# 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 Î”ADC â‰… Î”CBA.

Solution:

Given:
AB = CD and AD = BC.

To prove:

Consider Î”ADC and Î”CBA.
AB = CD {Given}
BC = AD {Given}
And AC = AC {Common side}
So,
By SSS congruence criterion, we have

Hence, proved.

### Question 2: In a Î”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 Î” PQR, PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP respectively

To prove:
LN = MN

Join 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 = MN

Hence, Proved.

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