Question

In the figure 1.76 $\angle$XLN = 100$°$, $\angle$YMN = 165$°$, $\angle$ZNM = 95$°$. find the measures of $\angle$MLN, $\angle$LMN, $\angle$LNM.

Answer 1

Given: $\angle \mathrm{XLN}=100°,\angle \mathrm{YMN}=165°\mathrm{and}\angle \mathrm{ZNM}=95°$

$$\begin{gathered} \mathrm{Here},\angle \mathrm{XLN}+\angle \mathrm{MLN}=180°(\mathrm{Linear}\mathrm{pair}) \\ \Rightarrow 100°+\angle \mathrm{MLN}=180°\left(\mathrm{Given}\right) \\ \Rightarrow \angle \mathrm{MLN}=80° \\ \mathrm{Similarly},\angle \mathrm{LMN}+\angle \mathrm{NMY}=180°(\mathrm{Linear}\mathrm{pair}) \\ \Rightarrow \angle \mathrm{LMN}+165°=180°\left(\mathrm{Given}\right) \\ \Rightarrow \angle \mathrm{LMN}=15° \\ \mathrm{Now},\angle \mathrm{MLN}+\angle \mathrm{LMN}+\angle \mathrm{LNM}=180°(\mathrm{Angle}\mathrm{sum}\mathrm{property}) \\ \Rightarrow 80°+15°+\angle \mathrm{LNM}=180° \\ \Rightarrow \angle \mathrm{LNM}=180°-\left(80°+15°\right)=180°-95° \\ \Rightarrow \angle \mathrm{LNM}=85° \end{gathered}$$