Question

ABCD is a cyclic quadrilateral in which BA and CD when produced meet in E and EA = ED. 

Prove that : 

(i) AD || BC 

(ii) EB = EC

Answer 1

Given that, 

ABCD is a cyclic quadrilateral in which 

(i) Since, 

EA = ED 

Then, 

∠EAD = ∠EDA  ...(i) 

(Opposite angles to equal sides) 

Since, 

ABCD is a cyclic quadrilateral 

Then, 

∠ABC + ∠ADC = 180° 

But, 

∠ABC + ∠EBC = 180° 

(Linear pair) 

Then, 

∠ADC = ∠EBC  ...(ii) 

Compare (i) and (ii), we get 

∠EAD = ∠EBC ...(iii) 

Since, 

corresponding angles are equal 

Then, 

BC ǁ AD 

(ii) From (iii), we have 

∠EAD = ∠EBC 

Similarly, 

∠EDA = ∠ECB .... (iv) 

Compare equations (i), (iii) and (iv), we get 

∠EBC = ∠ECB 

EB = EC 

(Opposite angles to equal sides)