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To solve for \( x \) in the given triangle, we can use the Law of Sines. The Law of Sines states:

\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\]

Given the triangle:
- \( \angle A = 50^\circ \)
- \( \angle B = 55^\circ \)
- \( \angle C = 180^\circ - 50^\circ - 55^\circ = 75^\circ \)
- The side opposite \( \angle C \) (10.0 cm) is given.

We want to find the side \( x \) opposite \( \angle B \).

Using the Law of Sines:

\[
\frac{x}{\sin(55^\circ)} = \frac{10.0}{\sin(75^\circ)}
\]

Solving for \( x \):

\[
x = 10.0 \times \frac{\sin(55^\circ)}{\sin(75^\circ)}
\]

Calculating the sine values:

\[
\sin(55^\circ) \approx 0.8192
\]
\[
\sin(75^\circ) \approx 0.9659
\]

Now, substituting these values:

\[
x = 10.0 \times \frac{0.8192}{0.9659} \approx 10.0 \times 0.8479 \approx 8.479 \text{ cm}
\]

Therefore, \( x \approx 8.48 \text{ cm} \).

Would you like further details or have any questions?

Here are 5 questions you might consider asking next:
1. Can you explain how the Law of Sines works in different types of triangles?
2. What are some common mistakes to avoid when using the Law of Sines?
3. How can we use the Law of Cosines to solve for unknown sides in a triangle?
4. Can you show how to verify the calculations using another method?
5. What are some practical applications of the Law of Sines in real life?

**Tip:** Always check the sum of the angles in a triangle to ensure they add up to 180 degrees as a quick verification step in your calculations.

Learn how to solve for an unknown side in a triangle using the Law of Sines. This problem involves calculating a side opposite a known angle using the sine ratios. Suitable for high school students studying geometry and trigonometry.

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