Determine whether the conditions \( \angle \mathrm{Q} \cong \angle \mathrm{N}, \angle \mathrm{P} \cong \angle \mathrm{M} \), and \( \overline{\mathrm{PQ}} \cong \overline{\mathrm{MN}} \) are sufficient to prove that \( \triangle \mathrm{PQR} \cong \triangle \mathrm{MNO} \) and justify your answer: Are the conditions \( \angle \mathrm{Q} \cong \angle \mathrm{N}, \angle \mathrm{P} \cong \angle \mathrm{M} \), and \( \overline{\mathrm{PQ}} \cong \overline{\mathrm{MN}} \) sufficient to prove that \( \triangle \mathrm{PQR} \cong \triangle \mathrm{MNO} \) ? No Yes Justify your answer with the congruence condition. Choose the correct answer below. AAS None ASA

To determine whether the conditions \( \angle \mathrm{Q} \cong \angle \mathrm{N}, \angle \mathrm{P} \cong \angle \mathrm{M} \), and \( \overline{\mathrm{PQ}} \cong \overline{\mathrm{MN}} \) are sufficient to prove that \( \triangle \mathrm{PQR} \cong \triangle \mathrm{MNO} \), we need to analyze the given conditions in the context of triangle congruence criteria. 1. **Identify the given conditions:** - \( \angle \mathrm{Q} \cong \angle \mathrm{N} \) - \( \angle \mathrm{P} \cong \angle \mathrm{M} \) - \( \overline{\mathrm{PQ}} \cong \overline{\mathrm{MN}} \) 2. **Determine the congruence criteria:** - The conditions provided include two angles and one side. This fits the criteria for the **Angle-Angle-Side (AAS)** congruence condition, which states that if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent. 3. **Conclusion:** - Since we have two angles and a side that is not between the angles, we can conclude that the triangles \( \triangle \mathrm{PQR} \) and \( \triangle \mathrm{MNO} \) are congruent by the AAS condition. Thus, the answer is **Yes**, the conditions are sufficient to prove that \( \triangle \mathrm{PQR} \cong \triangle \mathrm{MNO} \), and the justification is based on the **AAS** congruence condition.