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To find angle \(\beta\) in the triangle, we can use the Law of Sines. The Law of Sines states that:

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

### Given:
- \(\angle \alpha = 35^\circ\)
- \( \overline{\beta \gamma} = 8 \) (side opposite \(\alpha\))
- \( \overline{\alpha \gamma} = 6 \)

### Steps:
1. Use the Law of Sines:
\[
\frac{8}{\sin 35^\circ} = \frac{6}{\sin \beta}
\]

2. Solve for \(\sin \beta\):
\[
\sin \beta = \frac{6 \cdot \sin 35^\circ}{8}
\]

3. Calculate \(\beta\):
- Determine \(\sin \beta\).
- Use the inverse sine function (\(\sin^{-1}\)) to find \(\beta\).

I'll perform the calculations now.The measure of angle \(\beta\) is approximately \( 25.48^\circ \).

Would you like more details or have any other questions?

Here are some related questions to further explore this topic:

1. How can we find the length of the third side of the triangle?
2. What is the measure of angle \(\gamma\) in this triangle?
3. Can we confirm this solution using the Law of Cosines?
4. How do we classify this triangle based on its angles?
5. What is the area of this triangle?

**Tip**: Always double-check your results using another method, such as the Law of Cosines, to ensure accuracy.

Learn how to solve for angle β in a triangle using the Law of Sines. With a given angle of 35° and side lengths of 8 and 6, follow step-by-step calculations to determine the value of β.

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