Question : If $ABCD$ be a cyclic quadrilateral in which $\angle A=x^\circ,\angle B=7x^\circ,\angle C=5y^\circ,\angle D=y^\circ$, then $x:y$ is:
Option 1: 3 : 2
Option 2: 2 : 3
Option 3: 5 : 4
Option 4: 4 : 5
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Correct Answer: 2 : 3
Solution : Given, a cyclic quadrilateral $ABCD$ with $\angle A=x^\circ,\angle B=7x^\circ,\angle C=5y^\circ,\angle D=y^\circ$ We know that the sum of opposite angles of the cyclic quadrilateral is $180^\circ$. So, $\angle A + \angle C=\angle B+\angle D$ Or, $x+5y=7x+y$ Or, $7x-x=5y-y$ Or, $6x=4y$ Or, $x:y=2:3$ Hence, the correct answer is 2 : 3.
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