Given:

• $ABCD$ is a parallelogram.
• $E$ is the midpoint of $AB$.
• $DE$ bisects $\angle D$.

To Prove:

• $CE$ bisects $\angle C$.

Proof:

1. Basic properties of a parallelogram: In a parallelogram:

   • Opposite sides are equal, i.e., $AB = CD$ and $AD = BC$.
   • Opposite angles are equal, i.e., $\angle A = \angle C$ and $\angle B = \angle D$.

2. Midpoint and line bisector: Since $E$ is the midpoint of $AB$, we have: $AE = EB$ Also, $DE$ bisects $\angle D$, meaning: $\angle ADE = \angle CDE$

3. Considering triangles ADE and CDE:

   • In triangles $ADE$ and $CDE$:
     • $AD = BC$ (Opposite sides of a parallelogram are equal).
     • $DE = DE$ (Common side).
     • $\angle ADE = \angle CDE$ (Given that $DE$ bisects $\angle D$).

   By SAS (Side-Angle-Side) congruence criterion, triangles $ADE$ and $CDE$ are congruent: $\triangle ADE \cong \triangle CDE$

4. Consequences of congruence: From the congruence of triangles $ADE$ and $CDE$, we can deduce that: $\angle AEC = \angle BEC$ Therefore, $CE$ bisects $\angle C$, since the two adjacent angles at $E$ formed by $CE$ are equal.

Conclusion:

Thus, $CE$ bisects $\angle C$, as required.

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Would you like a more detailed explanation or clarification on any step?

Here are some related questions you might want to explore:

1. How do you prove the properties of a parallelogram?
2. What is the significance of angle bisectors in triangle congruence?
3. Can the midpoint theorem be applied in any quadrilateral?
4. What is the SAS criterion for congruence, and when is it used?
5. How do angle bisectors relate to the properties of cyclic quadrilaterals?

Tip: In any geometric proof, identifying congruent triangles can help establish equal angles or sides, which are crucial for proving bisectors and symmetry.