Question : PQRS is a cyclic quadrilateral. If $\angle P$ is 4 times $\angle R$, and $\angle S$ is 3 times $\angle Q$, then the average of $\angle Q$ and $\angle R$ is:
Option 1: 40.5°
Option 2: 45.7°
Option 3: 90°
Option 4: 81°
Correct Answer: 40.5°
Solution : According to the question $\angle P$ = 4 × $\angle R$ The sum of the opposite angles in a cyclic quadrilateral is 180°. So, $\angle P$ + $\angle R$ = 180° ⇒ 4$\angle R$ + $\angle R$ = 180° ⇒ $\angle R$ = 36° Now, $\angle S$ = 3 × $\angle Q$ So, $\angle S$ + $\angle Q$ = 180° ⇒ 3$\angle Q$ + $\angle Q$ = 180° ⇒ $\angle Q$ = 45° So, the required average = $\frac{\angle Q+ \angle R}{2}$ = $\frac{36° + 45°}{2}$ = $\frac{81°}{2}$ = 40.5° Hence, the correct answer is 40.5°.
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