In the given figure, if $x+y=w+z$, then prove that $\mathrm{AOB}$ is a line.

Solution:

It can be observed that,

$x+y+z+w=360^{\circ}$ (Complete angle)

It is given that,

$x+y=z+w$

$\therefore x+y+x+y=360^{\circ}$

$2(x+y)=360^{\circ}$

$x+y=180^{\circ}$

Since $x$ and $y$ form a linear pair, $A O B$ is a line.