Q21.ABCD is a parallelogram. AB is produced to E show that BE=AB.EF meets CB produced at F and is parallel to CA. prove that AF is equal to EC.

Here ABCD is a parallelogram, BE = AB , And AC||EF
To proof: AF = EC

In ΔABC and ΔFBE
AB = BE (given)
∠FEB = ∠CAB (Alternate interior angles)
∠FBE = ∠ABC (vertically opposite angle)
Hence ΔABC and ΔFBE are congruent by (ASA)
So by CPCT (BC = BF) (1)

In ΔABF and ΔBCE
∠ABF = ∠CBE  (Vertically opposite angle)
BC= BF {from (1)}
BE = AB (given)
Hence ΔABF and ΔBCE are congruent by SAS
So by CPCT AF = CE (Hence proved)


Cheers!