# Class 9 RD Sharma Solutions – Chapter 10 Congruent Triangles- Exercise 10.4

**Question 1: In figure, It is given that AB = CD and AD = BC. Prove that **Δ**ADC ≅ **Δ**CBA.**

**Solution:**

Given:

AB = CD and AD = BC.

To prove:

ΔADC ≅ ΔCBAConsider ΔADC and ΔCBA.

AB = CD {Given}

BC = AD {Given}

And AC = AC {Common side}

So,

By SSS congruence criterion, we have

ΔADC≅ ΔCBAHence, proved.

**Question 2: In a **Δ**PQR, if PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP respectively. Prove that LN = MN.**

**Solution:**

**Given: **

In Δ PQR, PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP respectively

To prove:

LN = MNJoin L and M, M and N, N and L

We have PL = LQ, QM = MR and RN = NP

[Since, L, M and N are mid-points of PQ, QR and RP respectively]

And also PQ = QR

PL = LQ = QM = MR = PN = LR —->(equation 1)MN || PQ and MN =

PQ{ Using mid-point theorem}

2

MN = PL = LQ —->(equation 2)Similarly, we have

LN || QR and LN =QR

2

LN = QM = MR —->(equation 3)From (equation 1), (equation 2) and (equation 3),

We have

PL = LQ = QM = MR = MN = LN

This implies, LN = MNHence, Proved.