Question

In Figure, \(∠\)PQR = \(∠\)PRQ, then prove that \(∠\)PQS = \(∠\)PRT.

Answer 1

Given: 

In the given figure, \( ST \) is a straight line and ray \( QP \) stands on it.

Step 1: Since \( ST \) is a straight line and ray \( QP \) stands on it:

\[ \angle PQS + \angle PQR = 180^\circ \quad \text{(Linear Pair)} \]

Therefore:

\[ \angle PQR = 180^\circ - \angle PQS \quad \text{(Equation 1)} \]

Step 2: Similarly, for the straight line \( ST \) and ray \( PR \) standing on it:

\[ \angle PRT + \angle PRQ = 180^\circ \quad \text{(Linear Pair)} \]

Thus:

\[ \angle PRQ = 180^\circ - \angle PRT \quad \text{(Equation 2)} \]

Step 3: It is given that \( \angle PQR = \angle PRQ \), so equating equations (1) and (2), we get:

\[ 180^\circ - \angle PQS = 180^\circ - \angle PRT \]

Therefore:

\[ \angle PQS = \angle PRT \]

Conclusion: We have proved that \( \angle PQS = \angle PRT \).