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Question
ABCD is a parallelogram. The sides AB and AD are produced to E and F respectively, such produced to E and F respectively, such that AB = BE and AD = DF.
Prove that: ΔBEC ≅ ΔDCF.
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Solution
ABCD is a parallelogram, The sides AB and AD are produced to E and F respectively,
such that AB = BE and AD = DF
We need to prove that ΔBEC ≅ ΔDCF.
Proof:
AB = DC ...[ Opposite sides of a parallelogram ] ...(1)
AB = BE ...[ Given ] ...(2)
From (1) and (2), We have
BE = DC ...(3)
AD = BC ...[ Opposite sides of a parallelogram ] ...(4)
AD = DF ....[Given] ...(5)
From (4) and (5), we have
BC = DF ...(6)
Since AD II BC, the corresponding angles are equal.
∴ ∠DAB = ∠CBE ...(7)
Since AB II DC, the corresponding angles are equal.
∴ ∠DAB = ∠FDC ...(8)
From (7) and (8), we have
∠CBE = ∠FDC
ln ΔBEC and ΔDCF
BF = DC ....[ from (3) ]
∠CBE = ∠FDC ...[ from (9) ]
BC = DF ....[ from (6) ]
∴ By Side-Angle-Side criterion of congruence,
ΔBEC ≅ ΔDCF
Hence proved.
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