Based on the provided image and information, you want to find the angle $BOC$ in the circle with $O$ as its center.

To solve this:

1. Identify Central and Inscribed Angles:

   • $\angle BDA$ is an inscribed angle that subtends arc $BC$.
   • $\angle CAD$ is also an inscribed angle that subtends arc $BD$.

2. Relate Inscribed Angles to Central Angles:

   • The central angle $\angle BOC$ subtends the same arc $BC$ as the inscribed angle $\angle BDA$. Recall that a central angle is twice the inscribed angle subtending the same arc.
   • Similarly, $\angle BOD$ is the central angle subtending the arc $BD$, which is twice $\angle CAD$.

3. Calculate $\angle BOC$ and $\angle BOD$:

   • $\angle BOC = 2 \times \angle BDA = 2 \times 25^\circ = 50^\circ$
   • $\angle BOD = 2 \times \angle CAD = 2 \times 32^\circ = 64^\circ$

4. Combine Angles to Find $\angle BOC$:

   • Since $O$ is the center, the full angle around point $O$ is $360^\circ$.
   • If $\angle BOC = 50^\circ$ and $\angle BOD = 64^\circ$, then $\angle COB$ is part of the remaining circle not covered by these two angles.

5. Confirm Calculation by Considering Arcs:

   • The arc covered by $\angle BOC$ and $\angle BOD$ together is $BC + BD$. Since $BD$ and $BC$ collectively cover the circle along with arc $CD$, we should verify the total.
   • $\angle BOC + \angle BOD + \angle COD = 360^\circ$ gives $\angle COD = 360^\circ - (50^\circ + 64^\circ) = 246^\circ$.

Since the text asks specifically for $\angle BOC$, we conclude it is $50^\circ$.

Would you like further details or have any other questions? Here are some additional topics you might find interesting:

1. How to calculate the remaining angles in the circle.
2. The difference between central and inscribed angles.
3. Applications of circle theorems in real-life situations.
4. How to construct a circle with given angles using a compass.
5. The properties of chords and tangents in relation to circle angles.

Tip: Always verify your calculations by ensuring that all angles around a point sum up to $360^\circ$ and that all angles in a triangle sum up to $180^\circ$.