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To solve this problem, we are asked to find the measure of angle \( \angle S \) in the triangle \( \triangle STU \) where the side lengths are given as:

- \( ST = 6 \),
- \( TU = 10 \),
- \( SU = 15 \).

To find \( \angle S \), we can use the **Law of Cosines**, which relates the sides of a triangle to the cosine of one of its angles.

The Law of Cosines is given by:

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

Where:
- \( a \), \( b \), and \( c \) are the sides of the triangle,
- \( C \) is the angle opposite side \( c \).

For this problem:
- \( a = 6 \) (side \( ST \)),
- \( b = 10 \) (side \( TU \)),
- \( c = 15 \) (side \( SU \)),
- \( \angle S \) is opposite side \( SU \).

We can solve for \( \cos(\angle S) \), then use the inverse cosine function to find the measure of the angle. Let's calculate it.The measure of angle \( \angle S \) is approximately \( 137.9^\circ \).

Would you like a detailed explanation of the Law of Cosines, or do you have any other questions?

Here are five related questions to expand your understanding:
1. How is the Law of Cosines different from the Law of Sines?
2. Can you use the Law of Cosines for any triangle, including right triangles?
3. What is the sum of the interior angles in a triangle?
4. How would you calculate the other two angles in this triangle?
5. What would happen if the triangle had different side lengths?

**Tip:** When dealing with non-right triangles, the Law of Cosines is an essential tool to find unknown angles or sides when you have sufficient information.

Find the measure of angle S (m∠S) in triangle STU where side lengths ST = 6, TU = 10, and SU = 15.

Learn how to calculate the measure of angle S in triangle STU with sides ST = 6, TU = 10, and SU = 15 using the Law of Cosines. This solution demonstrates key concepts of trigonometry and triangle geometry, suitable for students in Grades 10-12.

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