https://file.aiquickdraw.com/mathbot/file/17294287113488srhksb2.jpg

The triangle in the image appears to have multiple angles and side lengths labeled. Let me summarize the key information provided:

- The triangle is split into two smaller triangles.
- The top smaller triangle has:
  - An angle of \( 50^\circ \) at the top vertex.
  - Side lengths 4 and 5.
- The larger triangle includes both the top and bottom portions, and the full triangle has:
  - An angle \( 110^\circ \) in the middle.
  - A side of length 15.
  - An unknown angle \( \theta \).
  - The side labeled as \( x \).

We can solve for \( x \) or \( \theta \) depending on what is needed, using trigonometric relationships or the Law of Sines and Cosines.

### Steps for solving:
1. **Using the Law of Sines**: If you know one angle and the opposite side length in a triangle, you can relate the other angles and sides.
   
   \[
   \frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}
   \]

2. **Using the Law of Cosines**: To find side lengths when two sides and an included angle are given:

   \[
   c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
   \]

Let me know if you'd like to proceed with a specific calculation or if more information is needed! Would you like details on solving for \( x \), \( \theta \), or both?

Here are 5 relative questions to extend your understanding:
1. How can the Law of Sines help in solving triangles with non-right angles?
2. What is the Law of Cosines used for in triangle problems?
3. Why is the angle sum of any triangle always \( 180^\circ \)?
4. How does knowing one angle and one side in a triangle limit or aid solving?
5. How can trigonometric ratios (sine, cosine) be used to find unknown angles?

**Tip**: When solving for unknowns in a triangle, try to identify which formula simplifies the calculation the most based on the given data.

In the triangle, find the unknown angle θ and the side length x given the angles 50° and 110°, and side lengths 4, 5, and 15.

Learn how to find the unknown angle θ and side length x in a triangle using the Law of Sines and Law of Cosines. Given angles of 50° and 110°, and side lengths 4, 5, and 15, this problem demonstrates the application of key trigonometric concepts. Suitable for high school students.

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