Question

In figure, side $QR$ of $\Delta PQR$ is produced on both sides such that $\angle PQS=\angle PRT$. Prove that $PQ=PR$.

Answer 1

In $\Delta PQR$
$\angle PQS+\angle PQR={180}^0$ (linear pair angles)
$\angle PQS+\angle q={180}^0$
$\angle q={180}^0–\angle PQS$ …(i)
Also $\angle r={180}^0–\angle PRT$ …(ii)
But $\angle PQS=\angle PRT$ (given) …(iii)
From (iii), eqn (i) becomes
$\angle q={180}^0–\angle PRT$ …(iv)
Now from (ii) and (iv), we have
$\angle q=\angle r$
i.e., $\angle PQR=\angle PRQ$
$PR=PQ$ (converse of isosceles $A$ theorem)
Hence proved.