In the adjoining figure, ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA. Prove that

(i) △ABD ≅ △BAC

(ii) BD = AC

(iii) ∠ABD = ∠BAC

(i) In △ABD and △BAC,

AD = BC (Given)

∠BAD = ∠ABC (Given)

AB = AB (Common sides)

Hence, by SAS axiom △ABD ≅ △BAC.

(ii) As, △ABD ≅ △BAC.

We know that corresponding sides of congruent triangles are equal.

∴ BD = AC.

Hence, proved that BD = AC.

(iii) As, △ABD ≅ △BAC.

We know that corresponding angles of congruent triangles are equal.

∴ ∠ABD = ∠BAC.

Hence, proved that ∠ABD = ∠BAC.