In Fig, $10.18, P Q R S$ is quadrilateral and $T$ and $U$ are respectively points on $P S$ and $R S$ such that $P Q = R Q, ∠ P Q T = ∠ R Q U$ and $∠ T Q S = ∠ U Q S$ . Prove that $Q T = Q U$ .

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Solution

Let $P Q R S$ be a quadrilateral.

Let $T, U$ are points on $P S, R S$ respectively such that $P Q = R Q, ∠ P Q T = ∠ R Q U, ∠ T Q S = ∠ U Q S$

Since $∠ P Q T = ∠ R Q U, ∠ T Q S = ∠ U Q S$

$\Rightarrow ∠ P Q T + ∠ T Q S = ∠ R Q U + ∠ U Q S$

$\Rightarrow ∠ P Q S = ∠ R Q S$

In $Δ P Q S, Δ R Q S$ we have,

$P Q = R Q$ (given)
$Q S = Q S$ (common)
$∠ P Q S = ∠ R Q S$

Then $Δ P Q S ≅ Δ R Q S$ [SAS congruency]

$\Rightarrow ∠ P S Q = ∠ Q S R$ or $∠ T S Q = ∠ U S Q$

Again in $Δ T Q S, Δ S Q U$ we have,

$∠ T Q S = ∠ S Q U$ (given)
$Q S = Q S$ (common)
$∠ T S Q = ∠ U S Q$

Then $Δ T Q S ≅ Δ S Q U$ [ASA congruency]

$\Rightarrow Q T = Q U$

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In Fig, 10.18,PQRS is quadrilateral and T and U are respectively points on PS and RS such that PQ=RQ,∠PQT=∠RQU and ∠TQS=∠UQS. Prove that QT=QU.

In Fig, $10.18, P Q R S$ is quadrilateral and $T$ and $U$ are respectively points on $P S$ and $R S$ such that $P Q = R Q, ∠ P Q T = ∠ R Q U$ and $∠ T Q S = ∠ U Q S$ . Prove that $Q T = Q U$ .