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The image shows a geometric problem where you need to prove that \( AE = BE \). The diagram involves a parallelogram \( ABCD \) with a point \( E \) on side \( AB \) and a diagonal \( AC \).

### Given:
- \( ABCD \) is a parallelogram.
- \( E \) is a point on \( AB \).
- \( F \) is the intersection of diagonal \( AC \) with line \( DE \).

### To Prove:
\( AE = BE \).

### Proof Outline:
1. **Properties of a Parallelogram:**
   - In a parallelogram, opposite sides are equal and parallel. Hence, \( AB \parallel CD \) and \( AB = CD \), \( AD \parallel BC \) and \( AD = BC \).

2. **Triangles Sharing a Common Angle:**
   - Consider \( \triangle AFE \) and \( \triangle BFE \).
   - Since \( AB \parallel CD \) and \( DE \) intersects these parallel lines, angle \( \angle AEF = \angle BEF \) (Alternate Interior Angles).

3. **Triangles \( \triangle AFE \) and \( \triangle BFE \):**
   - \( \angle AFE = \angle BFE \) since they are vertically opposite angles.
   - \( EF \) is common to both triangles.

4. **Conclusion Using ASA Congruence:**
   - By the ASA (Angle-Side-Angle) Congruence Criterion, \( \triangle AFE \cong \triangle BFE \).
   - Therefore, \( AE = BE \), as corresponding parts of congruent triangles are equal.

### Conclusion:
\( AE \) is equal to \( BE \).

Would you like further details or have any questions? Here are some related questions to explore this topic further:

1. How does the property of alternate interior angles assist in proving congruence?
2. Why is the ASA Congruence Criterion used in this proof?
3. What other properties of parallelograms are useful in geometric proofs?
4. Can the same method be used to prove equal segments in other types of quadrilaterals?
5. What are the different criteria for triangle congruence?

**Tip:** When proving congruence or equality in geometry, always look for congruent triangles first, as this often simplifies the problem.

Learn how to prove AE = BE in parallelogram ABCD using geometric properties and triangle congruence. This problem involves understanding parallelogram properties and applying the ASA Congruence Criterion.

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