Question 1: In figure, It is given that AB = CD and AD = BC. Prove that ΔADC ≅ ΔCBA.
Solution:
Given:
AB = CD and AD = BC.
To prove:
ΔADC ≅ ΔCBA
Consider ΔADC and ΔCBA.
AB = CD {Given}
BC = AD {Given}
And AC = AC {Common side}
So,
By SSS congruence criterion, we have
ΔADC≅ ΔCBA
Hence, proved.
Question 2: In a ΔPQR, 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 = 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.