Question

In the given fig 2.64, $∆$MNS is an isosceles triangle in which MN = MS and ray MK bisects $\angle$RMS. Prove that ray MK || Side NS.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}\mathrm{MN}=\mathrm{MS} \\ \therefore \angle \mathrm{MNS}=\angle \mathrm{MSN}(\mathrm{Isosceles}\mathrm{angle}\mathrm{property}) \\ \mathrm{Let}\angle \mathrm{MNS}=\angle \mathrm{MSN}=y \\ \mathrm{Ray}\mathrm{MK}\mathrm{bisects}\angle \mathrm{RMS}.\mathrm{Now},\mathrm{let}\angle \mathrm{RMK}=\angle \mathrm{KMS}=x \\ \angle \mathrm{RMS}=\angle \mathrm{MNS}+\angle \mathrm{MSN}(\mathrm{Exterior}\mathrm{angle}\mathrm{property}) \\ \Rightarrow 2x=y+y \\ \Rightarrow 2x=2y \\ \Rightarrow x=y \\ \therefore \mathrm{Ray}\mathrm{MK}\parallel \mathrm{side}\mathrm{NS}(\mathrm{As}\mathrm{they}\mathrm{form}\mathrm{a}\mathrm{pair}\mathrm{of}\mathrm{alternate}\mathrm{angles}) \end{gathered}$$