The triangles ∆BAD ≅ ∆CDA are congruent.
The proof of the given problem is as follows:
Given:
BD ≅ AC
BA || AD
AD || DC
To prove:
∆BAD ≅ ∆CDA
Step 1: Identify congruent angles:
∠BAD ≅ ∠CDA: Since BA || AD and AD || DC, then ∠BAD and ∠CDA are alternate interior angles, which are congruent by definition.
∠ABD ≅ ∠ACD: Since BD ≅ AC, then triangles ABD and ACD are isosceles triangles. This means that their base angles are congruent. Therefore, ∠ABD ≅ ∠ACD.
Step 2: Apply the Angle-Angle-Side (AAS) Congruence Postulate:
We have two pairs of congruent angles (∠BAD ≅ ∠CDA and ∠ABD ≅ ∠ACD) and the congruent side AD between them. Therefore, by the AAS Congruence Postulate, we can conclude that:
∆BAD ≅ ∆CDA
Therefore, triangles BAD and CDA are congruent.