In Fig. 6.16, if $ x+y=w+z $, then prove that $ \mathrm{AOB} $ is a line.
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To do:

We have to prove that $AOB$ is a line.

Solution:

We know that,

The sum of the measures of the angles in linear pairs is always $180^o$.

So in order to prove that $AOB$ is a straight line, we have to prove that $x+y$ is a linear pair of $AOB$.

This implies,

$x+y=180^o$

We also know that,

The angles around a point give $360^o$.

This implies,

$x+y+w+z=360^o$

Since we have, 

$x+y=w+z$

We get,

$2x+y=360^o$

This implies,

$x+y=\frac{360^o}{2}$

$x+y=180^o$

Therefore, $x+y$ is the linear pair

This implies, that $AOB$ is a line.

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Updated on: 10-Oct-2022

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