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
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To prove that triangles \( \triangle ABC \) and \( \triangle PQR \) are congruent, we can use the Angle-Side-Angle (ASA) postulate. Given that \( AB = PQ \), we can focus on the corresponding angles. Since angle \( A = 40^\circ \) and angle \( B = 60^\circ \), we add them to find the third angle in triangle \( ABC \): \[ \angle C = 180^\circ - (40^\circ + 60^\circ) = 80^\circ \] Now, for triangle \( PQR \), we already have angle \( P = 4h \) and angle \( R = 80^\circ \). Since the sum of angles in a triangle is also \( 180^\circ \), angle \( Q \) can be determined as follows: \[ \angle Q = 180^\circ - (4h + 80^\circ) \] Setting \( 4h = 40^\circ \) gives us angle \( P = 40^\circ \) as well, leading us back to triangle \( ABC \). Thus, we have: - \( AB = PQ \) (Given) - \( \angle A = \angle P = 40^\circ \) - \( \angle B = \angle Q = 60^\circ \) This shows that both triangles share two angles and the included side \( AB \) and \( PQ \). Thus, by the ASA criterion for triangle congruence, it follows that \[ \triangle ABC \cong \triangle PQR \] is proven!
