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This is a problem about solving a triangle, given two sides and one angle. Let's summarize the information:

- Side \(a = 73\) m
- Side \(b = 70\) m
- Angle \( \alpha = 65^\circ \)

We need to solve for:
1. Angle \( \beta \) (opposite side \(b\)),
2. Angle \( \gamma \) (opposite side \(c\)),
3. Side \(c\).

### Step 1: Find Angle \( \beta \)
We can use the Law of Sines to solve for the unknown angles and sides. The Law of Sines states:

\[
\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}
\]

Let's solve for \( \beta \) first:

\[
\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} \implies \sin(\beta) = \frac{b \cdot \sin(\alpha)}{a}
\]

Substituting the given values:

\[
\sin(\beta) = \frac{70 \cdot \sin(65^\circ)}{73}
\]

Using a calculator:

\[
\sin(\beta) \approx \frac{70 \cdot 0.9063}{73} \approx 0.8687
\]

Now, solve for \( \beta \):

\[
\beta \approx \arcsin(0.8687) \approx 60.4^\circ
\]

### Step 2: Find Angle \( \gamma \)
Since the sum of angles in a triangle is always \(180^\circ\):

\[
\alpha + \beta + \gamma = 180^\circ
\]

\[
65^\circ + 60.4^\circ + \gamma = 180^\circ \implies \gamma = 180^\circ - 125.4^\circ = 54.6^\circ
\]

### Step 3: Find Side \(c\)
We now use the Law of Sines again to solve for side \(c\):

\[
\frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)} \implies c = \frac{a \cdot \sin(\gamma)}{\sin(\alpha)}
\]

Substituting the values:

\[
c = \frac{73 \cdot \sin(54.6^\circ)}{\sin(65^\circ)} \approx \frac{73 \cdot 0.8141}{0.9063} \approx 65.6 \text{ m}
\]

### Final Answers:
- \( \beta \approx 60.4^\circ \)
- \( \gamma \approx 54.6^\circ \)
- \( c \approx 65.6 \) m

Would you like further details on any of the steps, or have any questions? 

Here are five related questions to expand on the solution:
1. How would the Law of Cosines apply if we had two angles and one side?
2. How would you solve the triangle if one side and two angles were given?
3. What would happen if the sum of the given angles were more than \(180^\circ\)?
4. Can the Law of Sines be used for right triangles, and how does it compare to the Pythagorean Theorem?
5. How does rounding intermediate results affect the final answer?

**Tip:** Always be careful when using inverse trigonometric functions to ensure the angle is in the correct quadrant, especially when working with non-right triangles.

Solve the given triangle. Round angles to one decimal place and side measure to 3 significant figures. Given: a = 73 m, b = 70 m, α = 65.0°.

Learn how to solve a triangle given two sides (a = 73 m, b = 70 m) and one angle (α = 65°) using the Law of Sines. This problem covers essential trigonometric concepts and is suitable for high school students.

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