In the given arc AB= twice are BC and `angle AOB = 80 ^(@)` find:
`angle BOC`

Answer

To solve the problem, we need to find the measure of angle BOC given that arc AB is twice arc BC and angle AOB is 80 degrees.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We know that \( \angle AOB = 80^\circ \).
- We also know that arc AB is twice arc BC, which can be written as:
\[ \text{arc } AB = 2 \times \text{arc } BC \]

2. **Use the Property of Angles and Arcs**:
- The angle subtended by an arc at the center of the circle is proportional to the length of the arc. Therefore, we can write:
\[ \angle AOB = 2 \times \angle BOC \]
- This means that the angle AOB is twice the angle BOC.

3. **Substitute the Known Value**:
- We know that \( \angle AOB = 80^\circ \). Thus, we can substitute this value into the equation:
\[ 80^\circ = 2 \times \angle BOC \]

4. **Solve for Angle BOC**:
- To find \( \angle BOC \), we divide both sides of the equation by 2:
\[ \angle BOC = \frac{80^\circ}{2} = 40^\circ \]

5. **Conclusion**:
- Therefore, the measure of angle BOC is \( 40^\circ \).

### Final Answer:
\[ \angle BOC = 40^\circ \]