In the figure,if $x+y=w+z$,then prove that $AOB$ is a line.
(N/A) The sum of all the angles around a point is $360^\circ$.
Therefore,$x+y+z+w=360^\circ$.
We are given that $x+y=w+z$.
Substituting $(w+z)$ with $(x+y)$,we get:
$(x+y)+(x+y)=360^\circ$
$2(x+y)=360^\circ$
$x+y = \frac{360^\circ}{2} = 180^\circ$.
Since the sum of the adjacent angles $x$ and $y$ is $180^\circ$,they form a linear pair.
Therefore,$AOB$ is a straight line.
Therefore,$x+y+z+w=360^\circ$.
We are given that $x+y=w+z$.
Substituting $(w+z)$ with $(x+y)$,we get:
$(x+y)+(x+y)=360^\circ$
$2(x+y)=360^\circ$
$x+y = \frac{360^\circ}{2} = 180^\circ$.
Since the sum of the adjacent angles $x$ and $y$ is $180^\circ$,they form a linear pair.
Therefore,$AOB$ is a straight line.
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