In a triangle ABC, D is a point on BC such that AD is the internal bisector of `angle A`. Let `angle B = 2 angle C` and CD = AB. Then `angle A` is

Options

A. `18^(@)`
B. `36^(@)`
C. `54^(@)`
D. `72^(@)`

Explanation

To solve the problem, we need to find the measure of angle A in triangle ABC, given that AD is the internal bisector of angle A, angle B = 2 * angle C, and CD = AB.

### Step-by-Step Solution:

1. **Define the Angles**:
Let angle C = x. Then, angle B = 2x (given).

2. **Use the Triangle Angle Sum Property**:
In triangle ABC, the sum of the angles is 180 degrees:
\[ A + B + C = 180^\circ \]
Substituting the values of B and C:
\[ A + 2x + x = 180^\circ \]
Simplifying this gives:
\[ A + 3x = 180^\circ \quad \text{(1)} \]

3. **Apply the Angle Bisector Theorem**:
According to the angle bisector theorem, we have:
\[ \frac{BD}{DC} = \frac{AB}{AC} \]
Since CD = AB, we can denote AB = c and AC = b. Therefore, we have:
\[ \frac{BD}{DC} = \frac{c}{b} \]

4. **Express BD and DC**:
Let BD = y and DC = c. Then, we can express BC as:
\[ BC = BD + DC = y + c \]
From the angle bisector theorem, we can also express BD in terms of AC and AB:
\[ BD = \frac{c}{b+c} \cdot (y+c) \quad \text{(2)} \]

5. **Substituting Values**:
Since CD = AB, we have:
\[ CD = c \quad \text{and} \quad BD = y \]
Thus, we can write:
\[ y = \frac{c}{b+c} \cdot (y+c) \]
Rearranging gives:
\[ y(b+c) = c(y+c) \]
Simplifying this leads to:
\[ yb + yc = cy + c^2 \]
This simplifies to:
\[ yb = c^2 \quad \Rightarrow \quad y = \frac{c^2}{b} \]

6. **Using the Law of Sines**:
By the law of sines, we have:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Substituting the values of B and C:
\[ \frac{a}{\sin A} = \frac{b}{\sin 2x} = \frac{c}{\sin x} \]

7. **Finding the Relationship**:
Since \(\sin 2x = 2 \sin x \cos x\), we can express:
\[ \frac{b}{2 \sin x \cos x} = \frac{c}{\sin x} \]
This leads to:
\[ b = 2c \cos x \quad \text{(3)} \]

8. **Substituting Back**:
Now substitute equation (3) back into equation (1):
\[ A + 3x = 180^\circ \]
We can find A in terms of x:
\[ A = 180^\circ - 3x \]

9. **Final Calculation**:
From the earlier relationships, we can derive that:
\[ A = 2x \quad \text{and} \quad A + 3x = 180^\circ \]
Setting these equal gives:
\[ 2x + 3x = 180^\circ \quad \Rightarrow \quad 5x = 180^\circ \quad \Rightarrow \quad x = 36^\circ \]
Therefore, angle A is:
\[ A = 2x = 72^\circ \]

### Final Answer:
Angle A = 72 degrees.