1. **State the problem:** Given that ABCD is a parallelogram, CE bisects \(\angle DEB\), and \(\angle ADE \cong \angle ABE\), prove that \(\triangle DEC \cong \triangle BEC\). 2. **Recall important properties:** - Opposite angles of a parallelogram are congruent. - If congruent angles have congruent parts subtracted, the remaining angles are congruent. - An angle bisector divides an angle into two congruent angles. - The AAS (Angle-Angle-Side) congruence criterion states that if two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. 3. **Step 1:** Given: ABCD is a parallelogram, CE bisects \(\angle DEB\), and \(\angle ADE \cong \angle ABE\). 4. **Step 2:** \(\angle ADC \cong \angle ABC\) because opposite angles of a parallelogram are congruent. 5. **Step 3:** \(\angle EDC \cong \angle EBC\) because congruent angles with congruent parts subtracted remain congruent. 6. **Step 4:** \(\angle DEC \cong \angle BEC\) because CE bisects \(\angle DEB\), dividing it into two congruent angles. 7. **Step 5 (missing statement):** \(\overline{DC} \cong \overline{BC}\) because opposite sides of a parallelogram are congruent. 8. **Step 6:** \(\triangle DEC \cong \triangle BEC\) by AAS (two angles and the included side are congruent). **Final answer:** \(\overline{DC} \cong \overline{BC}\) because opposite sides of a parallelogram are congruent.