Question

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

Answer 1

Lines PQ and PR, intersect each other at O
$\because$ Vertically opposite angles are equal
$\therefore \angle POR=\angle QOS$ and $\angle ROQ=\angle POS$
$\angle POR:\angle ROQ=5:7$
Let $\angle POR=5x$ and $\angle ROQ=7x$
But $\angle POR+\angle ROQ={180}^{\circ }$ (Linear pair)
$\therefore 5x+7x={180}^{\circ }\Rightarrow 12x={180}^{\circ }$
$\Rightarrow x=\frac{{180}^{\circ }}{12}={15}^{\circ }$
$\therefore \angle POR=5x=5\times {15}^{\circ }={75}^{\circ }$
and $\angle ROQ=7x=7\times {15}^{\circ }={105}^{\circ }$
But $\angle QOS=POR={75}^{\circ }$
(Vertically opposite angles)
and $\angle POS=\angle ROQ={105}^{\circ }$