In the adjoining figure AC = AE, AB = AD and ∠BAD = ∠EAC, then BC is -

To determine the relationship between line segments BC and DE in the provided figure, we use the principles of triangle congruence.

Step-by-Step Solution
Analyze the Given Information:

AC = AE

AB = AD

Angle BAD = Angle EAC

Establish Equal Angles:

Add the common angle Angle DAC to both sides of the given equation (Angle BAD = Angle EAC).

This gives us: Angle BAD + Angle DAC = Angle EAC + Angle DAC.

Therefore, Angle BAC = Angle DAE.

Prove Triangle Congruence:

In triangles ABC and ADE:

AB = AD (Given)

Angle BAC = Angle DAE (Proved above)

AC = AE (Given)

By the Side-Angle-Side (SAS) congruence criterion, triangle ABC is congruent to triangle ADE.

Conclude the Result:

Since the triangles are congruent, their corresponding sides must be equal by CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Therefore, BC = DE.