In the figure, it is given that $B R = P C$ and $∠ A C B = ∠ Q P R$ and $A B ∥ P Q$ . Prove that $A C = Q R$ .

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Solution

Given $B R = P C$ and $∠ A C B = ∠ Q R P$ and $A B | | P Q$
Join the points $P$ and $B$
$\Rightarrow B C = P R$
Then $∠ C B A = ∠ R P Q$
$∠ A C B = ∠ Q R P$ (given )
$∴$ In $△ A B C$ and $△ P Q R$
$∠ B A C = ∠ Q P R$
$∴△ A B C ≅ P Q R$ ...... [By AAA condition of similarity]
$\Rightarrow A C = Q R$

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