SSC CPO 2019 PYQ — Question 106

The correct answer is Option 2 i.e. 75º

Given that ABCD is a cyclic quadrilateral. Sides AB and DC, when produced, meet at E, and sides BC and AD, when produced, meet at F. If ∠BFA = 60^o and ∠AED = 30^o as shown in the figure below.

∠ADC = 180º - ∠ABC (Sum of opposite angles of a cyclic quadrilateral is 180º)…………..(1)

∠BCD = 180º - ∠BAD (Sum of opposite angles of a cyclic quadrilateral is 180º)……………(2)

Also, ∠ADC + ∠FDC = 180º (linear pair)

So, ∠FDC = 180º - ∠ADC

From (1)

∠FDC =  ∠ABC………………(3)

Now, , ∠BCD+ ∠FCD = 180º (linear pair)

So, ∠FCD = 180º - ∠BCD

From (2)

∠FCD = ∠BAD ………………(4)

In triangle FDC

∠FDC + ∠FCD + ∠DFC = 180º

⇒ ∠ABC + ∠BAD + 60º =  180º

⇒ ∠ABC + ∠BAD = 120º ……………………….(5)

In triangle AED

⇒ ∠EAD + ∠ADE + ∠AED = 180º

⇒ 180º - ∠ABC + ∠BAD =  180º - 30º

⇒ - ∠ABC + ∠BAD =  30º …………………(6)

Subtract (6) from (5)

2∠ABC  = 150º

⇒ ∠ABC  = 75º