# 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 Fig. 7.40). Show that:

(i) Δ ABM ≅ Δ PQN

(ii) Δ ABC ≅ Δ PQR

**Solution:**

Given: AB = PQ, AM = PN, BM = QN

**(i)** In ΔABC, AM is the median to BC.

∴ BM = 1/2 BC

In ΔPQR, PN is the median to QR.

∴ QN = 1/2 QR

It is given that BC = QR

∴ 1/2 BC = 1/2 QR

∴ BM = QN … (1)

In ΔABM and ΔPQN,

AB = PQ (Given)

BM = QN [From equation (1)]

AM = PN (Given)

∴ ΔABM ≅ ΔPQN (Using SSS congruence criterion)

⇒ ∠ABM = ∠PQN (By CPCT)

⇒ ∠ABC = ∠PQR … (2)

**(ii)** In Δ ABC and Δ PQR ,

AB = PQ (Given)

∠ABC = ∠PQR [From Equation (2)]

BC = QR (Given)

∴ ΔABC ≅ ΔPQR (By SAS congruence rule)

**Video Solution:**

## 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 Fig. 7.40). Show that: (i) Δ ABM ≅ Δ PQN (ii) Δ ABC ≅ Δ PQR

### NCERT Maths Solutions Class 9 - Chapter 7 Exercise 7.3 Question 3:

**Summary:**

If 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, then ΔABM ≅ ΔPQN by SSS congruence and ΔABC ≅ ΔPQR by SAS congruence.