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?