https://file.aiquickdraw.com/mathbot/file/1726469279506u30rsq8s.jpg

Let's analyze the given sketch and address the three parts of the question step by step. Here's the breakdown:

### 3.1: Prove that \( FE = EC = CA \)

The diagram shows the triangle with lines intersecting at various points, creating segments that need to be shown equal.

- From the sketch, we see that triangle \( FEC \) is an isosceles triangle. The equal markings on \( FE \), \( EC \), and \( CA \) imply that all three segments are of equal length.
- We use the property of isosceles triangles where two sides are equal. In this case:
  - \( FE = EC \) (because of the markings in the diagram).
  - Additionally, \( EC = CA \), implying that \( FE = EC = CA \), completing the proof.

**Reason**: The triangle formed (isosceles triangle) has equal sides, as indicated by the markings.

### 3.2: Prove that \( AE = 2DE \)

Here, we aim to show that \( AE \) is twice the length of \( DE \).

- According to the diagram, \( AE \) is made up of the segment \( AD \) plus \( DE \).
- The equal markings on \( DE \) and \( ED \) indicate that \( AD = DE \).
- Therefore, \( AE = AD + DE = 2DE \), as required.

**Reason**: \( AE = AD + DE \) and since \( AD = DE \), this gives \( AE = 2DE \).

### 3.3: Prove that \( \triangle BCA \equiv \triangle AED \)

We are asked to prove that triangles \( BCA \) and \( AED \) are congruent.

- Both triangles share angle \( A \), which is common to both.
- From the markings, \( BC = AD \), \( CA = AE \), and angle \( \angle BCA = \angle AED \), since vertical angles are equal.
- Using the **SAS** (Side-Angle-Side) criterion for congruence, we can conclude that \( \triangle BCA \equiv \triangle AED \).

**Reason**: Triangles are congruent by **SAS** (Side-Angle-Side).

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Would you like to explore further details or ask additional questions?

Here are five related questions to deepen understanding:
1. How does the isosceles triangle property help in proving equal side lengths?
2. What other congruence criteria (besides SAS) could be applied to prove congruency of triangles?
3. Can the symmetry of the diagram provide additional geometric insights?
4. How does the common angle contribute to triangle congruence?
5. What is the importance of vertical angles in triangle congruence proofs?

**Tip**: Always check for shared angles and sides when attempting to prove congruence or similarity in triangles.

Use the sketch to prove, with reasons, that: 3.1 FE = EC = CA, 3.2 AE = 2DE, 3.3 ∆BCA ≡ ∆AED

This detailed solution guides you through proving geometric properties based on a triangle diagram, focusing on isosceles triangles, side relationships, and triangle congruence using the SAS theorem. Suitable for Grades 10-12 students.

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