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To determine which postulate or theorem can be used to prove that \(\triangle PQR \sim \triangle STU\), we need to examine the given information in the triangles.

We have:
- \(\angle PQR\) and \(\angle STU\) are corresponding angles.
- The sides given are \(PQ = 15\), \(QR = 8\), \(ST = 9\), and \(TU = 8\).

### Checking the Similarity Criteria:
1. **Angle-Angle (AA) Similarity Postulate:** 
   - We need two pairs of corresponding angles to be congruent.
   - From the given diagram, we only see one pair of corresponding angles marked, so we cannot use AA postulate here.

2. **Side-Side-Side (SSS) Similarity Theorem:**
   - We need all three pairs of corresponding sides to be proportional.
   - Only two pairs of sides are given, so we cannot use SSS theorem here.

3. **Side-Angle-Side (SAS) Similarity Theorem:**
   - We need one pair of corresponding angles to be congruent and the sides including these angles to be proportional.
   - We see that \(\angle PQR = \angle STU\) and the sides around these angles are \(PQ\) and \(QR\) for \(\triangle PQR\) and \(ST\) and \(TU\) for \(\triangle STU\).
   - Let's check if the sides are proportional:
     \[
     \frac{PQ}{ST} = \frac{15}{9} = \frac{5}{3}
     \]
     \[
     \frac{QR}{TU} = \frac{8}{8} = 1
     \]
   - The sides are not proportional, so we cannot use SAS similarity theorem.

Given that none of the provided pairs of sides and angles satisfy the proportionality needed for SAS similarity theorem or the required criteria for AA and SSS, it seems there is an issue with the given diagram and provided sides. Please verify the diagram for any missing or additional information.

Would you like more details on this explanation, or do you have any questions?

Here are some more questions you might find interesting:

1. How can you prove triangles are similar using the AA similarity postulate?
2. Can you explain how the SSS similarity theorem works?
3. What is the difference between congruent and similar triangles?
4. How do you use the SAS similarity theorem to prove triangle similarity?
5. What are some real-life applications of triangle similarity?

**Tip:** When checking for triangle similarity, ensure that all necessary criteria for the selected theorem or postulate are met, including the proportionality of sides and congruence of angles.

Learn how to determine triangle similarity using the Angle-Angle (AA) Similarity Postulate, Side-Side-Side (SSS) Similarity Theorem, and Side-Angle-Side (SAS) Similarity Theorem. Explore a detailed example of determining similarity between triangles using given information. Suitable for students in Grades 9-12.

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