Question

In the given figure, $\frac{QT}{PR}=\frac{QR}{QS}$ and $\angle 1=\angle 2$.

Which of the following is true?

• A. $△PQS\sim △TQR$
• B. $△PSQ\sim △TQR$
• C. $△QPS\sim △TQR$
• D. $△PQS\sim △QTR$

Answer 1

The correct option is A $△PQS\sim △TQR$

$\angle 1=\angle 2$ (given)

$\Rightarrow$ PR = PQ $...\left(i\right)$

(Sides opposite to equal angles in $△QRP$)

Also, $\frac{QT}{PR}=\frac{QR}{QS}$ (given) ...(ii)

From (i) and (ii), we have

$\frac{QT}{PQ}=\frac{QR}{QS}$

$\Rightarrow \frac{QP}{QT}=\frac{QS}{QR}$ ...(iii)

Now, in $△PQR$ and $△TQR$,

$\angle Q=\angle Q$ (Common in both the triangles)

$\frac{PQ}{TQ}=\frac{QS}{QR}$

$\therefore △PQS\sim △TQR$ (by SAS similarity criterion)