ABCD is a paralle

To Prove: BF = BC

Given: AD is produced to E, DE = DC

EC is produced to meet AB in F.

Concept Used:

Opposite sides of a parallelogram are equal.

ASA Congruence: The ASA rule states that. If two angles and the included side of one triangle are equal to two angles and included a side of another triangle, then the triangles are congruent.

Diagram:

Explanation:

In Δ ACE, D, and O are mid points of AE and AC respectively.

∴ DO||EC

⇒ OB||CF

⇒ AB = BF ….(i)

⇒ DC = BF [AB = DC as ABCD is a parallelogram]

In Δ’s EDC and CBF, we have

⇒ DC = BC

⇒ EDC = CBF

and ECD = CFB

So, by ASA congruence criterion, we have

ΔEDC ≅ ΔCBF

⇒ DE = BC

⇒ DC = BC

⇒ AB = BC

⇒ BF = BC [AB = BF from (i)]

Hence, Proved.