Since, x,y,w and z are angles at a single point,
x + y + w + z = \(360^\circ\)
But it is given that, x + y = w + z
\(\because \) x + y + x + y = \(360^\circ\)
\(\Rightarrow \) 2(x + y) = \(360^\circ\)
\(\Rightarrow \) (x + y) = \(180^\circ\)
Hence, by converse of linear pair axiom, it is proved that AOB is a straight line.