Discussion
In the following figure, lines PQ and RS intersect each other at point O. If ∠ POR : ∠ ROQ = 5 : 7, then the value of ∠ POS will be :
Answer: A
The value of ∠ POS is 105°.
∠ POR and ∠ ROQ lie on the straight line PQ, so they add up to 180°. Splitting 180° in the ratio 5 : 7:
\[ \begin{aligned} \angle POR &= \tfrac{5}{12} \times 180^\circ = 75^\circ \\ \angle ROQ &= \tfrac{7}{12} \times 180^\circ = 105^\circ \end{aligned} \]
∠ POS and ∠ ROQ are vertically opposite angles, so ∠ POS = 105°. 75° is ∠ POR (and its opposite ∠ SOQ), not ∠ POS; 150° and 210° double the parts.
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