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.