In `triangleABC,angleA=30^(@),a=6 and c=8`. Which of the following must be true?

Options

A. `0^(@) lt angleC lt 90^(@)`
B. `90^(@) lt angleC lt 135^(@)`
C. `45^(@) lt angleC lt 135^(@)`
D. `0^(@) lt angleC lt 45^(@)` or `135^(@) lt angleC lt 180^(@)`

Explanation

To solve the problem, we need to find the possible values of angle C in triangle ABC given that angle A = 30°, side a = 6, and side c = 8. We will use the sine rule to find the relationship between the sides and angles of the triangle.

### Step-by-Step Solution:

1. **Write down the Sine Rule:**
The sine rule states that:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Here, \( a \) is the side opposite angle \( A \), \( b \) is the side opposite angle \( B \), and \( c \) is the side opposite angle \( C \).

2. **Substitute the known values:**
We know:
- \( A = 30^\circ \)
- \( a = 6 \)
- \( c = 8 \)

Using the sine rule, we can set up the equation:
\[ \frac{6}{\sin 30^\circ} = \frac{8}{\sin C} \]

3. **Calculate \(\sin 30^\circ\):**
We know that:
\[ \sin 30^\circ = \frac{1}{2} \]
Therefore, substituting this into the equation gives:
\[ \frac{6}{\frac{1}{2}} = \frac{8}{\sin C} \]
Simplifying the left side:
\[ 6 \times 2 = 12 \]
So we have:
\[ 12 = \frac{8}{\sin C} \]

4. **Cross-multiply to solve for \(\sin C\):**
Cross-multiplying gives:
\[ 12 \sin C = 8 \]
Dividing both sides by 12:
\[ \sin C = \frac{8}{12} = \frac{2}{3} \]

5. **Find angle C:**
To find angle C, we take the inverse sine:
\[ C = \sin^{-1}\left(\frac{2}{3}\right) \]
Using a calculator, we find:
\[ C \approx 41.18^\circ \]

6. **Consider the second possible angle:**
Since \(\sin C = \frac{2}{3}\) can also correspond to an obtuse angle:
\[ C = 180^\circ - 41.18^\circ \approx 138.82^\circ \]

7. **Determine the range of angle C:**
Thus, angle C can either be approximately \(41.18^\circ\) or \(138.82^\circ\). Therefore, the possible ranges for angle C are:
- From \(0^\circ\) to \(90^\circ\) (acute angle)
- From \(90^\circ\) to \(180^\circ\) (obtuse angle)

### Conclusion:
The possible values for angle C are approximately \(41.18^\circ\) and \(138.82^\circ\). Thus, the correct option that must be true is that angle C can vary from \(0^\circ\) to \(45^\circ\) or from \(135^\circ\) to \(180^\circ\).