ABCD is a parallelogram. The sides AB and AD are produced to E and F respectively such that AB = BE and AD = DF. Prove that :

△ BEC ≅ △ DCF

Parallelogram ABCD with sides AB and AD are produced to E and F, respectively are is shown below:

Given,

AD = DF ……….(1)

AB = BE ……….(2)

We know that,

Opposite sides of parallelogram are equal.

∴ AD = BC ……..(3)

and,

AB = CD ……….(4)

From equations (1) and (3), we get :

⇒ BC = DF

From equations (2) and (4), we get :

⇒ BE = CD

We know that,

Opposite angles of a parallelogram are equal.

∴ ∠ABC = ∠ADC = x (let)

From figure,

Since, AE is a straight line.

⇒ ∠CBE + ∠ABC = 180°

⇒ ∠CBE + x = 180°

⇒ ∠CBE = 180° - x.

Since, AF is a straight line.

⇒ ∠CDF + ∠ADC = 180°

⇒ ∠CDF + x = 180°

⇒ ∠CDF = 180° - x.

In △ BEC and △ DCF,

⇒ BE = CD (Proved above)

⇒ BC = DF (Proved above)

⇒ ∠CBE = ∠CDF (Both equal to 180° - x)

∴ △ BEC ≅ △ DCF (By S.A.S. axiom).