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To solve for \( x \) in the triangle \( ABC \) using either the Law of Sines or the Law of Cosines, let's analyze the information given:

- Angle \( A = 30^\circ \)
- Angle \( C = 80^\circ \)
- Side \( AC = 10 \)

### Step 1: Find Angle \( B \)
Since the sum of angles in a triangle is \( 180^\circ \):
\[
B = 180^\circ - 30^\circ - 80^\circ = 70^\circ
\]

### Step 2: Apply the Law of Sines
The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
Here, \( a = x \), \( b = 10 \), \( \angle A = 30^\circ \), \( \angle B = 70^\circ \), and \( \angle C = 80^\circ \).

Using:
\[
\frac{x}{\sin 80^\circ} = \frac{10}{\sin 70^\circ}
\]

Solving for \( x \):
\[
x = \frac{10 \cdot \sin 80^\circ}{\sin 70^\circ}
\]

### Step 3: Calculate \( x \)
Using approximate values:
- \( \sin 80^\circ \approx 0.985 \)
- \( \sin 70^\circ \approx 0.94 \)

\[
x \approx \frac{10 \cdot 0.985}{0.94} \approx 10.48
\]

So, \( x \approx 10.48 \).

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

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Here are 5 related questions for further exploration:

1. How would the answer change if angle \( C \) were different?
2. Can we use the Law of Cosines to find \( x \) in this case? How?
3. How does the Law of Sines apply when all angles and only one side are known?
4. What is the significance of angle sum in a triangle?
5. Why do we use approximations for trigonometric values?

**Tip:** Remember to check if you have enough known angles and sides to determine if the Law of Sines or the Law of Cosines is more suitable.

Use the law of sines or the law of cosines to solve for x in the triangle ABC.

Learn how to solve for the side length x in a triangle using the Law of Sines. Given angles 30° and 80°, and side length 10, find x using trigonometric principles suitable for high school students.

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