Question

The measures of the angles of a quadrilateral taken in order are as 1:2:3:4. Prove that it is a trapezium.

Answer 1

By the angle addition property of a quadrilateral, for quadrilateral PQRS, we have:
$\angle \mathrm{P}+\angle \mathrm{Q}+\angle \mathrm{R}+\angle \mathrm{S}=360°$

$$\begin{gathered} \mathrm{Let}: \\ \angle \mathrm{P}=x,\angle \mathrm{Q}=2x,\angle \mathrm{R}=3x\mathrm{and}\angle \mathrm{S}=4x \\ \mathrm{Now},\mathrm{we}\mathrm{have}: \\ x+2x+3x+4x=360° \\ \Rightarrow 10x=360° \\ \Rightarrow x=36° \\ \mathrm{Thus},\mathrm{the}\mathrm{measures}\mathrm{of}\mathrm{the}\mathrm{angles}\mathrm{of}\mathrm{quadrilateral}\mathrm{PQRS}\mathrm{are}: \\ \angle \mathrm{P}=36° \\ \angle \mathrm{Q}=2\times 36°=72° \\ \angle \mathrm{R}=3\times 36°=108° \\ \mathrm{and}\angle \mathrm{S}=4\times 36°=144° \\ \mathrm{Here}, \\ \angle \mathrm{Q}+\angle \mathrm{R}=180° \\ \mathrm{It}\mathrm{means}\mathrm{that}\mathrm{the}\mathrm{angles}\mathrm{on}\mathrm{the}\mathrm{same}\mathrm{side}\mathrm{of}\mathrm{a}\mathrm{transversal}\mathrm{line}\mathrm{are}\mathrm{supplementary}. \\ \therefore \mathrm{SR}\parallel \mathrm{PQ} \\ \mathrm{Hence},\mathrm{we}\mathrm{can}\mathrm{say}\mathrm{that}\mathrm{PQRS}\mathrm{is}\mathrm{a}\mathrm{trapezium} \end{gathered}$$