Question

ABCD is parallelogram , O is the mid point of AB . DA is produced and CO is produced met at E. Prove that AD is equal to AE.

Answer 1

 ∵ O is mid-point of AB.

∴ OA = OB

Also, DA is produced to point E

∴ AE || DA

and DA || CB (c ABCD is a parallelogram)

Now, in triangles Δ AOE & Δ BOC, we have

∠AOE = ∠BOC (Vertically opposite angles)

∠EAO = ∠OBC (∵ AE || CB and AB is transversal , so alternative interior angles)

∠AEO = ∠OCB (∵ AE || CB and alternative interior angles)

Also, OA = OB (∵ O is mid-point of AB)

∴ ΔAOE ≅ ΔBOC (By AAS congruent criteria)

⇒ AE = BC (Property of congruent triangles)

⇒ AE = BC = AD (∵ BC = AD as ABCD is parallelogram)

∴ AE = AD

Hence Proved