# Ex.6.3 Q4 Triangles Solution - NCERT Maths Class 10

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## Question

In Figure \begin{align}\frac{QR}{QS}=\frac{QT}{PR}\end{align} and $$\angle 1 = \angle 2$$. Show that $$\Delta PQS\sim{\ }\Delta TQR$$ .

Diagram

## Text Solution

Reasoning:

As we know if one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

This criterion is referred to as the SAS (Side–Angle–Side) similarity criterion for two triangles.

Steps:

In $$\triangle PQR$$

\begin{align} \angle 1&=\angle 2\\ \Rightarrow \quad PR &= PQ \quad \text{(In a triangle sides opposite to equal angles are equal)} \end{align}

In $$\Delta PQR$$ and $$\Delta TQR$$

\begin{align} \angle PQS \,&=\, \angle TQR = \angle 1 \qquad \text{(same angle)}\\ \frac {{QR}}{QS} &=\frac {{QT}}{PQ} \qquad (\because \; \text{PR =PQ})\\\Rightarrow \triangle PQR& - \triangle TQR \qquad (\because\text{ SAS criterion})\end{align}

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