A B C D is a quadrilateral in which A D=B C and D A B= C B A : Pr… - Vidyadip

Question

$A B C D$ is a quadrilateral in which $A D=B C$ and $\angle D A B=\angle C B A$ : Prove that:
i. $\triangle A B D \cong \triangle B A C$
ii. $B D=A C$
iii. $\angle ABD =\angle BAC$

✓

Answer

In quadrilateral $ACBD$ , we have $AD = BC$ and $\angle DAB =\angle CBA$
i. In $\triangle ABC$ and $\triangle BAC$,
$A D=B C$ (Given)
$\angle DAB =\angle CBA$ (Given)
$AB = AB$ (Common)
$\triangle A B D \cong \triangle B A C \ldots[B y$ $SAS$ Congruence]
ii. Since $\triangle A B D \cong \triangle B A C$
$\Rightarrow BD = AC$ [By $C.P.C.T.]$
iii.Since $\triangle ABD ≅ \triangle BAC$
$\Rightarrow \angle ABD = \angle BAC$ [By $C.P.C.T.]$

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