To determine the rigid transformation that maps triangle \( \triangle ABC \) to triangle \( \triangle EDC \), we can analyze the properties given in the question:
- Both triangles share vertex \( C \).
- Sides \( BC \) and \( CD \) are congruent.
- Angles \( A \) and \( E \) are congruent.
- Angle \( C \) is congruent in both triangles.
Given that the triangles are congruent by the criteria of Angle-Side-Angle (ASA), the transformation that maps one triangle to the other will be either a reflection, rotation, or translation.
Since the triangles share a common vertex \( C \) and sides \( BC \) and \( CD \) are congruent, the most likely transformation is a reflection over the line that bisects angle \( C \), or a rotation around point \( C \) if the triangles are positioned such that a rotation aligns them.
However, since the problem does not specify rotation or the position of each vertex in detail, and both triangles share a common side and vertex, the most suitable option is typically reflection.
Therefore, the answer is: reflection.