Two sides AB and BC and median AM of one triangle ABC are respectively

1.	Two sides AB and BC and median AM of one triangle ABC are respectively equal to side PQ and QR and median PN of ∆PQR (see figure). Show that(i) ∆ABM = ∆PQN(ii) ∆ABC = ∆PQR
Answer» (i) In ∆’s ABM and PQN AB = PQ (given) AM = PN (given) and BC = QR (given) => 1/2BC = 1/2QR => BM = QN ∴ ∆ABM ≅ ∆PQN (by SSS congruency rule) (ii) In ∆ABC and ∆PQR ∵ ∆ABM ≅ ∆PQN [proved in (i)] => ∠B = ∠Q (by c.p.c.t) AB = PQ (given) BC = QR (given) ∴ ∆ABC ≅ ∆PQR (by SAS congruency rule)

Two sides AB and BC and median AM of one triangle ABC are respectively equal to side PQ and QR and median PN of ∆PQR (see figure). Show that(i) ∆ABM = ∆PQN(ii) ∆ABC = ∆PQR

Answer»

(i) In ∆’s ABM and PQN

AB = PQ (given)

AM = PN (given)

and BC = QR (given)

=> 1/2BC = 1/2QR

=> BM = QN

∴ ∆ABM ≅ ∆PQN

(by SSS congruency rule)

(ii) In ∆ABC and ∆PQR

∵ ∆ABM ≅ ∆PQN [proved in (i)]

=> ∠B = ∠Q (by c.p.c.t)

AB = PQ (given)

BC = QR (given)

∴ ∆ABC ≅ ∆PQR

(by SAS congruency rule)