To solve this problem, we need to use the Law of Sines to find the unknown angles of the triangle. Let's break it down step by step.

Given information:

• $C = 45^\circ 20'$ (converted to decimal: $C = 45 + \frac{20}{60} = 45.333^\circ$)
• $b = 25.1 \, \text{m}$
• $c = 31.6 \, \text{m}$

We need to find the angles $A$ and $B$.

Step 1: Use the Law of Sines

The Law of Sines states: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$ We can rearrange this to solve for the unknown angles: $\frac{\sin(A)}{a} = \frac{\sin(C)}{c}$ and $\frac{\sin(B)}{b} = \frac{\sin(C)}{c}$

Step 2: Find $\sin(C)$

First, we calculate $\sin(C)$: $\sin(C) = \sin(45.333^\circ) \approx 0.7071$

Step 3: Use the Law of Sines to find angle $B$

$\frac{\sin(B)}{25.1} = \frac{0.7071}{31.6}$ $\sin(B) = \frac{25.1 \times 0.7071}{31.6} \approx 0.5614$ Now, solve for angle $B$: $B = \sin^{-1}(0.5614) \approx 34.37^\circ$

Step 4: Find angle $A$

Since the sum of angles in a triangle is 180°, we can now find angle $A$: $A = 180^\circ - C - B = 180^\circ - 45.333^\circ - 34.37^\circ \approx 100.3^\circ$

Conclusion:

• Angle $A \approx 100.3^\circ$
• Angle $B \approx 34.4^\circ$

Thus, the correct choice is A, as there is only one possible set of remaining angles.

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Would you like any further explanation or assistance with another problem?

Here are some related questions you can explore:

1. How do you use the Law of Cosines in a triangle to solve for unknown sides or angles?
2. What is the difference between the Law of Sines and the Law of Cosines?
3. Can a triangle exist with given side lengths but without a valid angle solution? (Discuss the ambiguous case)
4. How does the angle sum property of triangles help when solving for missing angles?
5. What is the formula for the area of a triangle when two sides and the included angle are known?

Tip: When working with trigonometry, always ensure your angle measures are in the correct unit (degrees or radians) before performing calculations!