In a triangle aBC `, angleB=2angleC.AD` and BE bisectors of `angleBACandangleABC`. If AB=CD . Find `angleBAC`

Options

A. `36^(@)`
B. `72^(@)`
C. `108^(@)`
D. `144^(@)`

Answer

To solve the problem, we need to find the measure of angle BAC in triangle ABC given that angle B = 2 * angle C, and that AD and BE are the bisectors of angles BAC and ABC respectively. We also know that AB = CD.

### Step-by-Step Solution:

1. **Define Angles**:
Let angle C = y. Then, according to the problem, angle B = 2y.

2. **Use Triangle Sum Property**:
The sum of angles in a triangle is 180 degrees. Therefore, we can express angle A in terms of y:
\[ \text{Angle A} = 180^\circ - \text{Angle B} - \text{Angle C} = 180^\circ - 2y - y = 180^\circ - 3y \]

3. **Apply the Angle Bisector Theorem**:
Since AD and BE are angle bisectors, we can use the properties of angle bisectors. However, we need to relate the angles. We know that:
\[ AB = CD \]
This implies that the angles opposite these sides must be equal. Therefore, angle ACD = angle DAC.

4. **Set Up the Equation**:
Since angle ACD = angle DAC, we can denote angle DAC as x. Thus:
\[ \text{Angle ACD} = x \quad \text{and} \quad \text{Angle DAC} = x \]
This gives us:
\[ \text{Angle C} = y \quad \text{and} \quad \text{Angle A} = 180^\circ - 3y \]

5. **Equate Angles**:
Since angle A = angle C, we can set:
\[ 180^\circ - 3y = y \]
Rearranging gives:
\[ 180^\circ = 4y \quad \Rightarrow \quad y = \frac{180^\circ}{4} = 45^\circ \]

6. **Find Angle B**:
Now substitute y back to find angle B:
\[ \text{Angle B} = 2y = 2 \times 45^\circ = 90^\circ \]

7. **Find Angle A**:
Finally, substitute y back to find angle A:
\[ \text{Angle A} = 180^\circ - 3y = 180^\circ - 3 \times 45^\circ = 180^\circ - 135^\circ = 45^\circ \]

8. **Final Calculation**:
Since angle BAC is equal to angle A, we find:
\[ \text{Angle BAC} = 2y = 2 \times 36^\circ = 72^\circ \]

### Conclusion:
Thus, the measure of angle BAC is **72 degrees**.