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To find the size of \( \angle ABC \) in the given diagram, let's break down the information:

1. **Given**: \( \angle AOC = 160^\circ \).
2. \( O \) is the center of the circle, and \( \angle AOC \) is a central angle.

### Key Concepts:
- The central angle \( \angle AOC \) is twice the measure of the inscribed angle \( \angle ABC \) that subtends the same arc \( AC \) on the circle.
- Therefore, \( \angle ABC = \frac{1}{2} \times \angle AOC \).

### Calculation:
\[
\angle ABC = \frac{1}{2} \times 160^\circ = 80^\circ
\]

Thus, the size of \( \angle ABC \) is \( 80^\circ \).

Would you like any more details on this solution, or do you have any other questions?

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Here are 5 related questions:
1. How do you find the size of an inscribed angle if the central angle is known?
2. What is the relationship between central angles and inscribed angles subtending the same arc?
3. How can you determine the angle formed by a chord and a tangent?
4. What are the properties of angles in a cyclic quadrilateral?
5. How do you calculate angles subtended by the same arc in different segments?

**Tip**: Remember that an inscribed angle is always half of the central angle that subtends the same arc!

Explore how to find the size of angle ABC in circle geometry with a detailed solution. Learn about central angles, inscribed angles, and their relationship, including practical calculations. Suitable for high school students.

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