Question

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

Answer 1

As Given, $x+y=w+z$
To Prove: $AOB$ is a line or $x+y={180}^{\circ }$ (linear pair.)
According to the question,
$x+y+w+z={360}^{\circ }$ $\mid$ Angles around a point.
$(x+y)+(w+z)={360}^{\circ }$
$(x+y)+(x+y)={360}^{\circ }$ $\mid$ Given $x+y=w+z$

$2(x+y)={360}^{\circ }$
$(x+y)={180}^{\circ }$
Hence, $x+y$ makes a linear pair.

Therefore, $AOB$ is a straight line.