In the figure, two sides AB, BC and the median AD of ΔABC are respectively equal to two sides PQ, QR and median PS of ΔPQR. Prove that

(i) ΔADB ≅ ΔPSQ;

(ii) ΔADC ≅ ΔPSR.

Does it follow that triangles ABC and PQR are congruent?

(i) In Δ ADB and Δ PQS

AB = PQ

AD = PS

BC = QR

Since D and S are midpoints of BC and QR

⇒ 2DC = 2SR

⇒ DC = SR

So Δ ADB and Δ PQS are congruent to each other by S.S.S. axiom of congruency

(ii) In Δ ADC and Δ PSR

AD = PS

BC = QR

Since D and S are midpoints of BC and QR

⇒ 2BD = 2QS

⇒ BD = QS

∠ADB = ∠PSO(Corresponding parts of Congruent triangles)

⇒ 180⁰-∠ADB = 180⁰-∠PSO

∠ADC = ∠PSR

So Δ ADC and Δ PSR are congruent to each other by S.A.S. axiom of congruency

Yes it follows Δ ABC and Δ PQR are congruent because Δ ABC is the sum of Δ ADB and Δ ADC and Δ PQR is the sum of Δ PQS and Δ PSR