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The problem describes the measurement of three horizontal angles around a point \( P \) with angles \( \angle APD = 12^\circ 31'50'' \), \( \angle BPC = 29^\circ 37'30'' \), and \( \angle CPD = 47^\circ 36'30'' \). Additionally, the total measured angle \( \angle APD \) is given as \( 97^\circ 37'00'' \). The goal is to determine the most probable values of the angles.

### Step-by-Step Solution

1. **Determine the sum of the given angles:**

   Add the given angles \( \angle APD \), \( \angle BPC \), and \( \angle CPD \):

   \[
   12^\circ 31'50'' + 29^\circ 37'30'' + 47^\circ 36'30'' = 97^\circ 45'50''
   \]

2. **Calculate the discrepancy (error):**

   The total measured angle \( \angle APD \) is \( 97^\circ 37'00'' \), so the discrepancy \( e \) is:

   \[
   e = 97^\circ 45'50'' - 97^\circ 37'00'' = 8'50''
   \]

3. **Distribute the error:**

   Since each angle has an equal weight, the error is equally distributed across the three angles:

   \[
   \text{Correction per angle} = \frac{8'50''}{3} \approx 2'56''
   \]

4. **Apply the corrections:**

   Subtract this correction from each angle:

   - Corrected \( \angle APD = 12^\circ 31'50'' - 2'56'' = 12^\circ 28'54'' \)
   - Corrected \( \angle BPC = 29^\circ 37'30'' - 2'56'' = 29^\circ 34'34'' \)
   - Corrected \( \angle CPD = 47^\circ 36'30'' - 2'56'' = 47^\circ 33'34'' \)

### Final Answer
The most probable values of the angles are:

- \( \angle APD = 12^\circ 28'54'' \)
- \( \angle BPC = 29^\circ 34'34'' \)
- \( \angle CPD = 47^\circ 33'34'' \)

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

Here are some related questions to consider:

1. What are horizontal angles, and how are they used in surveying?
2. How does one perform error distribution in angle measurements?
3. What are common sources of errors in angle measurements in surveying?
4. How would you calculate corrections if the angles had different weights?
5. How does this method of error distribution relate to the method of least squares?

**Tip:** When working with angles, especially in surveying, always ensure to maintain precision in your measurements to avoid cumulative errors.

Explore how to analyze angle measurements around a point and distribute errors using trigonometric principles. This problem involves determining accurate angles based on measured discrepancies. Suitable for advanced high school and college students.

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