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To determine the number of possible solutions for this triangle, we can apply the Law of Sines since we have information about two sides and an angle that is not between them, which is a typical ambiguous case in trigonometry.

The given information is:
- \( m \angle B = 73^\circ \)
- \( a = 7 \)
- \( b = 5 \)

### Step-by-Step Solution:
1. **Identify the Ambiguous Case**: This is a case of SSA (Side-Side-Angle), where we know two sides and an angle that is not between them. In such cases, there may be:
   - No solution,
   - One solution,
   - Or two possible solutions.

2. **Calculate the Sine of the Known Angle**: Since we have angle \( B \) and sides \( a \) and \( b \), we can use the Law of Sines to find angle \( A \) by calculating the ratio:
   \[
   \sin A = \frac{a \sin B}{b}
   \]
   
3. **Plugging in Values**:
   \[
   \sin A = \frac{7 \sin 73^\circ}{5}
   \]
   Calculating \( \sin 73^\circ \approx 0.9563 \):
   \[
   \sin A = \frac{7 \times 0.9563}{5} = 1.339
   \]

4. **Interpret the Result**:
   - Since \( \sin A = 1.339 \) is greater than 1, it is impossible to satisfy this condition because the sine of an angle cannot exceed 1. 
   - Therefore, there are **no possible solutions** for this triangle.

### Conclusion:
This triangle has **no solutions** because the calculated value of \( \sin A \) exceeds 1, meaning there is no angle \( A \) that satisfies this setup.

Would you like further details or have questions on this?

#### Related Questions
1. How would the solution change if \( m \angle B \) was smaller?
2. What is the ambiguous case in triangle trigonometry?
3. How does the Law of Sines help determine the number of solutions?
4. How would this problem change if we were given the angle \( A \) instead?
5. What happens if both given sides are smaller than the side opposite the known angle?

#### Tip:
In SSA cases, always check the sine value calculation, as it determines whether the triangle has 0, 1, or 2 possible solutions.

For the triangle which has m∠B = 73°, a = 7, b = 5, how many possible solutions does this triangle have?

Explore how to determine the number of possible solutions for a triangle given m∠B = 73°, a = 7, and b = 5. This trigonometry problem applies the Law of Sines to address the SSA ambiguous case. Suitable for students in Grades 10-12.

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