Question

In Fig 3.18 Seg SR and seg QT are the altitude of isosceles triangle PQR where seg PQ $\cong$ seg PR show that the altitudes are congruent.

Answer 1

$$\begin{gathered} \mathrm{Given}:\mathrm{SR}\mathrm{and}\mathrm{QT}\mathrm{are}\mathrm{the}\mathrm{altitudes}\mathrm{of}\mathrm{the}\mathrm{triangle}\Rightarrow \angle \mathrm{QTP}=\angle \mathrm{RSP}=90° \\ seg\mathrm{PQ}\simeq seg\mathrm{PR}\mathrm{or}\mathrm{PQ}=\mathrm{PR} \\ \mathrm{In}△\mathrm{QTP}\mathrm{and}△\mathrm{RSP}, \\ \angle \mathrm{QTP}=\angle \mathrm{RSP}=90°\left(\mathrm{given}\right) \\ \angle \mathrm{QPT}=\angle \mathrm{RPS}\left(\mathrm{common}\right) \\ \mathrm{PQ}=\mathrm{PR}\left(\mathrm{given}\right) \\ \therefore △\mathrm{QTP}\cong △\mathrm{RSP}(\mathrm{AAS}\mathrm{test}) \\ \mathrm{QT}=\mathrm{RS}(\mathrm{by}\mathrm{c}.\mathrm{s}.\mathrm{c}.\mathrm{t}) \\ \text{Or},\mathrm{the}\mathrm{altitudes}\mathrm{of}\mathrm{the}\mathrm{triangle}\mathrm{are}\mathrm{congruent}\mathrm{to}\mathrm{each}\mathrm{other}. \end{gathered}$$