Question

$PQRS$ is a square and $\Delta TSR$ is an isosceles triangle with $TS=TR$. Prove that $PT=QT$.

Answer 1

As opposite angles to equal sides of a triangle are equal, $\Rightarrow \angle TSR=\angle TRS$

In $\Delta TSP$ and $\Delta TRQ,$

TS = TR (Given)

$\angle TSP=\angle TRQ(TSP\angle ={90}^{\circ }+\angle TSR={90}^{\circ }+\angle TRS=\angle TRQ)$

SP = RQ (sides of square are equal)

$\Rightarrow$ By S.A.S concurrency, $\Delta$TSP $\cong \Delta$ TRQ

As in congruent triangles , corresponding sides are equal, $\Rightarrow$ PT = TQ