In two triangles PQR and LMN, PQ = QR, `angleP=angleMandQR=LN`, then which of the following is true?

Options

A. Triangles are congruent only
B. Triangles are isosceles only
C. Triangles are both congruent and isosceles
D. can't be determined

Answer

To solve the problem, we need to analyze the information given about the two triangles, PQR and LMN, and determine the relationship between them based on the properties of congruence.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- We have two triangles: PQR and LMN.
- The following conditions are provided:
- PQ = QR (This indicates that triangle PQR is isosceles with sides PQ and QR being equal).
- ∠P = ∠M (Angle P in triangle PQR is equal to angle M in triangle LMN).
- QR = LN (This indicates that side QR in triangle PQR is equal to side LN in triangle LMN).

2. **Understanding Triangle Properties**:
- Since PQ = QR, triangle PQR is isosceles.
- The property of isosceles triangles states that the angles opposite to the equal sides are also equal. Therefore, if PQ = QR, then ∠Q = ∠R.

3. **Analyzing the Angles**:
- We know that ∠P = ∠M.
- Since ∠Q and ∠R are equal (because triangle PQR is isosceles), we can denote them as ∠Q = ∠R = x.
- The sum of angles in triangle PQR is 180 degrees, so:
- ∠P + ∠Q + ∠R = 180
- ∠M + x + x = 180
- ∠M + 2x = 180

4. **Comparing the Triangles**:
- We have established that triangle PQR has two equal angles (due to it being isosceles) and one angle equal to angle M in triangle LMN.
- However, we do not have enough information about the sides of triangle LMN or the other angles in triangle LMN to definitively conclude that the two triangles are congruent.

5. **Conclusion**:
- Since we cannot establish that the two triangles are congruent based on the given information, we conclude that we cannot determine the relationship between the two triangles definitively.
- Therefore, the correct answer is: **Triangles can't be determined**.