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The problem in the image asks the following:

In triangle \( \triangle ABC \), the internal bisectors of \( \angle B \) and \( \angle C \) meet at \( P \), and the external bisectors of \( \angle B \) and \( \angle C \) meet at \( Q \). The value of \( \angle BPC + \angle BQC \) is:
- (a) \( 90^\circ \)
- (b) \( 180^\circ \)
- (c) \( 270^\circ \)
- (d) \( 360^\circ \)

### Solution:

This question involves the concept of angle bisectors and properties of angles in a triangle. Here’s how to approach it:

1. **Angle Bisectors**: 
   - The internal angle bisectors of a triangle meet at a point known as the incenter (P in this case).
   - The external angle bisectors meet at a point known as the excenter (Q in this case).

2. **Summing the Angles**:
   - \( \angle BPC \) is the angle formed at the incenter by the internal bisectors.
   - \( \angle BQC \) is the angle formed at the excenter by the external bisectors.

A known geometric result states that the sum of \( \angle BPC \) (formed by the internal bisectors) and \( \angle BQC \) (formed by the external bisectors) in any triangle is always \( 180^\circ \).

### Conclusion:
Therefore, the value of \( \angle BPC + \angle BQC \) is \( \boxed{180^\circ} \).

Do you want further clarification or more details on this solution?

Here are 5 related questions you might want to explore:
1. What is the difference between an incenter and an excenter of a triangle?
2. How do you construct the internal and external angle bisectors in a triangle?
3. What is the significance of the incenter in a triangle's properties?
4. Can the sum of these angles \( \angle BPC + \angle BQC \) ever be greater than \( 180^\circ \)?
5. What are the other centers of a triangle, and how are they different?

**Tip:** In triangle geometry, understanding key points like the centroid, incenter, circumcenter, and orthocenter, as well as their properties, can simplify many complex problems!

In a triangle ABC, the internal bisectors of ∠B and ∠C meet at P, and the external bisectors of ∠B and ∠C meet at Q. Then the value of ∠BPC + ∠BQC is?

Solve the problem of finding the sum of ∠BPC and ∠BQC in triangle ABC, where P is the incenter and Q is the excenter. Learn how to apply geometric theorems related to angle bisectors.

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