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Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see Figure). Show that:

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Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see Figure). Show that:

(i) ΔABM ΔPQN

(ii) ΔABC ΔPQR

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Given parameters are:

AB = PQ,

BC = QR and

AM = PN

(i) 1/2 BC = BM and 1/2 QR = QN (Since AM and PN are medians)

BC = QR

1/2 BC = 1/2 QR

⇒ BM = QN

In ΔABM and ΔPQN,

AM = PN and AB = PQ (As given in the question)

BM = QN (Already proved)

∴ ΔABM ΔPQN by SSS congruency.

(ii) In ΔABC and ΔPQR,

AB = PQ and BC = QR (As given in the question)

ABC = PQR (by CPCT)

So, ΔABC ΔPQR by SAS congruency.

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