1. Given two triangles ABC and PQR, we are asked to find the condition for them to be congruent under the given angle equalities. 2. The problem states: - $\angle ABC = \angle PQR$ - $\angle ACB = \angle PQR$ - $\angle ABC = \angle PRQ$ - $\angle ACB = \angle PRQ$ 3. To prove two triangles congruent, we use criteria such as SAS (Side-Angle-Side), ASA (Angle-Side-Angle), or AAS (Angle-Angle-Side). 4. Notice that $\angle ABC$ and $\angle ACB$ correspond to angles at vertices B and C in triangle ABC, and $\angle PQR$ and $\angle PRQ$ correspond to angles at vertices Q and R in triangle PQR. 5. Since $\angle ABC = \angle PQR$ and $\angle ACB = \angle PRQ$, the two triangles have two pairs of equal angles. 6. By the Angle-Angle-Side (AAS) congruence rule, if two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent. 7. The side between these angles in triangle ABC is $AB$ or $AC$, and in triangle PQR is $PQ$ or $PR$. 8. Given the tick marks on sides $AC$ and $PR$ indicating $AC = PR$, the congruence condition is: $$\triangle ABC \cong \triangle PQR \quad \text{by AAS with} \quad \angle ABC = \angle PQR, \quad \angle ACB = \angle PRQ, \quad AC = PR.$$