Yes, this is indeed an SSA (Side-Side-Angle) situation, which can lead to the ambiguous case in triangle determination. Let's carefully re-evaluate the SSA scenario for the given triangle:

### Given:
- \( \angle A = 58^\circ \)
- \( a = 10.5 \) cm (side opposite \( \angle A \))
- \( b = 11.4 \) cm (another side of the triangle)

### Steps to Determine the Number of Possible Triangles

1. **Calculate the Height of the Triangle:**
   - Using the given angle and side \( a \), we calculate the height \( h \) from vertex \( C \) perpendicular to side \( AB \).
   \[
   h = a \cdot \sin(A)
   \]
   \[
   h = 10.5 \cdot \sin(58^\circ) \approx 10.5 \cdot 0.848 \approx 8.904 \text{ cm}
   \]

2. **Compare the Height with Side \( b \):**
   - \( b = 11.4 \) cm
   - Since \( b > h \), we need to further analyze the comparison between \( b \) and \( a \).

3. **Analyze the Lengths of Sides \( a \) and \( b \):**
   - If \( b \le a \), there could be two possible triangles.
   - If \( b > a \), there is exactly one possible triangle.

### Evaluation:

- Given \( h \approx 8.904 \) cm and \( b = 11.4 \) cm, we note that \( b > h \).
- Now, comparing \( b \) with \( a \):
  \[
  b (11.4 \text{ cm}) > a (10.5 \text{ cm})
  \]

Since \( b > a \), only one triangle can be formed.

### Detailed Steps:

1. **Height Calculation:**
   \[
   h = 10.5 \cdot \sin(58^\circ) \approx 8.904 \text{ cm}
   \]

2. **Comparison of \( b \) and \( h \):**
   - \( b = 11.4 \) cm, which is greater than \( h = 8.904 \) cm.
   - Since \( b > h \), there is a possibility for a triangle.

3. **Final Comparison of \( b \) and \( a \):**
   - \( b (11.4 \text{ cm}) > a (10.5 \text{ cm}) \).
   - This confirms that there is exactly one possible triangle.

### Conclusion:

**Exactly one triangle** can be formed with the given measurements.

Would you like further details or have any questions about this solution? Here are some follow-up questions you might consider:

1. How do we use the Law of Sines in an SSA scenario?
2. What are the specific conditions for zero, one, or two triangles in SSA cases?
3. How can we visualize the ambiguous case with geometric drawings?
4. How does the Law of Cosines apply differently from the Law of Sines?
5. What are some examples of ambiguous cases in real-life applications?

**Tip:** When dealing with SSA cases, always calculate the height and compare it with the given sides to determine the possible number of triangles accurately.

Explore the SSA (Side-Side-Angle) situation in triangles and its ambiguous case. This problem discusses how to determine the number of possible triangles with specific side lengths and angles using the Law of Sines. Suitable for high school students studying geometry.

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