https://file.aiquickdraw.com/mathbot/file/1726974289808vsfopc05.jpg

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 \( 180^\circ \).

O is the centre of the circle, angle BDA is 25°, angle CAD is 32°, and then angle BOC is

Calculate the central angle BOC given inscribed angles BDA = 25° and CAD = 32°. This geometry problem involves using circle theorems, such as the central and inscribed angle theorem. Suitable for students in Grades 8-10.

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