Abcd Is Cyclic Quadrilateral In Which Ab And Cd When Produced Meet In E And Ea Ed

ABCD is a cyclic quadrilateral in which AB and CD when produced meet in E and EA = ED. Prove that : (i) AD\\BC (ii) EB = EC

Answers:

(i) In △AED, we have

EA = ED   [given] 

⇒ ∠1 = ∠2  

[angles opposite to equal sides are equal]

Also, ∠3 = ∠2  [external angle of cyclic quad. = interior opposites angle]

 ∠1 = ∠2

and  ∠3 = ∠2

⇒  ∠1 = ∠3 

These are correspondence angles.

∴ AD || BC 

(ii) ∠1 = ∠4 [external angle of cyclic quad. = Interior opposite angle]

But ∠1 = ∠3  [proved above]

∠1 = ∠4 

and ∠1 = ∠3 

⇒  ∠3 = ∠4 

In  △BEC, we have 

 ∠3 = ∠4 ⇒ BE = CE  [side opposite to equal angles are equal]