1. **State the problem:** Given that $\angle FCE \cong \angle DEC$ and $\angle FCD \cong \angle DEF$, prove that quadrilateral $CDEF$ is a parallelogram. 2. **Recall the definition of a parallelogram:** A quadrilateral is a parallelogram if both pairs of opposite sides are parallel. 3. **Use the given angle congruences:** Since $\angle FCE \cong \angle DEC$ and $\angle FCD \cong \angle DEF$, these pairs of angles are alternate interior angles formed by transversal $FE$ intersecting lines $CD$ and $DE$. 4. **Apply the Alternate Interior Angles Theorem:** If alternate interior angles are congruent, then the lines are parallel. Therefore, $CD \parallel DE$ and $CF \parallel EF$. 5. **Conclude that $CDEF$ is a parallelogram:** Since both pairs of opposite sides are parallel, by definition, $CDEF$ is a parallelogram. **Final answer:** Quadrilateral $CDEF$ is a parallelogram because both pairs of opposite sides are parallel as shown by the congruent alternate interior angles.