Question

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

Open in App
Solution

## 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, $\angle \mathrm{A}=4\mathrm{y}+20\phantom{\rule{0ex}{0ex}}\angle \mathrm{B}=3\mathrm{y}-5\phantom{\rule{0ex}{0ex}}\angle \mathrm{C}=-4\mathrm{x}\phantom{\rule{0ex}{0ex}}\angle \mathrm{D}=-7\mathrm{x}+5$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$\begin{array}{rcl}3\mathrm{y}-3\mathrm{x}-\left(3\mathrm{y}-7\mathrm{x}\right)& =& 120-180\\ 3\mathrm{y}-3\mathrm{x}-3\mathrm{y}+7\mathrm{x}& =& -60\\ 4\mathrm{x}& =& -60\\ \mathrm{x}& =& -15\end{array}$Substituting this value in equation $\left(1\right)$, we get,$\begin{array}{rcl}\mathrm{y}-\mathrm{x}& =& 40\\ y-\left(-15\right)& =& 40\\ y+15& =& 40\\ y& =& 25\end{array}$Step $2:$ Finding the angles :On substituting the values of $\mathrm{x}$ and $\mathrm{y}$, we get,$\angle \mathrm{A}=4\mathrm{y}+20=4×25+20={120}^{\circ }\phantom{\rule{0ex}{0ex}}\angle \mathrm{B}=3\mathrm{y}-5=3×25-5={70}^{\circ }\phantom{\rule{0ex}{0ex}}\angle \mathrm{C}=-4\mathrm{x}=-4×\left(-15\right)={60}^{\circ }\phantom{\rule{0ex}{0ex}}\angle \mathrm{D}=-7\mathrm{x}+5=-7×\left(-15\right)+5={110}^{\circ }$Hence, the angles of the cyclic quadrilateral are $\angle A={120}^{\circ },\angle B={70}^{\circ },\angle C={60}^{\circ },\angle D={110}^{\circ }$.

Suggest Corrections
1