In `Delta ABC and Delta LMN, AB=LM, AC = LN and angle B = angle M`. Then, the

Options

A. triangles will be congruent if `angleB` is a right angle
B. triangles are congruent only if AB = AC
C. triangles cannot be congruent
D. triangles must be congruent

Answer

To determine the congruence of triangles ABC and LMN given that \( AB = LM \), \( AC = LN \), and \( \angle B = \angle M \), we can follow these steps:

### Step 1: Identify Given Information
We have the following information:
- \( AB = LM \)
- \( AC = LN \)
- \( \angle B = \angle M \)

### Step 2: Apply the SAS Congruence Criterion
The Side-Angle-Side (SAS) congruence criterion states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

In our case:
- We have two sides \( AB \) and \( AC \) from triangle ABC that are equal to \( LM \) and \( LN \) from triangle LMN respectively.
- The angle \( \angle B \) is equal to \( \angle M \).

### Step 3: Check the Included Angle
However, to apply the SAS criterion correctly, we need to ensure that the angle \( \angle B \) is the included angle between the sides \( AB \) and \( AC \). In triangle ABC, the included angle between sides \( AB \) and \( AC \) is \( \angle A \), not \( \angle B \).

### Step 4: Conclusion
Since the angle \( \angle B \) is not the included angle between the two sides \( AB \) and \( AC \) in triangle ABC, we cannot use the SAS criterion to prove that triangles ABC and LMN are congruent. Therefore, we conclude that the triangles cannot be proven to be congruent based on the given information.

### Final Answer
The correct conclusion is that triangles ABC and LMN cannot be congruent.

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