Question

ABCD is a quadrilateral in which AD = BC
and ∠DAB = ∠CBA. Prove that
(i) ΔABD ≅ ΔBAC (ii) BD = AC
(iii) ∠ABD = ∠BAC

Answer 1

To Prove:

(1) ∆ABD ≅ ∆BAC

(2) BD = AC

(3) ∠ABD = ∠BAC

Proof:

(i) In ∆ABD and ∆BAC

AD = BC (Data) ∠DAB = ∠CBA

(Data) AB is common SAS postulate.

∴ ∆ABD ≅ ∆BAC.

(ii) ∆ABD ≅ ∆BAC.

∵ Corresponding sides are equal.

∴ BD = AC.

(iii) ∆ABD ≅ ∆BAC.

(Proved) Equal sides of Adjacent angles are equal.

As we have AD = BC, Adjacent angle for AD is ABD Adjacent angle for BC is BAC

∴∠ABD = ∠BAC.

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Answer 2

Answer:

Solution:

As per given in the question,

∠DAB = ∠CBA and AD = BC.

(i) ΔABD and ΔBAC are similar by SAS congruency as

AB = BA (common arm)

∠DAB = ∠CBA and AD = BC (given)

So, triangles ABD and BAC are similar

i.e. ΔABD ≅ ΔBAC. (Hence proved).

(ii) As it is already proved,

ΔABD ≅ ΔBAC

So,

BD = AC (by CPCT)

(iii) Since ΔABD ≅ ΔBAC

So, the angles,

∠ABD = ∠BAC (by CPCT).