Question

If $ABCD$ is a cyclic quadrilateral and $\angle A=4y+{20}^o;\angle B=3y-5^o;\angle C=-4x,\angle D=-7x+5^o$, then find the value of $x$ and $y$.

Answer 1

We know, in a cyclic quadrilateral, the sum of opposite angles are supplementary, i.e. ${180}^o$.

Here, $ABCD$ is a cyclic quadrilateral.

Then, $\angle A+\angle C=\angle B+\angle D={180}^o$

$⟹4y+{20}^o+(-4x)=3y-5^o+(-7x+5^o)={180}^o$

$⟹4y+{20}^o-4x=3y-7x={180}^o$.

From the two equations, we get,

$y-x={40}^o,3y-7x={180}^o$.

Then, $4x=-{60}^o$ $⟹x=-{15}^o$.

and $y={40}^o+x={40}^o+(-{15}^o)={25}^o$.

$⟹x=-{15}^o$ and $y={25}^o$.