Question

In given figure 5.3 $\square$ PQRS is an Isosceles trapezium. Side PQ || side SR and $∆$RQE is an equilateral triangle; find the measures of all the angles of the trapezium.

Answer 1

$$\begin{gathered} \mathrm{Since}△\mathrm{REQ}\mathrm{is}\mathrm{an}\mathrm{equilateral}\mathrm{triangle},\angle \mathrm{RQP}=\angle \mathrm{REQ}=60°. \\ \angle \mathrm{SPQ}=\angle \mathrm{RQP}\left(\mathrm{Base}\mathrm{angles}\mathrm{of}\mathrm{isosceles}\mathrm{trapezium}\mathrm{are}\mathrm{congruent}\right) \\ \therefore \angle \mathrm{REQ}=\angle \mathrm{SPQ}=60°(\mathrm{Corresponding}\mathrm{angles}) \end{gathered}$$

$$\begin{gathered} \angle \mathrm{P}+\angle \mathrm{S}=180°(\mathrm{PQ}\parallel \mathrm{SR}\mathrm{and}\mathrm{angles}\mathrm{on}\mathrm{the}\mathrm{same}\mathrm{side}\mathrm{of}\mathrm{a}\mathrm{transversal}\mathrm{line}\mathrm{are}\mathrm{supplementary}.) \\ \mathrm{Or}, \\ \angle \mathrm{S}=180°-60°=120° \\ \mathrm{Similarly}, \\ \angle \mathrm{R}+\angle \mathrm{Q}=180° \\ \angle \mathrm{R}=180°-60°=120° \\ \mathrm{Therefore},\mathrm{angles}\mathrm{of}\mathrm{the}\mathrm{trapezium}\mathrm{are}60°,60°,120°\mathrm{and}120°. \end{gathered}$$