The sides AB and AC of `triangle ABC` are produced to points D and E, respectiely. The bisectors of `angleCBD " and " angleBCE` meet at P. If `angle A = 82^(@)`, then the measure of `angle P` is :

Options

A. `54^(@)`
B. `60^(@)`
C. `36^(@)`
D. `49^(@)`

Answer

To solve the problem, we need to find the measure of angle P where the angle bisectors of angles CBD and BCE meet. Given that angle A = 82 degrees, we can use the theorem related to the angle bisectors in this configuration.

### Step-by-Step Solution:

1. **Understand the Configuration**:
- We have triangle ABC with angle A = 82 degrees.
- The sides AB and AC are extended to points D and E respectively.
- We need to find angle P, which is formed by the intersection of the angle bisectors of angles CBD and BCE.

2. **Identify Angles**:
- Let angle ABC = B and angle ACB = C.
- By the triangle angle sum property, we know:
\[ A + B + C = 180^\circ \]
- Substituting the known value:
\[ 82 + B + C = 180 \]
\[ B + C = 98^\circ \]

3. **Using the Angle Bisector Theorem**:
- The theorem states that if we have two angle bisectors meeting at a point, the angle formed at that point (angle P) can be calculated using the formula:
\[ \angle P = 90^\circ - \frac{1}{2} \angle A \]

4. **Substituting the Value of Angle A**:
- Now substituting the value of angle A into the formula:
\[ \angle P = 90^\circ - \frac{1}{2} \times 82^\circ \]
\[ \angle P = 90^\circ - 41^\circ \]
\[ \angle P = 49^\circ \]

5. **Final Answer**:
- Therefore, the measure of angle P is:
\[ \angle P = 49^\circ \]