In Fig. $P Q R = P R Q,$ then prove that $P Q S = P R T .$

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Solution

Given, $∠ P Q R = ∠ P R Q$
Considering $S T$ is a straight line, so the sum of all angles made on it is $180 ° .$
$\Rightarrow ∠ P Q S + ∠ P Q R = 180 ° . . . (i)$
also, $∠ P R Q + ∠ P R T = 180 ° . . . (i i)$
On equating both the equations because RHS of both the equation is equal. So, LHS will also be equal.
$\Rightarrow ∠ P Q S + ∠ P Q R = ∠ P R Q + ∠ P R T$
$\Rightarrow ∠ P Q S + ∠ P Q R = ∠ P Q R + ∠ P R T$
$\Rightarrow ∠ P Q S = ∠ P R T$ [ $∠ P Q R = ∠ P R Q$ ]
Hence proved that $∠ P Q S = ∠ P R T .$

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