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.

Answer 1

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.