ABCD is a cyclic quadrilateral. AB and DC when produced meet at E. Prove that ΔEBC and ΔEDA are similar.

Given that ABCD is a cyclic quadrilateral and AB and DC are produced to meet at E.

We have to prove that ΔEBC and ΔEDA are similar.

Proof:

We know that in a cyclic quadrilateral, opposite angles are supplementary.

⇒∠ABC + ∠ADC = 180° … (1)

Also, ∠ABC + ∠EBC = 180° … (2) [linear pair]

From (1) and (2),

⇒∠ABC + ∠ADC = ∠ABC + ∠EBC

⇒∠ADC = ∠EBC … (3)

Similarly,

⇒∠BAD + ∠BCD = 180° … (4)

Also, ∠BCD + ∠BCE = 180° … (5) [linear pair]

From (4) and (5),

⇒∠BAD + ∠BCD = ∠BCD + ∠BCE

⇒∠BAD = ∠BCE … (6)

And ∠BEC = ∠AED … (7)

From (3), (6) and (7),

∴ ΔEBC and ΔEDA are similar.

Hence proved