https://file.aiquickdraw.com/mathbot/file/1730139188834113bxbwq.jpg

Let's analyze the problem.

### **Explanation:**
- The center of the circle is at \(O\).
- The angle at \(O\) is given as \(56^\circ\).
- We are asked to determine the measure of \(\angle ABC\), which is an **inscribed angle** that subtends the same arc as the **central angle** \(\angle AOC\).

### **Concept:**
In a circle:
- A **central angle** (like \(\angle AOC\)) is twice the measure of the **inscribed angle** (like \(\angle ABC\)) that subtends the same arc.

### **Calculation:**
Given:
\[
\angle AOC = 56^\circ
\]
Using the inscribed angle theorem:
\[
\angle ABC = \frac{1}{2} \times \angle AOC = \frac{1}{2} \times 56^\circ = 28^\circ
\]

Thus, the measure of \(\angle ABC\) is:
\[
\boxed{28^\circ}
\]

Do you need more clarification or a detailed breakdown of the steps?

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Here are **5 related questions** to deepen your understanding:
1. What is the relationship between central and inscribed angles in a circle?
2. How do we compute the angle subtended by a semicircle?
3. If \(\angle AOC\) was \(90^\circ\), what would be the value of \(\angle ABC\)?
4. How would the solution change if \(O\) were not the center of the circle?
5. Can you prove the inscribed angle theorem geometrically?

**Tip:** Always remember that the inscribed angle is half of the corresponding central angle in circle problems.

The center of the circle in the figure is at O. Determine the measure of ∠ABC.

Determine the measure of angle ∠ABC in a circle with center at O and given central angle ∠AOC = 56°. Learn how to apply the Inscribed Angle Theorem to find the answer. Suitable for students in Grades 8-10.

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