D and E are the mid-points of the side AB and AC, respectively, of `Delta`ABC. DE is produced to F. To prove that CF is equal and parallel to DA, we need an additional information which is

Options

A. `angleDAE=angleEFC`
B. AE=EF
C. DE=EF
D. `angleADE=angleECF`

Explanation

To solve the problem, we need to prove that CF is equal and parallel to DA given that D and E are the midpoints of sides AB and AC of triangle ABC, and DE is extended to point F. We will also identify the additional information required to complete this proof.

### Step-by-Step Solution:

1. **Draw Triangle ABC**:
- Start by sketching triangle ABC with vertices A, B, and C.

2. **Identify Midpoints D and E**:
- Mark point D as the midpoint of side AB and point E as the midpoint of side AC. This means that AD = DB and AE = EC.

3. **Extend DE to Point F**:
- Draw line segment DE and extend it to point F.

4. **Join CF**:
- Connect point C to point F by drawing line segment CF.

5. **Assume DE = EF**:
- For the proof, we need to assume that DE is equal to EF. This assumption will help us establish congruence between triangles ADE and CFE.

6. **Establish Congruence**:
- We will show that triangles ADE and CFE are congruent using the following criteria:
- AE = EC (since E is the midpoint of AC).
- Angle AED = Angle CEF (by vertically opposite angles).
- DE = EF (our assumption).
- By the Side-Angle-Side (SAS) congruence criterion, we can conclude that triangle ADE is congruent to triangle CFE.

7. **Apply CPCT**:
- Since the triangles are congruent, by Corresponding Parts of Congruent Triangles (CPCT), we have:
- AD = CF
- Angle ADE = Angle CFE

8. **Identify Parallel Lines**:
- Angles ADE and CFE are alternate interior angles formed by transversal DF. If these angles are equal, then by the Alternate Interior Angles Theorem, lines AD and CF are parallel.

9. **Conclusion**:
- Therefore, we have proved that CF is equal to AD and that CF is parallel to AD, given the assumption that DE = EF.

### Additional Information Required:
To complete the proof, the additional information we need is:
- **DE = EF** (This is the assumption that must hold true for the proof to work).