To find the angle $\angle ADB$, we can use the properties of a circle and the given information:

1. Given Information:

   • $A, B, C,$ and $D$ are points on the circle.
   • $O$ is the center of the circle.
   • $AOC$ is a diameter.
   • $\angle BAC = 55^\circ$.

2. Step-by-Step Solution:

   • Since $AOC$ is a diameter of the circle, $\angle BAC$ is an inscribed angle that subtends the arc $AC$.
   • By the Inscribed Angle Theorem, the angle at the circumference (like $\angle BAC$) is half the angle at the center for the same arc. Since $\angle BAC = 55^\circ$, the central angle $\angle BOC$ that subtends the same arc $AC$ would be $2 \times 55^\circ = 110^\circ$.
   • $\angle ADB$ also subtends the same arc $AC$ as $\angle BAC$.
   • Therefore, $\angle ADB = \angle BAC = 55^\circ$ because angles subtended by the same arc in a circle are equal.

3. Conclusion:

   • $\angle ADB = 55^\circ$.

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

Related Questions:

1. What is the Inscribed Angle Theorem, and how is it applied in circle geometry?
2. How do central and inscribed angles compare when subtending the same arc?
3. How do we find angles in cyclic quadrilaterals using circle properties?
4. What other properties can be derived when a diameter subtends a right angle?
5. How can we calculate angles subtended by different segments in a circle?

Tip: Always remember that an inscribed angle subtending the diameter of a circle is a right angle (90°).