MCQ
Lines $PQ$ and $RS$ intersect at $O.$ If $\angle\text{POS}=2\angle\text{SOQ},$ then the four angles at $O$ are:
  • A
    $30^\circ , 30^\circ , 120^\circ , 180^\circ $
  • $60^\circ , 60^\circ , 120^\circ , 120^\circ $
  • C
    $60^\circ , 90^\circ , 90^\circ , 120^\circ $
  • D
    $30^\circ , 60^\circ , 90^\circ , 180^\circ$

Answer

Correct option: B.
$60^\circ , 60^\circ , 120^\circ , 120^\circ $

$PQ$ and $RS$ intersect at $O.$ then,
$\angle\text{POS}=\angle\text{QOR}$ (opposite angles)
$\angle\text{SOQ}=\angle\text{POR}$ (opposite angles)
Given, $\angle\text{POS}=2\angle\text{SOQ}$
Sum of all angles $= 360$
$\angle\text{POS}+\angle\text{SOQ}+\angle\text{QOR}+\angle\text{ROP}=360$
$6\angle\text{SOQ}=360$
$\angle\text{SOQ}=60$
Hence, the four angles $= 60^\circ , 60^\circ , 120^\circ , 120^\circ .$

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