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The given problem asks us to prove that triangles \( \Delta ACF \) and \( \Delta CBE \) are congruent, given the following conditions:

1. \( \triangle ABC \) is an isosceles triangle with \( AC = BC \).
2. Point \( D \) is an extension of line segment \( AB \).
3. Point \( E \) lies on line \( CD \), and \( BE = DE \).
4. Point \( F \) lies on line \( CD \) such that \( AF \parallel BE \).

Let's break down the information and approach the proof:

### Step 1: Identify and label the relevant angles and sides.
- Since \( AC = BC \) (isosceles triangle), we know that \( \angle CAB = \angle CBA \).
- \( AF \parallel BE \) implies that corresponding angles are equal, i.e., \( \angle CAF = \angle CBE \).

### Step 2: Check the given side equality.
- We are given that \( BE = DE \), which helps in relating parts of the triangles involved.

### Step 3: Prove the triangles are congruent.
To prove that \( \Delta ACF \cong \Delta CBE \), we can use the **ASA (Angle-Side-Angle)** congruence criterion:
1. \( \angle CAF = \angle CBE \) (corresponding angles due to parallel lines).
2. \( \angle ACF = \angle BCE \) (angles opposite to equal sides in the isosceles triangle).
3. Side \( AC = BC \) (given by the isosceles property).

Thus, by ASA, the triangles \( \Delta ACF \) and \( \Delta CBE \) are congruent.

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Would you like further details or have any questions? 

Here are 5 related questions:
1. What other methods could we use to prove triangle congruence?
2. How does the parallel condition \( AF \parallel BE \) help in proving the congruence?
3. Could \( \Delta ACF \) and \( \Delta CBE \) be proven congruent using any other properties of isosceles triangles?
4. Why is ASA (Angle-Side-Angle) suitable for this proof over other congruence rules like SSS or SAS?
5. What additional information would invalidate the congruence between \( \Delta ACF \) and \( \Delta CBE \)?

**Tip:** When proving triangle congruence, always look for shared angles, sides, or parallel lines to simplify your approach.

Prove that triangles ΔACF and ΔCBE are congruent given that ΔABC is an isosceles triangle with AC = BC, D is an extension of AB, E lies on CD with BE = DE, and F lies on CD such that AF || BE.

Learn how to prove that triangles ΔACF and ΔCBE are congruent using the ASA (Angle-Side-Angle) criterion. This problem involves geometric concepts such as isosceles triangles, parallel lines, and properties of angles. Suitable for high school students in Grades 9-12.

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