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To determine the measure of angle \( \alpha \) to the nearest degree, we will use the Law of Cosines. This approach is useful when we know the lengths of all three sides of a triangle.

The Law of Cosines states:
\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\]

Here:
- \(a = 6.0\)
- \(b = 5.4\)
- \(c = 5.2\)

We are looking to find angle \( \alpha \), which is opposite side \( c \).

Let's rearrange the Law of Cosines to solve for \( \cos(\alpha) \):
\[
\cos(\alpha) = \frac{a^2 + b^2 - c^2}{2ab}
\]

Substitute the given values:
\[
\cos(\alpha) = \frac{6.0^2 + 5.4^2 - 5.2^2}{2 \cdot 6.0 \cdot 5.4}
\]

Calculate each term:
\[
6.0^2 = 36.0
\]
\[
5.4^2 = 29.16
\]
\[
5.2^2 = 27.04
\]

So:
\[
\cos(\alpha) = \frac{36.0 + 29.16 - 27.04}{2 \cdot 6.0 \cdot 5.4} = \frac{38.12}{64.8} \approx 0.588
\]

Now find \( \alpha \) using the inverse cosine function:
\[
\alpha = \cos^{-1}(0.588) \approx 54.06^\circ
\]

Therefore, to the nearest degree, the measure of angle \( \alpha \) is \( 54^\circ \).

Would you like further details or have any questions about this solution? Here are some follow-up questions you might consider:

1. How do we derive the Law of Cosines?
2. Can the Law of Cosines be used for any type of triangle?
3. What are the differences between the Law of Sines and the Law of Cosines?
4. How can the Law of Cosines help in finding the third side of a triangle?
5. What are some practical applications of the Law of Cosines in real life?

**Tip:** When using trigonometric functions, ensure your calculator is set to the correct angle unit (degrees or radians) as required by the problem.

Learn how to find the measure of angle alpha in a triangle using the Law of Cosines. Given the lengths of all three sides of a triangle, this solution demonstrates step-by-step calculations to find angle alpha to the nearest degree. Suitable for high school students.

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