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To solve for the angle \( \theta \) using the Law of Cosines, we'll use the following formula:

\[
\cos \theta = \frac{a^2 + b^2 - c^2}{2ab}
\]

where:
- \( a = 124.5 \)
- \( b = 61.3 \)
- \( c = 163 \)

### Step 1: Substitute the values

\[
\cos \theta = \frac{(124.5)^2 + (61.3)^2 - (163)^2}{2 \times 124.5 \times 61.3}
\]

### Step 2: Calculate the squared values and simplify

1. \( (124.5)^2 = 15500.25 \)
2. \( (61.3)^2 = 3757.69 \)
3. \( (163)^2 = 26569 \)

Substitute these into the formula:

\[
\cos \theta = \frac{15500.25 + 3757.69 - 26569}{2 \times 124.5 \times 61.3}
\]

### Step 3: Simplify the numerator and the denominator

1. Numerator: \( 15500.25 + 3757.69 - 26569 = -4311.06 \)
2. Denominator: \( 2 \times 124.5 \times 61.3 = 15257.7 \)

So:

\[
\cos \theta = \frac{-4311.06}{15257.7} \approx -0.2826
\]

### Step 4: Find \( \theta \) using the inverse cosine function

\[
\theta = \cos^{-1}(-0.2826)
\]

Calculating this gives:

\[
\theta \approx 106^\circ
\]

### Answer:
The value of \( \theta \), rounded to the nearest degree, is approximately **106°**.

Would you like more details on any part of this solution, or do you have any other questions?

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Here are some related questions that may help expand your understanding:

1. How does the Law of Cosines compare to the Pythagorean Theorem?
2. Can the Law of Cosines be used for all triangles, including right triangles?
3. What other methods can be used to find an angle in a triangle if all side lengths are known?
4. How can the Law of Cosines be rearranged to solve for a side length instead of an angle?
5. What are the practical applications of the Law of Cosines in real-life scenarios?

**Tip:** When using trigonometric functions, always check if your calculator is in the correct mode (degrees or radians) to match the problem requirements.

Use the Law of Cosines to determine the indicated angle θ. (Assume a = 124.5, b = 61.3, and c = 163. Round your answer to the nearest degree.)

Solve for the angle θ in a triangle with sides a = 124.5, b = 61.3, and c = 163 using the Law of Cosines. This problem involves trigonometric principles and provides a step-by-step solution. Suitable for high school students in Grades 10-12.

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