In the below quadrilateral ABCD, if AD = BC and $∠ D A B$ = $∠ C B A$ , then what is the relation between $∠ A B D$ and $∠ B A C$ ?

$∠ A B D$ = $∠ B A C$

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$∠ A B D$ < $∠ B A C$

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$∠ A B D$ > $∠ B A C$

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$∠ D A B$ = $∠ B A C$

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Solution

The correct option is A
$∠ A B D$ = $∠ B A C$

In $Δ A B D and Δ B A C$ ,
AD = BC (Given)
$∠ B A D$ = $∠ C B A$ (Given)
AB = BA (Side common to both triangles)
Hence $Δ A B D ≅ Δ B A C$
(by SAS congruence condition).
Thus, $∠ A B D$ = $∠ B A C$
(corresponding parts of congruent triangles).

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