ABCD is a rectangle. X and Y are points on sides AD and BC respectively such that AY = BX. Prove that BY = AX and ∠BAY = ∠ABX.

In the adjoining figure, ∠ABC = ∠ACB, D and E are points on the sides AC and AB respectively such that BE = CD. Prove that

(i) △EBC ≅ △DCB

(ii) △OEB ≅ △ODC

(iii) OB = OC.

In the adjoining figure, ∠BCD = ∠ADC and ∠BCA = ∠ADB. Show that

(i) △ACD ≅ △BDC

(ii) BC = AD

(iii) ∠A = ∠B

In the adjoining figure, BA ⊥ AC, DE ⊥ DF such that BA = DE and BF = EC. Show that △ABC ≅ △DEF.