In the given `AB= BC = DC and angle AOB = 50^(@)`
` angle BOD`

Answer

To solve the problem step by step, we will use the properties of angles and equal sides in the context of circles.

### Step-by-Step Solution:

1. **Identify Given Information:**
- We are given that \( AB = BC = DC \).
- We are also given that \( \angle AOB = 50^\circ \).

2. **Understand the Implications of Equal Sides:**
- Since \( AB = BC \), the angles opposite these equal sides must also be equal. Therefore, \( \angle AOB = \angle BOC \).
- Similarly, since \( BC = DC \), we have \( \angle BOC = \angle COD \).

3. **Set Up the Equal Angles:**
- From the above, we can conclude:
- \( \angle BOC = \angle AOB = 50^\circ \)
- \( \angle COD = \angle BOC = 50^\circ \)

4. **Calculate Angle BOD:**
- The angle \( \angle BOD \) is the sum of angles \( \angle BOC \) and \( \angle COD \):
\[ \angle BOD = \angle BOC + \angle COD \]
- Substituting the values we found:
\[ \angle BOD = 50^\circ + 50^\circ = 100^\circ \]

5. **Final Answer:**
- Therefore, \( \angle BOD = 100^\circ \).