To solve for \(\angle A\) in \(\triangle ABC\) using the Law of Sines and considering the **ambiguous case** (SSA - Side-Side-Angle), we have:

- \(\angle B = 75^\circ\)
- \(BC = 31\) meters (side \(a\))
- \(AC = 6\) meters (side \(b\))

We want to find \(m\angle A\).

### Step-by-Step Solution Using Law of Sines and Ambiguous Case:

1. **Apply the Law of Sines:**

\[
\frac{\sin A}{a} = \frac{\sin B}{b}
\]

Substitute the known values:

\[
\frac{\sin A}{31} = \frac{\sin 75^\circ}{6}
\]

2. **Calculate \(\sin 75^\circ\):**

\[
\sin 75^\circ = \sin(90^\circ - 15^\circ) = \cos 15^\circ
\]

We know \(\cos 15^\circ \approx 0.9659\) (from standard trigonometric values).

Thus:

\[
\sin 75^\circ \approx 0.9659
\]

3. **Solve for \(\sin A\):**

\[
\frac{\sin A}{31} = \frac{0.9659}{6}
\]

\[
\sin A = 31 \times \frac{0.9659}{6}
\]

\[
\sin A \approx 31 \times 0.1609833
\]

\[
\sin A \approx 4.9905
\]

### Recognizing the Ambiguous Case:

Here we encounter an issue because \(\sin A > 1\), which is not possible for a valid angle. Let's address this by acknowledging:

The ambiguous case involves **two potential solutions** or **no valid solution** when:

- We have two sides and a non-included angle.
- The \(\sin\) function can lead to multiple triangles under certain angle and side combinations.

In our scenario:

1. **Check for valid or invalid configurations:**

Using the given:

\[
\frac{\sin B}{b} = \frac{\sin 75^\circ}{6}
\]

Already established correct initial calculation path.

Given angle: 

1. Confirm **side condition alignment** by valid approach-check.
2. **Interpret solution results based on valid calculation steps.**

**Direct next steps if correct detail base:**

\[
A = 180^\circ - (B + \angle C)
\]

Thus review given facts again and solve accurately given setups!

**Ask if need deeper steps on this or specific triangle-checks again?**

Explore how to solve a triangle problem using the Law of Sines and addressing the ambiguous case. Given angles and side lengths, discover the steps to find angle A using trigonometric calculations. Suitable for high school students learning trigonometry.

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