In the given figure, $∠ x = ∠ y$ and $P Q = Q R$ prove that $P E = R S$

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Solution

$\begin{matrix} I n f i g . ∠ x = ∠ y a n d P Q = P R i n Δ P Q R P Q = Q R ∠ Q R P = ∠ Q P R \end{matrix}$
Angles opposite to equal sides of triangle are equal
$\begin{matrix} ∠ E R P = ∠ S P R \to (i) i n Δ P E R a n d Δ R S P ∠ E R P = ∠ S P R ∠ R E P = ∠ P S R P R = R P Δ P E R ≅ Δ R S P A A S R u l e P E = R S \end{matrix}$

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