In a triangle ABC,D is the mid - point of BC, AD is produced upto E so that DE = AD.
Prove that :
`AB=EC`

Answer

To prove that \( AB = EC \) in triangle \( ABC \) where \( D \) is the midpoint of \( BC \) and \( AD \) is extended to \( E \) such that \( DE = AD \), we can follow these steps:

### Step-by-Step Solution:

1. **Draw Triangle ABC**:
- Begin by sketching triangle \( ABC \) with points \( A \), \( B \), and \( C \).

2. **Identify Midpoint D**:
- Mark point \( D \) as the midpoint of side \( BC \). This means \( BD = DC \).

3. **Extend AD to Point E**:
- Extend line segment \( AD \) to point \( E \) such that \( DE = AD \).

4. **Label the Given Information**:
- We know that \( DE = AD \) (given) and \( BD = DC \) (since \( D \) is the midpoint).

5. **Identify Vertically Opposite Angles**:
- Note that angles \( \angle ADB \) and \( \angle CDE \) are vertically opposite angles. Therefore, \( \angle ADB = \angle CDE \).

6. **Form Triangles ABD and DEC**:
- Consider triangles \( ABD \) and \( DEC \).

7. **Establish Congruence**:
- We have the following:
- \( BD = DC \) (as \( D \) is the midpoint)
- \( AD = DE \) (given)
- \( \angle ADB = \angle CDE \) (vertically opposite angles)
- By the Side-Angle-Side (SAS) congruence criterion, we can conclude that \( \triangle ABD \cong \triangle DEC \).

8. **Apply CPCTC**:
- Since the triangles are congruent, their corresponding parts are equal. Thus, we have \( AB = EC \).

### Conclusion:
We have proved that \( AB = EC \).