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We are given a triangle \( \triangle VTU \) where:
- \( \angle T = 98^\circ \),
- \( \angle U = 35^\circ \),
- \( \overline{TU} = 20 \), and
- We are tasked with finding \( u \), which is the length of side \( VT \).

To find \( u \), we can apply the **Law of Sines**, which states:

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

Here, \( a \), \( b \), and \( c \) are the lengths of the sides opposite to angles \( A \), \( B \), and \( C \), respectively.

### Step 1: Calculate \( \angle V \)
Since the sum of the angles in a triangle is \( 180^\circ \), we can find \( \angle V \) as follows:

\[
\angle V = 180^\circ - 98^\circ - 35^\circ = 47^\circ
\]

### Step 2: Apply the Law of Sines
Using the Law of Sines, we know:

\[
\frac{u}{\sin 35^\circ} = \frac{20}{\sin 47^\circ}
\]

Solve for \( u \):

\[
u = \frac{20 \cdot \sin 35^\circ}{\sin 47^\circ}
\]

I'll compute the result.The length of side \( u \) (which is side \( VT \)) is approximately **15.7** units.

Would you like further details or have any questions about this solution? Here are five related questions to expand on this topic:

1. How does the Law of Sines apply to non-right triangles?
2. Can the Law of Sines be used when two sides and an angle are given?
3. What other methods can be used to solve triangles apart from the Law of Sines?
4. How do you calculate the area of a triangle using side lengths and angles?
5. What is the Law of Cosines, and when would it be applied?

**Tip:** When solving for unknown sides or angles in triangles, always check if the Law of Sines or the Law of Cosines is more applicable based on the given information.

Find the length of side u in triangle VTU, where angle T = 98°, angle U = 35°, and side TU = 20.

Learn how to find the length of side u in triangle VTU using the Law of Sines. Given that angle T = 98°, angle U = 35°, and side TU = 20, this problem demonstrates the application of trigonometric principles and the Law of Sines theorem. Suitable for high school students (Grades 9-12).

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