Question

In fig 2.42, $△$SMA $~$$△$RTQ. Using the information given in the figure, find the measurement of the remaining angles and sides.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}△\mathrm{SMA}\hspace{0.17em}~△\mathrm{RTQ} \\ \mathrm{By}\mathrm{similarity}\mathrm{correspondence},\mathrm{we}\mathrm{have}: \\ \frac{\mathrm{SM}}{\mathrm{RT}}=\frac{\mathrm{MA}}{\mathrm{TQ}}=\frac{\mathrm{SA}}{\mathrm{RQ}}\mathrm{and}\angle \mathrm{S}\cong \angle \mathrm{R},\angle \mathrm{M}\cong \angle \mathrm{T},\angle \mathrm{A}\cong \angle \mathrm{Q} \\ \Rightarrow \frac{8}{6}=\frac{10}{\mathrm{TQ}};\angle \mathrm{M}=\angle \mathrm{T}=110°\mathrm{and}\angle \mathrm{A}=\angle \mathrm{Q}=27° \\ \mathrm{Also},\mathrm{TQ}=7.5 \\ \mathrm{In}△\mathrm{SMA},\mathrm{we}\mathrm{have}: \\ \angle \mathrm{S}+\angle \mathrm{M}+\angle \mathrm{A}=180° \\ \Rightarrow \angle \mathrm{S}+110°+27°=180° \\ \Rightarrow \angle \mathrm{S}=43°=\angle \mathrm{R} \\ \mathrm{Thus},\angle \mathrm{S}=\angle \mathrm{R}=43°,\mathrm{TQ}=7.5,\angle \mathrm{A}=27°\mathrm{and}\angle \mathrm{T}=110° \end{gathered}$$