Question

The vertical angle of isosceles triangle is of 60$°$. Find the base angles. Hence prove that the triangle is equilateral triangle.

Answer 1

$$\begin{gathered} \text{Suppose},\mathrm{in}△\mathrm{ABC},\mathrm{AB}=\mathrm{AC}. \\ \Rightarrow \angle \mathrm{B}=\angle \mathrm{C}\left(\text{A}\mathrm{ngle}\text{s}\mathrm{opposite}\mathrm{to}\mathrm{equal}\mathrm{sides}\mathrm{are}\mathrm{equal}\right) \\ \mathrm{Given}\angle \mathrm{A}=60°. \\ \mathrm{We}\mathrm{know}\mathrm{that} \\ \angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180° \\ \Rightarrow 60°+2\angle \mathrm{B}=180° \\ \Rightarrow 2\angle \mathrm{B}=120° \\ \Rightarrow \angle \mathrm{B}=60°=\angle \mathrm{C} \\ \mathrm{Since}\mathrm{all}\mathrm{the}\mathrm{angles}\mathrm{are}60°,\mathrm{it}\mathrm{is}\mathrm{an}\mathrm{equilateral}\mathrm{triangle}. \end{gathered}$$