Question

In given figure, IF $\frac{QT}{PR}=\frac{QR}{QS}$ AND $\angle 1=\angle 2$. Prove that $△PQS\sim △TQR$.

Answer 1

We have,

$\frac{QT}{PR}=\frac{QR}{QS}$ [Given]

$\Rightarrow$ $\frac{QT}{QR}=\frac{PR}{QS}$.......(i)

We also have,

$\angle 1=\angle 2$ [Given]

$\Rightarrow$PR=PQ [Sides opposite to equal angles are equal].....(ii)

From (i) and (ii), we get

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

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

Thus, in triangles PQS and TQR, we have

$\frac{PQ}{QT}=\frac{QS}{QR}$ and $\angle PQS=\angle TQR=\angle Q$

So, by SAS-criterion of similarity, we have

$△PQS\sim △TQR$. $\left[Henceproved\right]$