.

### Step-by-Step Solution:

1. **Identify Given Values:**
- \( PQ = 5 \, \text{cm} \)
- \( PR = 4 \, \text{cm} \)
- \( \angle P = 30^\circ \)
- \( LM = 5 \, \text{cm} \)
- \( QR = MN \)

2. **Use Congruence of Triangles:**
Since \( \Delta PRQ \) is congruent to \( \Delta LMN \), we can match their corresponding sides and angles. The congruence can be represented as:
- \( PQ \) corresponds to \( LM \)
- \( PR \) corresponds to \( LN \)
- \( QR \) corresponds to \( MN \)

3. **Set Up Corresponding Sides:**
From the congruence:
- \( PQ = LM \)
- \( PR = LN \)
- \( QR = MN \)

4. **Substitute Known Values:**
- We know \( PQ = 5 \, \text{cm} \) and \( LM = 5 \, \text{cm} \) (this is given).
- We also know \( PR = 4 \, \text{cm} \).

5. **Find LN:**
Since \( PR \) corresponds to \( LN \) and \( PR = 4 \, \text{cm} \), it follows that:
\[ LN = PR = 4 \, \text{cm} \]

6. **Conclusion:**
Therefore, the length of \( LN \) is \( 4 \, \text{cm} \).

### Final Answer:
\[ LN = 4 \, \text{cm} \]