Question

In Fig. If $x+y=w+z$ then prove that $AOB$ is a line.

Answer 1

Given $x+y=z+w\cdots \left(1\right)$

From Angle at a point theorem

Sum of all angles at a point is ${360}^{\circ }$

$x+y+z+w=360x+y+x+y=3602(x+y)=360x+y=180$

$x,y$ are adjacent angles and form ${180}^{\circ }$ together so $AOB$ is a straight line