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We are given the vertices of two triangles, \( \triangle ABC \) and \( \triangle DEF \), and we are tasked with determining which transformations of \( \triangle ABC \) will map it onto \( \triangle DEF \), making them congruent. Let's analyze this step by step:

**Vertices of \( \triangle ABC \)**:
- \( A(-6, 2) \)
- \( B(-4, 6) \)
- \( C(-2, 2) \)

**Vertices of \( \triangle DEF \)**:
- \( D(1, -2) \)
- \( E(3, -6) \)
- \( F(5, -2) \)

### Step 1: Check Possible Reflections and Translations
We need to identify the transformation that makes these triangles congruent, starting with the listed options:

1. **Reflection across the x-axis**:  
   Reflecting across the x-axis changes the sign of the \( y \)-coordinates. So, the reflection of \( \triangle ABC \) would have vertices:
   - \( A'( -6, -2 ) \)
   - \( B'( -4, -6 ) \)
   - \( C'( -2, -2 ) \)

   Comparing this with the coordinates of \( \triangle DEF \), we see that the vertices \( D(1, -2) \), \( E(3, -6) \), and \( F(5, -2) \) are translations of \( A', B', C' \) but shifted horizontally.

2. **Horizontal translation**:  
   For the transformation to match the vertices exactly, we can try moving \( A', B', C' \) horizontally to the right or left by a unit or more, as required in the options.

### Step 2: Analyze Options
Now we will test each option:

- **Option A**: Reflection across the x-axis and translation 1 unit right.  
   Reflecting across the x-axis gives \( A'( -6, -2 ) \), \( B'( -4, -6 ) \), \( C'( -2, -2 ) \), and then translating 1 unit right:
   - \( A'': (-5, -2) \)
   - \( B'': (-3, -6) \)
   - \( C'': (-1, -2) \)
   
   This does **not** match \( \triangle DEF \).

- **Option B**: Reflection across the y-axis and translation 1 unit left.  
   Reflecting across the y-axis gives:
   - \( A'( 6, 2 ) \)
   - \( B'( 4, 6 ) \)
   - \( C'( 2, 2 ) \)
   
   Translating 1 unit left gives:
   - \( A'': (5, 2) \)
   - \( B'': (3, 6) \)
   - \( C'': (1, 2) \)

   This also does **not** match \( \triangle DEF \).

- **Option C**: Reflection across the x-axis, then reflection across the y-axis, and translation 1 unit right.  
   First, reflect across the x-axis:
   - \( A'( -6, -2 ) \)
   - \( B'( -4, -6 ) \)
   - \( C'( -2, -2 ) \)
   
   Then reflect across the y-axis:
   - \( A''( 6, -2 ) \)
   - \( B''( 4, -6 ) \)
   - \( C''( 2, -2 ) \)
   
   Finally, translate 1 unit right:
   - \( A'''( 7, -2 ) \)
   - \( B'''( 5, -6 ) \)
   - \( C'''( 3, -2 ) \)

   This matches the vertices of \( \triangle DEF \), so **Option C is correct**.

- **Option D**: Reflection across the x-axis, then reflection across the y-axis, and translation 1 unit left.  
   Starting with the same steps as in Option C, but translating 1 unit left gives:
   - \( A'''( 5, -2 ) \)
   - \( B'''( 3, -6 ) \)
   - \( C'''( 1, -2 ) \)

   This **exactly matches** the vertices of \( \triangle DEF \), so **Option D is also correct**.

### Step 3: Conclusion
The correct answers are **C** and **D**.

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Would you like further clarification on any of these transformations?

Here are some related questions:
1. What other transformations can be applied to triangles to make them congruent?
2. How do you reflect a point across the x-axis and y-axis?
3. How do you perform a 180° rotation about the origin?
4. Can reflections and rotations be combined to map any two congruent figures?
5. How can you verify congruence between two triangles using transformation sequences?

**Tip**: When reflecting across both axes, remember that the x-axis changes the sign of the y-coordinates, while the y-axis changes the sign of the x-coordinates!

The vertices of ∆ABC are (-6, 2), (-4, 6), and (-2, 2), and the vertices of ∆DEF are (1, -2), (3, -6), and (5, -2). Which of these sequences of transformations of ∆ABC exhibit the congruence between the two triangles?

Analyze the congruence between two triangles by applying transformations such as reflections and translations. This problem involves understanding geometric transformations and their role in congruence. Suitable for students in Grades 9-11.

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