The problem involves a circle with center $O$ and a quadrilateral inscribed in the circle. You're asked to find the measure of $\angle ABD$ given that $\angle ACD = 47^\circ$.

Step-by-Step Solution:

1. Identify the Known Angle:

   • $\angle ACD = 47^\circ$ is given.

2. Use the Inscribed Angle Theorem:

   • According to the inscribed angle theorem, an inscribed angle is half the measure of the central angle subtended by the same arc.
   • $\angle ABD$ and $\angle ACD$ subtend the same arc $AD$.

3. Relationship Between $\angle ABD$ and $\angle ACD$:

   • Since both $\angle ABD$ and $\angle ACD$ subtend the same arc $AD$, $\angle ABD$ must be equal to $\angle ACD$.
   • Therefore, $\angle ABD = \angle ACD = 47^\circ$.

Answer:

$\angle ABD = 47^\circ$.

Would you like to go over the details, or do you have any questions?

Here are 5 related questions to consider:

1. How would the angle $\angle ABD$ change if the measure of $\angle ACD$ were different?
2. What is the central angle corresponding to arc $AD$ if $\angle ACD = 47^\circ$?
3. If the circle's center $O$ were connected to points $A$ and $D$, what would be the measure of $\angle AOD$?
4. What happens to $\angle ABD$ if $D$ is moved along the circle, closer to or farther from $A$?
5. How would you find the measure of $\angle BCD$ given only $\angle ABD$ and the properties of the circle?

Tip: When working with inscribed angles, remember that angles subtending the same arc are always equal. This is a powerful property that simplifies many problems involving circles.