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

Answer

To solve the problem, we need to find the measure of angle AOC given that AB = BC = DC and angle AOB = 50°.

### Step-by-Step Solution:

1. **Identify Given Information**:
- We have three equal sides: AB = BC = DC.
- Angle AOB = 50°.

2. **Use Properties of Isosceles Triangles**:
- Since AB = BC, triangle AOB is isosceles. Therefore, the angles opposite to equal sides are also equal.
- This means angle AOB = angle BOC = angle COD.

3. **Set Up the Equal Angles**:
- Let angle BOC = angle COD = x.
- We know angle AOB = 50°, so we can write:
- angle AOB = 50°,
- angle BOC = x,
- angle COD = x.

4. **Sum of Angles Around Point O**:
- The sum of angles around point O must equal 360°:
- angle AOB + angle BOC + angle COD = 360°.
- Substitute the known values:
- 50° + x + x = 360°.

5. **Solve for x**:
- Combine like terms:
- 50° + 2x = 360°.
- Subtract 50° from both sides:
- 2x = 310°.
- Divide by 2:
- x = 155°.

6. **Find Angle AOC**:
- Angle AOC is the sum of angle AOB and angle BOC:
- angle AOC = angle AOB + angle BOC = 50° + 155°.
- Therefore, angle AOC = 205°.

### Final Answer:
Angle AOC = 205°.

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