Question

ABCD is a quadrilateral in which AD = BC and ∠ DAB = ∠ CBA (see Fig. 7.17). Prove that 

(i) ∆ ABD ≅ ∆ BAC 

(ii) BD = AC 

(iii) ∠ ABD = ∠ BAC.

Answer 1

We are given two triangles \( \triangle ABD \) and \( \triangle BAC \). Our objective is to prove that these two triangles are congruent and to demonstrate the equality of certain sides and angles.

Step 1: Understanding the given information: 

In the triangle \( \triangle ABD \), we are given the following facts:

• Side \( AD \) is equal to side \( BC \): \[ AD = BC \quad \text{(Given)} \]
• Angle \( \angle DAB \) is equal to angle \( \angle CBA \): \[ \angle DAB = \angle CBA \quad \text{(Given)} \]
• Side \( AB \) is common to both triangles: \[ AB = BA \quad \text{(Common side)} \]

Step 2: Applying the SAS (Side-Angle-Side) Congruence Rule:

We now use the SAS congruence rule to prove that the two triangles are congruent. The SAS rule states that if two sides and the included angle of one triangle are respectively equal to the corresponding two sides and the included angle of another triangle, then the two triangles are congruent.

In our case:

• Side \( AD \) is equal to side \( BC \): \( AD = BC \)
• Angle \( \angle DAB \) is equal to angle \( \angle CBA \): \( \angle DAB = \angle CBA \)
• Side \( AB \) is common to both triangles: \( AB = BA \)

Therefore, by the SAS congruence rule, we can conclude that:

\[ \triangle ABD \cong \triangle BAC \quad \text{(By SAS congruence rule)} \]

Step 3: Applying CPCT (Corresponding Parts of Congruent Triangles):

Since we have established that the two triangles are congruent, we can apply the principle of CPCT, which states that corresponding parts of congruent triangles are equal. Therefore, from \( \triangle ABD \cong \triangle BAC \), we deduce the following:

• The side \( BD \) is equal to side \( AC \): \[ BD = AC \quad \text{(By CPCT)} \]
• The angle \( \angle ABD \) is equal to angle \( \angle BAC \): \[ \angle ABD = \angle BAC \quad \text{(By CPCT)} \]

Conclusion: By applying the SAS congruence rule and the CPCT property, we have proven that:

• The sides \( BD \) and \( AC \) are equal.
• The angles \( \angle ABD \) and \( \angle BAC \) are equal.

This demonstrates that \( \triangle ABD \) and \( \triangle BAC \) are congruent, and certain corresponding parts of these triangles are indeed equal.