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

Answer

To solve the problem step by step, we will find the measure of angle AOD given that AB = BC = DC and angle AOB = 50°.

### Step-by-step Solution:

1. **Identify Given Information:**
- We know that AB = BC = DC.
- We are given that angle AOB = 50°.

2. **Use the Property of Angles Opposite to Equal Sides:**
- Since AB = BC = DC, the angles opposite these equal sides will also be equal.
- Therefore, angle AOB = angle BOC = angle COD.

3. **Set Up the Equation:**
- Since angle AOB = 50°, we can conclude:
- angle BOC = 50°
- angle COD = 50°

4. **Calculate Angle AOD:**
- Angle AOD is the sum of angles AOB, BOC, and COD.
- Thus, angle AOD = angle AOB + angle BOC + angle COD.
- Substituting the values we have:
- angle AOD = 50° + 50° + 50°.

5. **Perform the Addition:**
- angle AOD = 150°.

### Final Answer:
- Therefore, angle AOD = 150°.