Question

In a $∆$PQR, if $\angle$P $-$ $\angle$Q = 42$°$ and $\angle$Q - $\angle$R = 21$°$, find $\angle$P, $\angle$Q and $\angle$R.

Answer 1

$$\begin{gathered} \mathrm{In}△\mathrm{PQR},\mathrm{we}\mathrm{have}, \\ \angle \mathrm{P}-\angle \mathrm{Q}=42° \\ \therefore \angle \mathrm{P}=\angle \mathrm{Q}+42°...\left(\mathrm{i}\right) \\ \mathrm{Also},\angle \mathrm{Q}-\angle \mathrm{R}=21° \\ \Rightarrow \angle \mathrm{R}=\angle \mathrm{Q}-21°...\left(\mathrm{ii}\right) \\ \mathrm{We}\mathrm{know}\mathrm{that}\angle \mathrm{P}+\angle \mathrm{Q}+\angle \mathrm{R}=180°.....\left(\mathrm{iii}\right) \\ \mathrm{Substituting}\mathrm{the}\mathrm{values}\mathrm{from}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right)\mathrm{in}\left(\mathrm{iii}\right),\mathrm{we}\mathrm{get}, \\ 42°+\angle \mathrm{Q}+\angle \mathrm{Q}+\angle \mathrm{Q}-21°=180° \\ \Rightarrow 21°+3\angle \mathrm{Q}=180° \\ \Rightarrow 3\angle \mathrm{Q}=159° \\ \Rightarrow \angle \mathrm{Q}=53° \\ \mathrm{Now},\mathrm{from}\left(\mathrm{i}\right),\mathrm{we}\mathrm{have}, \\ \angle \mathrm{P}=42°+53°=95° \\ \mathrm{Also},\mathrm{from}\left(\mathrm{ii}\right),\mathrm{we}\mathrm{have}, \\ \angle \mathrm{R}=53°-21°=32° \end{gathered}$$