Question : PQRS is a cyclic quadrilateral. If $\angle$P is three times of $\angle$R and $\angle$S is four times of $\angle$Q, then the sum of $\angle$S + $\angle$R will be:
Option 1: $169^{\circ}$
Option 2: $171^{\circ}$
Option 3: $187^{\circ}$
Option 4: $189^{\circ}$
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Correct Answer: $189^{\circ}$
Solution :
$\angle$P = 3$\angle$R
$\angle$S = 4$\angle$Q
As PQRS is a cyclic quadrilateral,
$\angle$P + $\angle$R = $180^{\circ}$
⇒ 3$\angle$R + $\angle$R = $180^{\circ}$
⇒ 4$\angle$R = $180^{\circ}$
$\therefore$ $\angle$R = $45^{\circ}$
⇒ $\angle$P = 3 × $45^{\circ}$ = $135^{\circ}$
$\angle$S + $\angle$Q = $180^{\circ}$
⇒ 4$\angle$Q + $\angle$Q = $180^{\circ}$
⇒ 5$\angle$Q = $180^{\circ}$
$\therefore$ $\angle$Q = $36^{\circ}$
⇒ $\angle$S $= 4 × 36^{\circ}=144^{\circ}$
So, $\angle$S + $\angle$R $=144^{\circ}+45^{\circ}=189^{\circ}$
Hence, the correct answer is $189^{\circ}$.
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