Math Problem Statement

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

Solution

We are given a parallelogram ABCDABCD in the diagram and need to prove the relation:

ABBE=CFFB.\frac{AB}{BE} = \frac{CF}{FB}.

Steps to solve:

  1. Identify triangles: Look at triangles ABEABE and CFBCFB, 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 ABDCAB \parallel DC (since ABCDABCD is a parallelogram) and that the line EFEF acts as a transversal between these two parallel lines. This creates alternate interior angles:

    • ABE=CFB\angle ABE = \angle CFB (since ABDCAB \parallel DC),
    • AEB=BFC\angle AEB = \angle BFC.

    These angle correspondences suggest that triangles ABEABE and CFBCFB are similar by AA similarity (Angle-Angle).

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

    ABBE=CFFB.\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 ABBE=CFFB\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!

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Math Problem Analysis

Mathematical Concepts

Geometry
Similar Triangles
Proportions

Formulas

AA similarity criterion
Proportion of sides in similar triangles

Theorems

AA similarity theorem
Properties of parallelograms

Suitable Grade Level

Grades 9-10