Are two triangles congruent if two sides and an angle of one triangle are respectively equal to two sides and an angle of the other? If not then under what conditions will they be congruent?

Answer

To determine whether two triangles are congruent given that two sides and an angle of one triangle are respectively equal to two sides and an angle of the other triangle, we need to analyze the conditions for congruence.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We have two triangles, let's call them Triangle ABC and Triangle PQR.
- We know that two sides and an angle of Triangle ABC are equal to two sides and an angle of Triangle PQR.

2. **Identifying the Congruence Criteria**:
- The congruence criterion we are considering here is the Side-Angle-Side (SAS) criterion.
- According to the SAS criterion, if two sides of one triangle are equal to two sides of another triangle, and the angle between those two sides is also equal, then the triangles are congruent.

3. **Analyzing the Given Angles**:
- If we assume that AB = PQ, BC = QR, and angle B = angle Q, then by the SAS property, Triangle ABC is congruent to Triangle PQR.
- However, if we consider AB = PQ, BC = QR, but instead take angle C = angle R (which is not the angle between the two sides), we cannot use the SAS criterion.

4. **Conclusion**:
- Therefore, the statement that two triangles are congruent if two sides and an angle of one triangle are respectively equal to two sides and an angle of the other triangle is **not necessarily true**.
- For the triangles to be congruent, the angle must be the included angle between the two equal sides.

5. **Final Statement**:
- The triangles will be congruent if the angle included between the two equal sides is also equal.