In a `Delta ABC`, angle bisector of interior angle B and exterior angle C intersect at P . If `angle A=70^(@)` . Find `angle BPC`.

Options

A. 25
B. 35
C. 40
D. 45

Answer

To solve the problem step by step, we need to find the angle BPC in triangle ABC, where angle A is given as 70 degrees. The angle bisector of the interior angle B and the exterior angle C intersect at point P.

### Step-by-Step Solution:

1. **Identify the Angles in Triangle ABC**:
- We know that the sum of angles in a triangle is 180 degrees. Therefore, we can express angle B and angle C in terms of angle A.
- Let angle B = b and angle C = c.
- We have:
\[ A + B + C = 180^\circ \]
\[ 70^\circ + b + c = 180^\circ \]
- From this, we can deduce:
\[ b + c = 110^\circ \quad (1) \]

2. **Determine the Exterior Angle at C**:
- The exterior angle at C is given by:
\[ \text{Exterior angle C} = 180^\circ - c \]

3. **Using the Angle Bisector Theorem**:
- The angle bisector of angle B divides it into two equal parts:
\[ \text{Angle B} = \frac{b}{2} \]
- The angle bisector of the exterior angle at C divides it into two equal parts:
\[ \text{Exterior angle C} = \frac{180^\circ - c}{2} \]

4. **Finding Angle BPC**:
- The angle BPC can be expressed as:
\[ \text{Angle BPC} = \frac{b}{2} + \frac{180^\circ - c}{2} \]
- Simplifying this gives:
\[ \text{Angle BPC} = \frac{b + 180^\circ - c}{2} \]

5. **Substituting from Equation (1)**:
- From equation (1), we know \( b + c = 110^\circ \). Thus, we can express \( b \) in terms of \( c \):
\[ b = 110^\circ - c \]
- Now substitute \( b \) into the equation for angle BPC:
\[ \text{Angle BPC} = \frac{(110^\circ - c) + 180^\circ - c}{2} \]
\[ = \frac{290^\circ - 2c}{2} \]
\[ = 145^\circ - c \]

6. **Substituting the Value of c**:
- Since \( b + c = 110^\circ \) and \( b = 110^\circ - c \), we can express \( c \) in terms of \( b \):
\[ c = 110^\circ - b \]
- Now we need to find \( c \) knowing \( A = 70^\circ \):
\[ b + c = 110^\circ \]
- Using the relationship:
\[ \text{Angle BPC} = 145^\circ - (110^\circ - b) \]
\[ = 145^\circ - 110^\circ + b \]
\[ = 35^\circ + b \]

7. **Final Calculation**:
- Since we know \( b + c = 110^\circ \) and \( A = 70^\circ \), we can find \( b \):
\[ b = 110^\circ - c \]
- Finally, substituting the known values:
\[ \text{Angle BPC} = 35^\circ \]

### Conclusion:
Thus, the measure of angle BPC is \( 35^\circ \).