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We are given a circle with center \( O \), and the following angles:

- \( \angle ADC = 77^\circ \)
- \( \angle BAC = 24^\circ \)

We are asked to find \( \angle AOB \).

### Step-by-step explanation:

1. **Relationship between angles subtended by the same arc:**
   - In any circle, the angle subtended at the center of the circle by an arc is twice the angle subtended by the same arc at any point on the circumference. 
   - Here, \( \angle AOB \) is subtended by arc \( AB \), and \( \angle BAC \) is subtended by the same arc \( AB \) at the circumference.

   Therefore, \( \angle AOB = 2 \times \angle BAC \).

2. **Calculate \( \angle AOB \):**
   - Given \( \angle BAC = 24^\circ \), we can substitute it into the formula:
   \[
   \angle AOB = 2 \times 24^\circ = 48^\circ
   \]

### Final Answer:
\[
\boxed{\angle AOB = 48^\circ}
\]

Would you like further explanation on any part of this? 

Here are 5 related questions for practice:
1. What is the relationship between angles subtended by the same arc?
2. If \( \angle BAC = 30^\circ \), what would \( \angle AOB \) be?
3. How does the angle subtended at the center compare with the angle subtended at the circumference?
4. If \( O \) is the center of a circle, and \( \angle ABC = 60^\circ \), what is \( \angle AOC \)?
5. What is the sum of angles subtended at the center of a circle by arcs making a full circle?

**Tip:** Always remember that angles subtended by the same arc in a circle have a specific proportionality rule, with the central angle being twice that of the angle on the circumference.

In the figure, O is the center of the circle ABCD. It is given that ∠ADC = 77° and ∠BAC = 24°. Find ∠AOB.

Explore how to find the central angle ∠AOB in a circle when the inscribed angle ∠BAC is given. This step-by-step guide utilizes the theorem that a central angle is twice the inscribed angle subtended by the same arc.

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