Question

Prove that the angles opposite to equal sides of a triangle are equal.

Answer 1

In triangle PQR, $PQ=QR.$
To prove: $\angle Q=\angle R$
Proof:
Draw a bisector of $\angle P$and let S be the point of intersection.

$$\begin{gathered} \mathrm{In}△\mathrm{QPS}\mathrm{and}△\mathrm{RPS},\mathrm{we}\mathrm{have}: \\ \mathrm{PQ}=\mathrm{PR}\left(\mathrm{Given}\right) \\ \angle \mathrm{QPS}=\angle \mathrm{RPS}(\mathrm{By}\mathrm{construction}) \\ \mathrm{PS}=\mathrm{PS}\left(\mathrm{Common}\right) \\ △\mathrm{QPS}\cong △\mathrm{RPS}(\mathrm{SAS}\mathrm{criterion}) \\ \angle \mathrm{PQS}=\angle \mathrm{PRS}\left(\mathrm{CPCT}\right) \end{gathered}$$
∴ $\angle Q=\angle R$
Hence Proved.