Question

From the figure, find all the four angles of the cyclic quadrilateral.

Answer 1

Here, $ABCD$ is a cyclic quadrilateral.

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

Here, $\angle A=y$ and $\angle C=2y$ are opposite angles.

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

$⟹$ $y+2y={180}^o$

$⟹$ $3y={180}^o$

$⟹$ $y={60}^o$.

Here, $\angle B=3x$ and $\angle D=2x$ are opposite angles.

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

$⟹$ $3x+2x={180}^o$

$⟹$ $5x={180}^o$

$⟹$ $x={36}^o$.

Hence, the four angles are:

$\angle A=y={60}^o$,

$\angle B=3x=3\left({36}^o\right)={108}^o$,

$\angle C=2y=2\left({60}^o\right)={120}^o$

and $\angle D=2x=2\left({36}^o\right)={72}^o$.