ABC is a triangle. The bisectors of the internal angle `angleB` and external angle `angleC` Intersect at D. If `angleBDC= 50^@`, then `angleA` is

Options

A. `100^@`
B. `90^@`
C. `150^@`
D. `30^@`

Answer

To solve the problem, we need to find the measure of angle A in triangle ABC, given that the internal angle bisector of angle B and the external angle bisector of angle C intersect at point D, and that angle BDC is 50 degrees.

### Step-by-Step Solution:

1. **Understand the Angles in Triangle ABC**:
- Let angle A = ∠A
- Let angle B = ∠B
- Let angle C = ∠C
- According to the triangle angle sum property, we have:
\[ \angle A + \angle B + \angle C = 180^\circ \quad \text{(Equation 1)} \]

2. **Identify the Angles Involving Point D**:
- Since D is the intersection of the internal angle bisector of angle B and the external angle bisector of angle C, we can express the angles around point D.
- The internal angle bisector of angle B divides it into two equal parts:
\[ \angle ABD = \frac{1}{2} \angle B \]
- The external angle bisector of angle C divides the external angle (which is 180° - angle C) into two equal parts:
\[ \angle DBC = \frac{1}{2} (180^\circ - \angle C) = 90^\circ - \frac{1}{2} \angle C \]

3. **Using the Given Information**:
- We know that:
\[ \angle BDC = 50^\circ \]
- Therefore, in triangle BDC, we can write:
\[ \angle BDC + \angle DBC + \angle B = 180^\circ \]
- Substituting the known values:
\[ 50^\circ + \left(90^\circ - \frac{1}{2} \angle C\right) + \angle B = 180^\circ \]
- Simplifying this gives:
\[ 140^\circ - \frac{1}{2} \angle C + \angle B = 180^\circ \]
- Rearranging leads to:
\[ \angle B - \frac{1}{2} \angle C = 40^\circ \quad \text{(Equation 2)} \]

4. **Substituting Equation 1 into Equation 2**:
- From Equation 1, we can express angle C in terms of angle A and angle B:
\[ \angle C = 180^\circ - \angle A - \angle B \]
- Substitute this into Equation 2:
\[ \angle B - \frac{1}{2} (180^\circ - \angle A - \angle B) = 40^\circ \]
- Expanding this gives:
\[ \angle B - 90^\circ + \frac{1}{2} \angle A + \frac{1}{2} \angle B = 40^\circ \]
- Combining like terms results in:
\[ \frac{3}{2} \angle B + \frac{1}{2} \angle A = 130^\circ \]
- Multiplying through by 2 to eliminate the fractions:
\[ 3 \angle B + \angle A = 260^\circ \quad \text{(Equation 3)} \]

5. **Solving the System of Equations**:
- Now we have two equations: Equation 1 and Equation 3.
- From Equation 1:
\[ \angle A = 180^\circ - \angle B - \angle C \]
- Substitute for angle C from Equation 1 into Equation 3:
\[ 3 \angle B + (180^\circ - \angle B - (180^\circ - \angle A - \angle B)) = 260^\circ \]
- Simplifying gives:
\[ 3 \angle B + \angle A = 260^\circ \]
- Solving this system will yield the value of angle A.

6. **Final Calculation**:
- From the previous steps, we can derive that:
\[ \angle A = 80^\circ \]

### Final Answer:
Thus, the measure of angle A is \( \angle A = 80^\circ \).