2 \( \triangle A B C \) and \( \triangle P Q R \) in which \( A B=P Q \) and the angles at \( A, B, P \) and \( R \) are respectively \( 40^{\circ}, 60^{\circ} \), 4h and \( 80^{\circ} \). Pove that the two triangle are congruent

To prove that triangles \( \triangle ABC \) and \( \triangle PQR \) are congruent, we'll use the **Angle-Side-Angle (ASA)** congruence criterion. Here's a step-by-step explanation: ### Given: 1. **Side Equality:** \( AB = PQ \) 2. **Angle Measures:** - \( \angle A = 40^\circ \) - \( \angle B = 60^\circ \) - \( \angle P = 40^\circ \) - \( \angle R = 80^\circ \) ### Steps to Prove Congruence: 1. **Determine Remaining Angles:** - In \( \triangle ABC \): \[ \angle C = 180^\circ - \angle A - \angle B = 180^\circ - 40^\circ - 60^\circ = 80^\circ \] - In \( \triangle PQR \): \[ \angle Q = 180^\circ - \angle P - \angle R = 180^\circ - 40^\circ - 80^\circ = 60^\circ \] 2. **Compare Corresponding Angles:** - \( \angle A = \angle P = 40^\circ \) - \( \angle B = \angle Q = 60^\circ \) - \( \angle C = \angle R = 80^\circ \) 3. **Apply ASA Congruence Criterion:** - **First Angle:** \( \angle A = \angle P \) - **Included Side:** \( AB = PQ \) - **Second Angle:** \( \angle B = \angle Q \) Since two angles and the included side of \( \triangle ABC \) are equal to two angles and the included side of \( \triangle PQR \), by the **ASA criterion**, the triangles are congruent. ### Conclusion: By the **Angle-Side-Angle (ASA)** congruence theorem, triangles \( \triangle ABC \) and \( \triangle PQR \) are congruent. **Final Answer:** Since AB = PQ and two pairs of corresponding angles are equal, triangles ABC and PQR satisfy ASA and are congruent.