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To solve this triangle problem using the Law of Sines, let’s summarize the given information:

- Side \( a = 61 \)
- Side \( b = 71 \)
- Angle \( A = 34^\circ \)

Our goal is to find:
1. Angle \( B \)
2. Angle \( C \)
3. Side \( c \)

### Step 1: Using the Law of Sines
The Law of Sines formula is:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

Since we know \( a \), \( b \), and \( A \), we can first solve for \( B \).

### Step 2: Solve for \( B \)
\[
\frac{\sin B}{b} = \frac{\sin A}{a}
\]
\[
\sin B = b \cdot \frac{\sin A}{a} = 71 \cdot \frac{\sin 34^\circ}{61}
\]

Calculating \( \sin 34^\circ \):
\[
\sin 34^\circ \approx 0.5592
\]
Then,
\[
\sin B \approx 71 \cdot \frac{0.5592}{61} \approx 0.6506
\]

Now, take the inverse sine to find \( B \):
\[
B \approx \arcsin(0.6506) \approx 40.6^\circ
\]

### Step 3: Solve for Angle \( C \)
Since the sum of angles in a triangle is \( 180^\circ \):
\[
C = 180^\circ - A - B
\]
\[
C \approx 180^\circ - 34^\circ - 40.6^\circ \approx 105.4^\circ
\]

### Step 4: Solve for Side \( c \)
Now, use the Law of Sines again to find \( c \):
\[
\frac{c}{\sin C} = \frac{a}{\sin A}
\]
\[
c = a \cdot \frac{\sin C}{\sin A} = 61 \cdot \frac{\sin 105.4^\circ}{\sin 34^\circ}
\]

Calculating \( \sin 105.4^\circ \):
\[
\sin 105.4^\circ \approx 0.9659
\]

Then,
\[
c \approx 61 \cdot \frac{0.9659}{0.5592} \approx 105.4
\]

### Summary of Results
- \( B \approx 40.6^\circ \)
- \( C \approx 105.4^\circ \)
- \( c \approx 105.4 \)

Let me know if you want further details or have any questions. 

---

Here are five additional questions to consider:

1. What would happen if angle \( A \) were larger?
2. How would the calculations change if side \( b \) were smaller than side \( a \)?
3. How is the Law of Cosines different from the Law of Sines, and when should each be used?
4. How does rounding affect the accuracy of our calculations in trigonometry?
5. How can we confirm if there are multiple solutions for a triangle?

**Tip:** For non-right triangles, the Law of Sines is typically used when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).

Consider a triangle ABC like the one below. Suppose that a = 61, b = 71, and A = 34°. (The figure is not drawn to scale.) Solve the triangle.

Learn how to solve triangle ABC using the Law of Sines, with given values a = 61, b = 71, and angle A = 34°. This solution includes step-by-step calculations for angles B, C, and side c, suitable for high school students in Grades 10-12.

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