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?

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.