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In this problem, we need to classify each transformation rule of triangle \( \triangle ABC \) to determine if it results in a congruent or non-congruent triangle \( \triangle A'B'C' \).

### Congruent Transformations:
Congruent transformations are those that preserve the size and shape of the figure, such as:
1. **Translations** (shifting the figure without changing its size or orientation).
2. **Rotations** (turning the figure around a point).
3. **Reflections** (flipping the figure over a line).

### Non-Congruent Transformations:
Non-congruent transformations change the size of the figure, which means the triangles will not be congruent. These include:
1. **Dilations** (changing the size of the figure but preserving its shape).
2. **Other non-isometric transformations** (such as compressions or expansions).

#### Let's classify each rule:

1. \( (x, y) \to (x + 5, y) \) – This is a **translation**, shifting the figure to the right by 5 units. **Congruent**.
2. \( (x, y) \to (5x, 5y) \) – This is a **dilation** (scaling the figure by a factor of 5). **Not congruent**.
3. \( (x, y) \to (0.5x, 0.5y) \) – This is a **dilation** (scaling down by a factor of 0.5). **Not congruent**.
4. \( (x, y) \to (x, -y) \) – This is a **reflection** across the x-axis. **Congruent**.
5. \( (x, y) \to (-x, -y) \) – This is a **rotation** by 180° around the origin. **Congruent**.

### Final Classification:

- **Congruent**:
   - \( (x, y) \to (x + 5, y) \)
   - \( (x, y) \to (x, -y) \)
   - \( (x, y) \to (-x, -y) \)

- **Not Congruent**:
   - \( (x, y) \to (5x, 5y) \)
   - \( (x, y) \to (0.5x, 0.5y) \)

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Would you like further details or have any questions about the classification?

### Related Questions:
1. What are other examples of congruent transformations?
2. How does a reflection preserve congruence in a figure?
3. How do you calculate the scale factor for dilations?
4. Can a dilation result in a congruent figure if the scale factor is 1?
5. What happens to angles and sides under a rotation?

### Tip:
When identifying congruence, remember that transformations preserving distances between points result in congruent figures.

Triangle ABC is mapped to triangle A'B'C' using each of the given rules. Which rules would result in triangle ABC being congruent or not congruent to triangle A'B'C'?

Explore how different transformation rules affect the congruence of triangle ABC to triangle A'B'C'. This problem addresses key geometric concepts including translations, rotations, reflections, and dilations, making it suitable for students in Grades 8-10.

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