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To solve for angle \( A \), angle \( C \), and side \( c \), we can use the Law of Cosines and the Law of Sines.

### Given:
- Angle \( B = 50^\circ \)
- Side \( AB = 10 \) (opposite angle \( C \))
- Side \( AC = 8 \) (opposite angle \( B \))

### 1. Find Angle \( A \)
We can start by using the **Law of Cosines** to find angle \( A \). The Law of Cosines states:

\[
\cos A = \frac{b^2 + c^2 - a^2}{2bc}
\]

Where:
- \( a = 10 \)
- \( b = 8 \)
- \( c = \) (side opposite angle \( A \))

To find \( A \), we need to first calculate side \( c \) using the Law of Cosines. 

### 2. Find Side \( c \)
Using the Law of Cosines:
\[
c^2 = a^2 + b^2 - 2ab \times \cos B
\]
\[
c = \sqrt{a^2 + b^2 - 2ab \times \cos B}
\]

Substituting the values into the equation:
\[
c = \sqrt{10^2 + 8^2 - 2 \times 10 \times 8 \times \cos 50^\circ}
\]

Once we find \( c \), we'll proceed to calculate angles \( A \) and \( C \).

Let me compute the values.### Final Results:
1. **Angle \( A \)**: \( 51.6^\circ \)
2. **Angle \( C \)**: \( 78.4^\circ \)
3. **Side \( c \)**: \( 7.8 \) units

If you have any more questions or need further explanation, feel free to ask!

Here are 5 related questions for further exploration:
1. How does the Law of Cosines help when only two sides and one angle are known?
2. Can you apply the Law of Sines to verify the calculated angles?
3. What would change in the calculations if the triangle were an obtuse triangle?
4. How would you find the area of this triangle using trigonometric methods?
5. What is the significance of the angle sum property of triangles in trigonometry?

**Tip**: Always check that the sum of your angles is \( 180^\circ \) to confirm that the triangle calculations are correct!

Learn how to solve for angle A, angle C, and side c in a triangle using the Law of Cosines and Law of Sines. Given angles and sides, find the missing angles and side lengths with step-by-step calculations. Suitable for high school students studying trigonometry.

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