Question

The angles P, Q, R and S of a $\square$PQRS are in the ratio 1:3:7:9, find the measure of each angle.

Answer 1

In $\square$PQRS,
By the angle addition property of a quadrilateral,
$\angle \mathrm{P}+\angle \mathrm{Q}+\angle \mathrm{R}+\angle \mathrm{S}=360°$
Now,
$$\begin{gathered} \mathrm{Let}\angle \mathrm{P}=x,\angle \mathrm{Q}=3x,\angle \mathrm{R}=7x\mathrm{and}\angle \mathrm{S}=9x,\mathrm{where}x\mathrm{is}\mathrm{any}\mathrm{multiple}\mathrm{of}\angle \mathrm{P},\angle \mathrm{Q},\angle \mathrm{R}\mathrm{and}\angle \mathrm{S}. \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ x+3x+7x+9x=360° \\ \Rightarrow 20x=360° \\ \Rightarrow x=18° \\ \mathrm{Thus},\mathrm{the}\mathrm{measures}\mathrm{of}\mathrm{the}\mathrm{angles}\mathrm{of}\mathrm{quadrilateral}\mathrm{PQRS}\mathrm{are}: \\ \angle \mathrm{P}=18° \\ \angle \mathrm{Q}=54° \\ \angle \mathrm{R}=126° \\ \mathrm{and}\angle \mathrm{S}=162° \end{gathered}$$