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We are given a triangle with the following values:

- Side \( a = 10 \)
- Side \( c = 15 \)
- Angle \( \beta = 97.9^\circ \)

### Step 1: Use the Law of Cosines to find side \( b \)
The Law of Cosines is given by:

\[
b^2 = a^2 + c^2 - 2ac \cdot \cos(\beta)
\]

Substituting the known values:

\[
b^2 = 10^2 + 15^2 - 2 \cdot 10 \cdot 15 \cdot \cos(97.9^\circ)
\]

Let's calculate \( b \).Using the Law of Cosines, side \( b \) is approximately \( 19.14 \).

### Step 2: Use the Law of Sines to find the remaining angles \( \alpha \) and \( \gamma \)

The Law of Sines states:

\[
\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}
\]

First, we'll find \( \alpha \):

\[
\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)}
\]

This implies:

\[
\sin(\alpha) = \frac{a \cdot \sin(\beta)}{b}
\]

Now let's calculate \( \alpha \).It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can still use the Law of Sines approach explained earlier to complete the calculation:

\[
\sin(\alpha) = \frac{a \cdot \sin(\beta)}{b}
\]

Once you compute \(\alpha\), you can find the third angle \( \gamma \) using:

\[
\gamma = 180^\circ - \alpha - \beta
\]

Would you like to proceed with more information or solve this step by step?

Here are 5 related questions:
1. How does the Law of Cosines apply to obtuse triangles?
2. What is the significance of the Law of Sines for non-right triangles?
3. How can you determine if a triangle is obtuse, acute, or right given its angles?
4. How would the solution change if angle \( \beta \) was less than 90 degrees?
5. How can you check if your solution to a triangle is correct?

**Tip**: Always ensure angles sum up to \( 180^\circ \) in a triangle!

Solve the triangle with a = 10, c = 15, and β = 97.9 degrees.

Learn how to solve a triangle with sides a=10, c=15, and angle β=97.9° using the Law of Cosines and Law of Sines. Includes detailed step-by-step instructions. Suitable for students in Grades 10-12.

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