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The problem describes a trapezoid \( TURS \), where \( TU \parallel QR \), and provides several lengths. We need to find the length of segment \( SU \).

### Given Information:
- \( QS = 35 \)
- \( TS = 15 \)
- \( QR = 28 \)
- \( SR = 21 \)
  
We are asked to find the length of \( SU \).

### Steps:
1. **Use the Trapezoid Proportionality Theorem**: Since \( TU \parallel QR \), we can apply the trapezoid midsegment theorem or use the property that triangles formed by diagonals of the trapezoid are proportional.

2. **Identify Similar Triangles**: Triangles \( TUS \) and \( QRS \) are similar by the AA similarity criterion because both share angle \( S \) and the angles formed by the parallel lines \( TU \) and \( QR \) are congruent.

3. **Set Up Proportion**: Since the triangles are similar, the sides are proportional. Thus, the ratio of corresponding sides is:
   \[
   \frac{SU}{SR} = \frac{TS}{QR}
   \]

4. **Substitute Known Values**:
   - \( SR = 21 \)
   - \( TS = 15 \)
   - \( QR = 28 \)

   \[
   \frac{SU}{21} = \frac{15}{28}
   \]

5. **Solve for \( SU \)**:
   \[
   SU = \frac{15}{28} \times 21 = \frac{15 \times 21}{28} = \frac{315}{28} = 11.25
   \]

Thus, the length of \( SU \) is **11.25 units**.

Would you like more details or clarifications? Here are five related questions you might find helpful:

1. How can we use properties of trapezoids in other types of geometric problems?
2. What is the trapezoid midsegment theorem, and when is it applicable?
3. Can the similarity of triangles be applied in more complex shapes?
4. How does proportional reasoning simplify calculations in geometry?
5. What other geometric figures share properties similar to trapezoids?

**Tip**: Whenever working with parallel lines in polygons, check for similar triangles to simplify your calculations!

In the trapezoid below, TU is parallel to QR. If QS = 35, TS = 15, QR = 28, and SR = 21, find the length of SU.

Learn how to calculate the length of segment SU in a trapezoid where TU is parallel to QR. This geometry problem applies the trapezoid proportionality theorem and AA similarity to solve for SU using given side lengths.

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