https://file.aiquickdraw.com/mathbot/file/1729095159991pf0oog5t.jpg

To solve this problem, we need to determine how triangle \( \triangle DEF \) maps onto \( \triangle D'E'F' \) and which transformation describes this correctly.

### Step 1: Analyzing the Points
From the diagram, the coordinates of the vertices of the triangles are approximately:

- \( D (-3, -2) \), \( E (2, -4) \), \( F (4, -5) \)
- \( D' (-3, 2) \), \( E' (2, 4) \), \( F' (4, 5) \)

### Step 2: Type of Transformation
Looking at the coordinates, the transformation appears to be a reflection. Specifically, notice the following relationships:

- The points of \( \triangle D'E'F' \) have the same \( x \)-coordinates as \( \triangle DEF \), but the \( y \)-coordinates are the opposite (positive vs. negative). This indicates that the reflection happened across the **x-axis**, where the \( y \)-coordinates change sign, but the \( x \)-coordinates remain the same.

### Step 3: Choosing the Correct Answer
Given this reflection across the x-axis, the correct description of the transformation is:

- **\( \triangle DEF \) is congruent to \( \triangle D'E'F' \) because you can map \( \triangle DEF \) to \( \triangle D'E'F' \) using a reflection across the x-axis, which is a rigid motion.**

Thus, the last option is the correct one.

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Would you like more details or have any further questions?

Here are 5 relative questions to expand your understanding:
1. What are the properties of a rigid motion?
2. How can we identify the line of reflection in transformations?
3. How do congruence and similarity differ in transformations?
4. How do other rigid motions, like rotations or translations, affect triangle mappings?
5. What are the real-world applications of rigid transformations?

**Tip:** In geometry, reflections across different axes or lines always preserve the size and shape, making the figures congruent.

Which statement correctly describes the relationship between \( \triangle DEF \) and \( \triangle D'E'F' \)?

Learn how to determine the relationship between triangles \( \triangle DEF \) and \( \triangle D'E'F' \) through a reflection. This problem explores concepts of congruence, rigid motions, and the Reflection Theorem, making it suitable for students in Grades 8-10.

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