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The question asks, "If \( \triangle ABC \sim \triangle PQR \), then which of the following is not always true?"

Since the triangles are similar, corresponding sides and angles are proportional. This means:

- \( AB \) corresponds to \( PQ \)
- \( BC \) corresponds to \( QR \)
- \( AC \) corresponds to \( PR \)

Now, let's analyze the options:

**A. \( BC \cdot PR = AC \cdot RQ \):**
- This is not a general rule for similar triangles. It does not directly relate to the proportionality of corresponding sides.

**B. \( AB \cdot PR = AC \cdot QP \):**
- This is also not a rule for similar triangles. \( AB \) corresponds to \( PQ \), not \( PR \), and this equation doesn't reflect correct proportionality.

**C. \( BC \cdot QR = AB \cdot PR \):**
- This could represent the proportionality between the sides, since \( BC \) corresponds to \( QR \) and \( AB \) to \( PQ \).

**D. \( BC \cdot QP = AB \cdot RQ \):**
- Similar reasoning applies here. \( BC \) corresponds to \( QR \), not \( QP \), making this unlikely.

The answer that is **not always true** is **A**, as it does not maintain the correct proportionality of corresponding sides in similar triangles.

Would you like more details or have any other questions?

Here are five related questions to explore:
1. What are the properties of similar triangles?
2. How do you prove triangles are similar?
3. Can the Pythagorean theorem be used with similar triangles?
4. How do angle bisectors relate to triangle similarity?
5. What is the difference between similar and congruent triangles?

**Tip**: Always match corresponding sides and angles when working with similar triangles.

If ΔABC ∼ ΔPQR, then which of the following is not always true?

Understand the concept of similar triangles with a question involving ΔABC and ΔPQR. Explore how corresponding sides are proportional and identify which equation may not always hold true. This problem is ideal for students in Grades 9-10 studying geometry.

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