Question

ABCD is a cyclic quadrilateral. Find the angles of the cyclic quadrilateral.

Answer 1

Finding the angles of cyclic quadrilateral:

Step $1:$Finding the values of $\mathrm{x}$ and $\mathrm{y}$,

In cyclic quadrilateral, the sum of the opposite angles is ${180}^0$.

From figure, we get,

$$\begin{gathered} \angle \mathrm{A}=4\mathrm{y}+20 \\ \angle \mathrm{B}=3\mathrm{y}-5 \\ \angle \mathrm{C}=-4\mathrm{x} \\ \angle \mathrm{D}=-7\mathrm{x}+5 \end{gathered}$$

Therefore,$\angle C+\angle A={180}^{\circ }$

$\begin{array}{rcl}4\mathrm{y}+20-4\mathrm{x}& =& 180\\ -4\mathrm{x}+4\mathrm{y}& =& 160\end{array}$

$\begin{array}{rcl}\mathrm{y}-\mathrm{x}& =& 40\end{array}$ …………………….$\left(1\right)$

And, $\angle \mathrm{B}+\angle \mathrm{D}={180}^{\circ }$

$\begin{array}{rcl}3\mathrm{y}-5-7\mathrm{x}+5& =& 180\end{array}$

$\begin{array}{rcl}3\mathrm{y}-7\mathrm{x}& =& 180\end{array}$ ……………….$\left(2\right)$

On multiplying $3$ to equation $\left(1\right)$, we get,

$\begin{array}{rcl}3\mathrm{y}-3\mathrm{x}& =& 120\end{array}$……………………………$\left(3\right)$

On subtracting equation $\left(2\right)$ from equation $\left(3\right)$, we get

$\begin{array}{rcl}3\mathrm{y}-3\mathrm{x}-(3\mathrm{y}-7\mathrm{x})& =& 120-180\\ 3\mathrm{y}-3\mathrm{x}-3\mathrm{y}+7\mathrm{x}& =& -60\\ 4\mathrm{x}& =& -60\\ \mathrm{x}& =& -15\end{array}$

Substituting this value in equation $\left(1\right)$, we get,

$\begin{array}{rcl}\mathrm{y}-\mathrm{x}& =& 40\\ y-(-15)& =& 40\\ y+15& =& 40\\ y& =& 25\end{array}$

Step $2:$ Finding the angles :

On substituting the values of $\mathrm{x}$ and $\mathrm{y}$, we get,

$$\begin{gathered} \angle \mathrm{A}=4\mathrm{y}+20=4\times 25+20={120}^{\circ } \\ \angle \mathrm{B}=3\mathrm{y}-5=3\times 25-5={70}^{\circ } \\ \angle \mathrm{C}=-4\mathrm{x}=-4\times (-15)={60}^{\circ } \\ \angle \mathrm{D}=-7\mathrm{x}+5=-7\times (-15)+5={110}^{\circ } \end{gathered}$$

Hence, the angles of the cyclic quadrilateral are $\angle A={120}^{\circ },\angle B={70}^{\circ },\angle C={60}^{\circ },\angle D={110}^{\circ }$.