Question

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

1. ΔABD ≅ ΔBAC
2. BD = AC
3. ∠ABD = ∠BAC

Answer 1

Given:

ABCD is a quadrilateral with AD = BC and ∠DAB = ∠CBA

To Prove:

1. ΔABD ≅ ΔBAC
2. BD = AC
3. ∠ABD = ∠BAC

Proof [Step-wise]:

1. Consider triangles ΔABD and ΔBAC.

2. AB is common to both triangles.   ...(Common side)

3. AD = BC.   ...(Given)

4. ∠DAB = ∠CBA.   ...(Given)

5. In ΔABD and ΔBAC, we have two sides and the included angle equal:

AD = BC

AB = AB

And ∠DAB = ∠CBA

Hence, by the SAS congruence criterion, 

ΔABD ≅ ΔBAC   ...(SAS congruence → corresponding parts equal)

6. From the congruence (CPCTC) corresponding sides are equal.

So BD = AC.

7. From the congruence (CPCTC) corresponding angles are equal.

So ∠ABD = ∠BAC.