Question

In the fig. 2.21, RAP is a triangle in which $\angle$R : $\angle$A: $\angle$P = 3:2:1 and RP $\perp$PS. Find the measures of $\angle$SPW.

Answer 1

$$\begin{gathered} \mathrm{In}△\mathrm{RAP},\mathrm{we}\mathrm{have}: \\ \angle \mathrm{R}:\angle \mathrm{A}:\angle \mathrm{P}=3:2:1 \\ \mathrm{Let}\mathrm{the}\mathrm{angles}\mathrm{be}\angle \mathrm{R}=3x,\angle \mathrm{A}=2x\mathrm{and}\angle \mathrm{P}=x \\ \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{angle}\mathrm{sum}\mathrm{property},\mathrm{we}\mathrm{have}: \\ \angle \mathrm{R}+\angle \mathrm{A}+\angle \mathrm{P}=180° \\ \Rightarrow 3x+2x+x=180° \\ \Rightarrow 6x=180° \\ \Rightarrow x=30° \\ \mathrm{Thus},\angle \mathrm{R}=3\times 30°=90° \\ \angle \mathrm{A}=2\times 30°=60° \\ \angle \mathrm{P}=30° \\ \mathrm{Since}\mathrm{APW}\mathrm{is}\mathrm{a}\mathrm{straight}\mathrm{line},\mathrm{we}\mathrm{have}: \\ \angle \mathrm{APR}+\angle \mathrm{RPS}+\angle \mathrm{SPW}=180° \\ \Rightarrow 30°+90°+\angle \mathrm{SPW}=180°(\because \mathrm{RP}\perp \mathrm{PS}) \\ \Rightarrow \angle \mathrm{SPW}=60° \end{gathered}$$