In the given figure, $R S = Q T$ and $Q S = R T .$ Then which of the following options is correct?

$P Q = P R$

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$P R = T R$

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$P Q = Q S$

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None of these

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Solution

The correct option is A
$P Q = P R$

In $Δ R S Q$ and $Δ Q T R$
$R S = Q T$ [given]
$R T = S Q$ [given]
$R Q = R Q$ [common]
Thus, $Δ R S Q ≅ Δ Q T R$ [by $S S S$ congruence rule]
$∠ 1 = ∠ 2$ [CPCT]
$∠ 3 = ∠ 4$ [vertically opposite angles]
$∠ 5 = ∠ 6$
Now, in $Δ P S Q$ and $Δ P T R$
$Q S = R T$ [given]

$∵ ∠ 1 + ∠ 7 = ∠ 2 + ∠ 8$ [supplementary angles]
So, $∠ 7 = ∠ 8$ [ $∵ ∠ 1 = ∠ 2$ ]
$∠ 6 = ∠ 5$ [already proved]

$∴ Δ P S Q ≅ Δ P T R$ [by $A A S$ congruence rule]
$\Rightarrow P Q = P R$ [CPCT]

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