Problem Breakdown:

We are given the following:

1. $AB \parallel CD$ (parallel lines)
2. $\angle A = 128^\circ$
3. $\angle E = 144^\circ$
4. The task is to find $\angle FCD$.

Solution:

Since $AB \parallel CD$, the properties of parallel lines and their transversals will be used.

Step 1: Analyze the diagram

• The given diagram shows a transversal $EF$ cutting $AB$ and $CD$.
• The angle $\angle E = 144^\circ$ is an exterior angle, while $\angle A = 128^\circ$ is an interior angle at the top.

Step 2: Relationship of angles

• Using the property of parallel lines, the sum of the interior angle on the same side of the transversal is $180^\circ$.
  Thus,
  $\angle A + \angle FCE = 180^\circ$
  Substituting $\angle A = 128^\circ$,
  $128^\circ + \angle FCE = 180^\circ$
  $\angle FCE = 52^\circ$

Step 3: Find $\angle FCD$

• The total straight angle at point $C$ is $180^\circ$, and it is divided into two angles: $\angle FCE$ and $\angle FCD$.
  Thus,
  $\angle FCD = 180^\circ - \angle FCE$
  Substituting $\angle FCE = 52^\circ$,
  $\angle FCD = 180^\circ - 52^\circ = 128^\circ$

Final Answer:

The value of $\angle FCD$ is $\mathbf{128^\circ}$.