In a `DeltaABC` if `b =a (sqrt3-1) and /_C =30^@` then the measure of the angle A is

Options

A. `15^(@)`
B. `45^(@)`
C. `75^(@)`
D. `105^(@)`

Answer

To solve the problem step by step, we will use the given information and apply the sine rule to find the measure of angle A in triangle ABC.

### Step-by-Step Solution:

1. **Given Information:**
- \( b = a(\sqrt{3} - 1) \)
- \( \angle C = 30^\circ \)

2. **Express \( \frac{a}{b} \):**
\[ \frac{a}{b} = \frac{1}{\sqrt{3} - 1} \]

3. **Rationalize the Denominator:**
To rationalize \( \frac{1}{\sqrt{3} - 1} \), multiply the numerator and denominator by \( \sqrt{3} + 1 \):
\[ \frac{1}{\sqrt{3} - 1} \cdot \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = \frac{\sqrt{3} + 1}{(\sqrt{3})^2 - (1)^2} = \frac{\sqrt{3} + 1}{3 - 1} = \frac{\sqrt{3} + 1}{2} \]
Thus,
\[ \frac{a}{b} = \frac{\sqrt{3} + 1}{2} \]

4. **Using the Sine Rule:**
According to the sine rule:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
From the first two parts, we have:
\[ \frac{a}{b} = \frac{\sin A}{\sin B} \]
Substituting the value of \( \frac{a}{b} \):
\[ \frac{\sin A}{\sin B} = \frac{\sqrt{3} + 1}{2} \]

5. **Finding \( \sin B \):**
Since \( \angle C = 30^\circ \), we can find \( \sin C \):
\[ \sin C = \sin 30^\circ = \frac{1}{2} \]
Now, we also know that:
\[ \frac{b}{\sin B} = \frac{a}{\sin A} \]
Therefore, we can express \( \sin B \) in terms of \( a \) and \( b \):
\[ \sin B = \frac{b \cdot \sin A}{a} \]

6. **Substituting \( b = a(\sqrt{3} - 1) \):**
\[ \sin B = \frac{a(\sqrt{3} - 1) \cdot \sin A}{a} = (\sqrt{3} - 1) \cdot \sin A \]

7. **Substituting into the Sine Rule:**
Now we can substitute \( \sin B \) back into the equation:
\[ \frac{\sin A}{(\sqrt{3} - 1) \cdot \sin A} = \frac{\sqrt{3} + 1}{2} \]
Simplifying gives:
\[ \frac{1}{\sqrt{3} - 1} = \frac{\sqrt{3} + 1}{2} \]

8. **Finding \( \angle B \):**
From the sine rule, we can find \( \sin B \):
\[ \sin B = \frac{1}{\sqrt{2}} \quad \Rightarrow \quad B = 45^\circ \]

9. **Finding \( \angle A \):**
Now we can find \( \angle A \):
\[ \angle A = 180^\circ - \angle B - \angle C = 180^\circ - 45^\circ - 30^\circ = 105^\circ \]

### Final Answer:
The measure of angle A is \( 105^\circ \).