Question : If $ABCD$ is a cyclic quadrilateral with $\angle A=50^{\circ},\angle B=80^{\circ}$, then $\angle C$ and $ \angle D$ are:
Option 1: $100^{\circ}$ and $130^{\circ}$
Option 2: $115^{\circ}$ and $115^{\circ}$
Option 3: $110^{\circ}$ and $120^{\circ}$
Option 4: $130^{\circ}$ and $100^{\circ}$
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Correct Answer: $130^{\circ}$ and $100^{\circ}$
Solution :
In a cyclic quadrilateral, the sum of each pair of opposite angles is $180^{\circ}$.
$⇒\angle A + \angle C = 180^{\circ}$
$⇒\angle B + \angle D = 180^{\circ}$
Given that $\angle A = 50^{\circ}$ and $\angle B = 80^{\circ}$
$\therefore\angle C = 180^{\circ} - \angle A = 180^{\circ} - 50^{\circ} = 130^{\circ}$
$\therefore \angle D = 180^{\circ} - \angle B = 180^{\circ} - 80^{\circ} = 100^{\circ}$
Hence, the correct answer is $130^{\circ}$ and $100^{\circ}$.
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