ABCD is cyclic quadrilateral. Sides AB and DC, when produced, meet at E, and sides BC and AD, when produced, meet at F. If $$\angle$$BFA = $$60^\circ$$ and $$\angle$$AED = $$30^\circ$$, then the measure of $$\angle$$ABC is:

From the given question we draw the diagram is given below 

from the above diagram $$\angle BFA = 60^\circ , $$\angle AFD = 30^\circ $$

then $$ \angle EBC + \angle ABC = 180^\circ $$ (straight line) .............(1)

$$ \angle ABC + \angle ADC = 180^\circ $$ (Opposite angle of cyclic quadrilateral)..... (2)  

from the above Equestion (1) and (2) 

$$ \angle EBC + \angle ABC = \angle ABC + \angle ADC $$

$$\angle EBC = \angle ADC $$ .......(3) 

$$ \angle DFC + \angle DCF + \angle CDF = 180^\circ $$ (angle sum property of a triangle) ....... (4) 

$$ \angle BCE + \angle CBE + \angle CEB = 180^\circ $$ (angle sum property of a triangle) .........(5)

from the equestion (4) and (5) 

$$ \angle DCF = \angle BCF $$ (Vertically Opposite angle) 

$$\angle DFC + \angle DCF + \angle CDF = \angle BCE + \angle CBF + \angle CEB $$

$$\Rightarrow \angle DFC + \angle CDF = \angle CBF + \angle CEB $$

$$\Rightarrow 60^\circ + 180^\circ - \angle EBC = \angle EBC + \angle CEB $$

$$\Rightarrow 60^\circ + 180^\circ = 2 \angle EBC + 30^\circ $$

$$\Rightarrow 2 \angle EBC = 210^\circ $$

$$\Rightarrow \angle EBC= 105^\circ $$

then $$\angle ABC + \angle EBC = 180^\circ $$

$$\Rightarrow \angle ABC + 105^\circ = 180^\circ $$

$$\Rightarrow \angle ABC = 180^\circ -105^\circ $$

$$\Rightarrow \angle ABC = 75^\circ $$ Ans