ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA. Prove that

(i) Δ ABD ≅ Δ BAC

(ii) BD = AC

(iii) ∠ABD = ∠BAC

Given :

AD = BC and ∠DAB = ∠CBA

(i) In △ ABD and △ BAC,

⇒ AD = BC (Given)

⇒ ∠DAB = ∠CBA (Given)

⇒ AB = BA (Common side)

∴ △ ABD ≅ △ BAC (By S.A.S. congruence rule)

Hence, proved that △ ABD ≅ △ BAC.

(ii) As,

△ ABD ≅ △ BAC,

We know that,

Corresponding parts of congruent triangles are equal.

∴ BD = AC (By C.P.C.T.)

Hence, proved that BD = AC.

(iii) As,

△ ABD ≅ △ BAC,

We know that,

Corresponding parts of congruent triangles are equal.

⇒ ∠ABD = ∠BAC (By C.P.C.T.)

Hence, proved that ∠ABD = ∠BAC.