In a `Delta ABC`, angle bisector of exterior `angle B & angle C` meets at point P, `angle BPC=65^(@)` then find `angleA` .

Answer

To solve for angle A in triangle ABC given the conditions of the problem, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have triangle ABC where the angle bisectors of the exterior angles at B and C meet at point P. We know that angle BPC = 65°.

2. **Using the Property of Exterior Angles**: The angle bisector of an exterior angle divides it into two equal parts. Therefore, if we denote the exterior angle at B as \( \angle B_{ext} \) and at C as \( \angle C_{ext} \), we can write:
- \( \angle B_{ext} = 180° - \angle B \)
- \( \angle C_{ext} = 180° - \angle C \)

3. **Setting Up the Angles**: Since P is the intersection of the angle bisectors of the exterior angles, we can express:
- \( \angle BPC = \frac{1}{2} \left( \angle B_{ext} + \angle C_{ext} \right) \)

4. **Substituting the Values**: We know that:
\[ \angle BPC = 65° \]
Therefore, we can set up the equation:
\[ 65° = \frac{1}{2} \left( (180° - \angle B) + (180° - \angle C) \right) \]

5. **Simplifying the Equation**: This simplifies to:
\[ 65° = \frac{1}{2} \left( 360° - (\angle B + \angle C) \right) \]
Multiplying both sides by 2 gives:
\[ 130° = 360° - (\angle B + \angle C) \]
Rearranging gives:
\[ \angle B + \angle C = 360° - 130° = 230° \]

6. **Using the Triangle Sum Property**: In triangle ABC, the sum of the angles is:
\[ \angle A + \angle B + \angle C = 180° \]
Substituting \( \angle B + \angle C = 230° \) into this equation gives:
\[ \angle A + 230° = 180° \]

7. **Solving for Angle A**: Rearranging gives:
\[ \angle A = 180° - 230° = -50° \]
This result is not possible, indicating a mistake in the interpretation of angles.

8. **Correcting the Approach**: Instead, we should use the relationship:
\[ 90° - \frac{\angle A}{2} = 65° \]
Rearranging gives:
\[ \frac{\angle A}{2} = 90° - 65° = 25° \]
Thus:
\[ \angle A = 25° \times 2 = 50° \]

### Final Answer:
\[ \angle A = 50° \]