Mathematics

In Fig., PQ and RS intersect each other at point O. If $$ \angle POR : \angle ROQ = 5:7,$$ find all the angles.


SOLUTION
We have. $$ \angle POR : \angle ROQ = 5 : 7 $$

So, Let $$\angle POR = 5x $$ and $$ \angle ROQ = 7x $$

Since $$ \angle POR $$ and $$ \angle ROQ $$ form a linear pair.

$$ \therefore \angle POR + \angle ROQ = 180^{\circ}\Rightarrow  5x+7x = 180^{\circ} \Rightarrow x = 15^{\circ}$$

$$ \therefore \angle POR = \left ( 5 \times 15 \right )^{\circ} = 75^{\circ}$$ and 

$$\angle ROQ = \left ( 7 \times 15 \right )^{\circ} = 105^{\circ}$$

But, $$ \angle QOS = \angle POR $$ and $$\angle POS = \angle ROQ$$ [vertically opposite angles]

$$ \therefore \angle QOS = 75^{\circ}$$ and $$\angle POS = 105^{\circ}$$
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