Question

pqrs is a cyclic quadrilateral. If <P=3x, <Q=y, <R​=x, <S=5y, find the measure of each of its angle

Answer 1

$$\begin{gathered} \mathrm{PQRS}\mathrm{is}\mathrm{a}\mathrm{cyclic}\mathrm{quadrilateral}. \\ \therefore \angle \mathrm{P}+\angle \mathrm{R}=180°\left(\mathrm{Sum}\mathrm{of}\mathrm{opposite}\mathrm{angles}\mathrm{of}\mathrm{cyclic}\mathrm{quadrilateral}\mathrm{is}180°\right) \\ \Rightarrow 3\mathrm{x}+\mathrm{x}=180° \\ \Rightarrow 4\mathrm{x}=180° \\ \Rightarrow \mathrm{x}=\frac{180°}{4}=45° \\ \angle \mathrm{P}=3\mathrm{x}=3\times 45°=135° \\ \angle \mathrm{R}=\mathrm{x}=45° \\ \mathrm{Also}, \\ \angle \mathrm{Q}+\angle \mathrm{S}=180°\left(\mathrm{Sum}\mathrm{of}\mathrm{opposite}\mathrm{angles}\mathrm{of}\mathrm{cyclic}\mathrm{quadrilateral}\mathrm{is}180°\right) \\ \Rightarrow \mathrm{y}+5\mathrm{y}=180° \\ \Rightarrow 6\mathrm{y}=180° \\ \Rightarrow \mathrm{y}=\frac{180°}{6}=30° \\ \angle \mathrm{Q}=\mathrm{y}=30° \\ \angle \mathrm{S}=5\mathrm{y}=5\times 30°=150° \end{gathered}$$