Question

In $\square$KLMN, $\angle$K = 30$°$, $\angle$M = 150$°$ and $\angle$N = 110$°$.
Point P is on the side KL as shown in fig. 5.8. Find the measures of $\angle$KLM and $\angle$MLP.

Answer 1

$$\begin{gathered} \mathrm{Consider}\mathrm{the}\mathrm{figure}\mathrm{given}\mathrm{in}\mathrm{the}\mathrm{question}. \\ \mathrm{By}\mathrm{the}\mathrm{angle}\mathrm{addition}\mathrm{property}\mathrm{of}\mathrm{a}\mathrm{quadrilateral},\mathrm{we}\mathrm{can}\mathrm{say}: \\ \mathrm{In}\mathrm{quadrilateral}\mathrm{KLMN},\mathrm{we}\mathrm{have}: \\ \angle \mathrm{NKL}+\angle \mathrm{KNM}+\angle \mathrm{NML}+\angle \mathrm{KLM}=360° \\ \because \angle \mathrm{NKL}=\angle \mathrm{K}=30°,\angle \mathrm{KNM}=\angle \mathrm{N}=110°\mathrm{and}\angle \mathrm{NML}=\angle \mathrm{M}=150° \\ \therefore 30°+110°+150°+\angle \mathrm{KLM}=360° \\ \Rightarrow 290°+\angle \mathrm{KLM}=360° \\ \Rightarrow \angle \mathrm{KLM}=360°-290° \\ \Rightarrow \angle \mathrm{KLM}=70° \\ \mathrm{Now}, \\ \angle \mathrm{KLM}+\angle \mathrm{MLP}=180°\left[\mathrm{Linear}\mathrm{pair}\right] \\ \Rightarrow 70°+\angle \mathrm{MLP}=180° \\ \Rightarrow \angle \mathrm{MLP}=180°-70° \\ \Rightarrow \angle \mathrm{MLP}=110° \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ \angle \mathrm{KLM}=70°\mathrm{and}\angle \mathrm{MLP}=110° \end{gathered}$$