In the given figure, AB // EC, AB = AC and AE bisects ∠DAC. Prove that:

(i) ∠EAC = ∠ACB

(ii) ABCE is a parallelogram.

(i) Given :

AB // EC, AB = AC and AE bisects ∠DAC.

To prove :

∠EAC = ∠ACB

Proof :

In triangle ABC and AEC,

AC = AC (Common)

AB = AC (Given)

∠ BAC = ∠ AEC

By Side Angle Side congruency,

Δ ABC ≅ Δ AEC

By using Corresponding Parts of Congruent Triangles,

∠EAC = ∠ACB

(ii) To prove :

ABCE is a parallelogram.

Proof :

∠EAC = ∠ACB (Proved)

Hence, AE // BC

AB // EC (Given)

Hence, ABCE is a parallelogram.