What transformations were made to △ABC to produce △XYZ ? Assess whether the two triangles are congruent. How do you know? Show your work. (2 points)

To assess the transformations from triangle \( \triangle ABC \) to triangle \( \triangle XYZ \) and determine if they are congruent, we first need to analyze the coordinates of each triangle:

The coordinates for triangle \( \triangle ABC \) are:
- \( A(-5, 2) \)
- \( B(-3, 2) \)
- \( C(-3, 5) \)

The coordinates for triangle \( \triangle XYZ \) are:
- \( X(0, -6) \)
- \( Y(0, -4) \)
- \( Z(-3, -4) \)

### Step 1: Determine Positions of Vertices
Let's plot the points and analyze their positions:

1. **Triangle \( \triangle ABC \):**
- Vertex \( A \) is at (-5, 2)
- Vertex \( B \) is at (-3, 2) (to the right of A)
- Vertex \( C \) is at (-3, 5) (above B)

2. **Triangle \( \triangle XYZ \):**
- Vertex \( X \) is at (0, -6)
- Vertex \( Y \) is at (0, -4) (above X)
- Vertex \( Z \) is at (-3, -4) (to the left of Y)

### Step 2: Calculate the Side Lengths

**For triangle \( \triangle ABC \):**
- Length \( AB = \sqrt{((-3) - (-5))^2 + (2 - 2)^2} = \sqrt{(2)^2 + (0)^2} = 2 \)
- Length \( BC = \sqrt{((-3) - (-3))^2 + (5 - 2)^2} = \sqrt{(0)^2 + (3)^2} = 3 \)
- Length \( CA = \sqrt{((-5) - (-3))^2 + (2 - 5)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \)

**For triangle \( \triangle XYZ \):**
- Length \( XY = \sqrt{(0 - 0)^2 + (-4 - (-6))^2} = \sqrt{(0)^2 + (2)^2} = 2 \)
- Length \( YZ = \sqrt{((-3) - 0)^2 + (-4 - (-4))^2} = \sqrt{(-3)^2 + (0)^2} = 3 \)
- Length \( ZX = \sqrt{(0 - (-3))^2 + (-6 - (-4))^2} = \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \)

### Step 3: Compare Side Lengths
- \( AB = XY = 2 \)
- \( BC = YZ = 3 \)
- \( CA = ZX = \sqrt{13} \)

Since \( \triangle ABC \) and \( \triangle XYZ \) have the same side lengths, the two triangles are congruent.

### Step 4: Identify Transformations
To determine what transformations were applied, we observe the following:

1. **Translation:**
- The triangles appear to be translated. We need to find out how much the points were moved:
- To go from \( A(-5, 2) \) to \( X(0, -6) \):
- Change in x: \( 0 - (-5) = 5 \)
- Change in y: \( -6 - 2 = -8 \)
- To go from \( B(-3, 2) \) to \( Y(0, -4) \):
- Change in x: \( 0 - (-3) = 3 \)
- Change in y: \( -4 - 2 = -6 \)
- To go from \( C(-3, 5) \) to \( Z(-3, -4) \):
- Change in x: \( -3 - (-3) = 0 \)
- Change in y: \( -4 - 5 = -9 \)

Given the translations are not consistent across all points, the transformation might not just be simple translation but perhaps involves reflecting or rotating after translation.

### Conclusion
The corresponding sides of triangles \( \triangle ABC \) and \( \triangle XYZ \) have equal lengths, proving that the two triangles are congruent by Side-Side-Side (SSS) congruence. The nature of transformations might involve a combination of translation and reflection or rotation as they are positioned in different quadrants.