Question

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

1. ∠EAC = ∠ACB
2. ABCE is a parallelogram.

Answer 1

ABCE is a quadrilateral in which AC is its diagonal and AB || EC, AB = AC

BA is produced to D

AE bisects ∠DAC

To prove:

(i) ∠EAC = ∠ACB

(ii) ABCE is a parallelogram

Proof:

(i) In ∆ABC and ∆AEC

AC = AC (common)

AB || CE (given)

∠BAC = ∠ACE (Alternate angle)

∆ABC = ∆AEC (SAS Axiom)

(ii) ∠BCA = ∠CAE (c.p.c.t.)

But these are alternate angles

AE || BC

But AB || EC (given)

∴ ABCE is a parallelogram