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We are given a parallelogram \( ABCD \) in the diagram and need to prove the relation:

\[
\frac{AB}{BE} = \frac{CF}{FB}.
\]

### Steps to solve:

1. **Identify triangles**: Look at triangles \( ABE \) and \( CFB \), as these are crucial to the proof. These triangles share angles and sides that we can use to relate the given segments.

2. **Similar triangles**: From the figure, observe that \( AB \parallel DC \) (since \( ABCD \) is a parallelogram) and that the line \( EF \) acts as a transversal between these two parallel lines. This creates alternate interior angles:

   - \( \angle ABE = \angle CFB \) (since \( AB \parallel DC \)),
   - \( \angle AEB = \angle BFC \).

   These angle correspondences suggest that triangles \( ABE \) and \( CFB \) are similar by **AA similarity** (Angle-Angle).

3. **Ratio of corresponding sides**: Since the triangles are similar, the corresponding sides are proportional:

   \[
   \frac{AB}{BE} = \frac{CF}{FB}.
   \]

   This proves the desired relation.

### Conclusion:

By using the properties of similar triangles and the parallel lines in the parallelogram, we have proved the equation \( \frac{AB}{BE} = \frac{CF}{FB} \).

Let me know if you want further details or explanations!

### 5 Related Questions:

1. What are the properties of a parallelogram that are useful in proofs?
2. How do we prove two triangles are similar using the AA similarity criterion?
3. What other ways can you prove triangles are similar, apart from AA similarity?
4. How are ratios of corresponding sides in similar triangles used in geometric proofs?
5. What role do transversal lines play in identifying similar triangles?

### Tip:
When working with parallelograms, always look for parallel lines and transversal lines to identify possible similar triangles!

In the given figure, ABCD is a parallelogram. Prove that AB/BE = CF/FB.

Learn how to prove that AB/BE = CF/FB in a parallelogram ABCD using similar triangles and the AA similarity criterion. This problem covers key geometric concepts, including properties of parallelograms and proportions of corresponding sides. Suitable for students in Grades 9-10.

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