Two chords AB and CD of a circle with centre O intersect at point P within the circle. `/_AOC + /_BOD = ?`

Options

A. `/_APC`
B. `2/_APC`
C. `3/2 /_APC`
D. None of these

Answer

To solve the problem, we need to find the sum of the angles \( \angle AOC + \angle BOD \) when two chords AB and CD intersect at point P within a circle.

### Step-by-step Solution:

1. **Draw the Circle and Chords**:
- Start by drawing a circle with center O.
- Draw two chords AB and CD that intersect at point P inside the circle.

2. **Label the Angles**:
- Let \( \angle AOC \) be denoted as \( \theta \).
- Let \( \angle BOD \) be denoted as \( \alpha \).

3. **Understanding the Angles at the Circumference**:
- According to the properties of circles, the angle subtended by a chord at the center of the circle is twice the angle subtended at any point on the circumference.
- Therefore, if \( \angle AOC = \theta \), then the angle subtended by chord AB at the circumference (let's denote it as \( \angle ABC \)) will be \( \frac{\theta}{2} \).
- Similarly, if \( \angle BOD = \alpha \), then the angle subtended by chord CD at the circumference (let's denote it as \( \angle BDC \)) will be \( \frac{\alpha}{2} \).

4. **Using Triangle Properties**:
- Consider triangle PBC, where \( \angle ABC = \frac{\theta}{2} \) and \( \angle BDC = \frac{\alpha}{2} \).
- The sum of angles in triangle PBC is \( 180^\circ \):
\[ \angle PBC + \angle ABC + \angle BDC = 180^\circ \]
- Substituting the known angles:
\[ \angle PBC + \frac{\theta}{2} + \frac{\alpha}{2} = 180^\circ \]

5. **Expressing the Angles**:
- Rearranging gives:
\[ \angle PBC = 180^\circ - \left(\frac{\theta}{2} + \frac{\alpha}{2}\right) \]
- This implies:
\[ \angle PBC = 180^\circ - \frac{\theta + \alpha}{2} \]

6. **Finding the Relationship**:
- From the properties of angles in a circle, we know that:
\[ \angle AOC + \angle BOD = 2 \cdot \angle PBC \]
- Therefore, substituting \( \angle PBC \):
\[ \angle AOC + \angle BOD = 2 \left(180^\circ - \frac{\theta + \alpha}{2}\right) \]
- Simplifying gives:
\[ \angle AOC + \angle BOD = 360^\circ - (\theta + \alpha) \]

7. **Final Result**:
- Hence, we conclude:
\[ \angle AOC + \angle BOD = 180^\circ \]

### Final Answer:
\[ \angle AOC + \angle BOD = 180^\circ \]