Question

Abcd is a parallelogram , AD is produced to E so that DE =DC and EC produced meets AB produced in F prove that BF =BC

Answer 1

Given: ABCD is a parallelogram. AD is produced to E such that DE = DC. EC is produced to intersect AB produced in F.

To prove: BF = BC

Proof:

In ΔDCE,

DE = DC (Given)

∴ ∠DCE = ∠DEC ...(1) (In a triangle, equal sides have equal angles opposite to them)

AB || CD (Opposite sides of the parallelogram are parallel)

∴ AF || CD (AB lies on AF)

AF || CD and EF is the transversal,

∴ ∠DCE = ∠BFC ... (2) (Pair of corresponding angles)

From (1) and (2), we get

∠DEC = ∠BFC

In ΔAFE,

∠AFE = ∠AEF (∠DEC = ∠BFC)

∴ AE = AF (In a triangle, equal angles have equal sides opposite to them)

⇒ AD + DE = AB + BF

⇒ BC + AB = AB + BF (AD = BC, DE = CD and CD = AB ⇒ AB = DE)

⇒ BC = BF