Question

Triangles A B C and X Y Z are shown. Angles B A C and Y X Z are congruent. The lengths of A B and Y X are both 28 centimeters and the lengths of A C and X Z are both 32 centimeters. Point A of triangle A B C pointing to the left. Point X of triangle X Y Z is pointing down.
How can ΔABC be mapped to ΔXYZ?

First, translate vertex A to vertex
.

Next,
ΔABC to align the sides and angles.

Answer 1

To map triangle ΔABC to triangle ΔXYZ based on the given information, we can follow these steps:

1. **Translate Vertex A to Vertex X**:
We start by moving point A of triangle ABC to point X of triangle XYZ. This can be achieved by translating the entire triangle ΔABC such that point A aligns with point X.

2. **Rotate ΔABC**:
After translating point A to point X, we need to rotate triangle ΔABC. Since we know that angles BAC and YXZ are congruent, we can rotate triangle ABC around point X until angle BAC aligns with angle YXZ.

3. **Reflect (if necessary)**:
Based on the orientation of the triangles, you may need to reflect triangle ΔABC over a line to ensure that the triangles are congruent in terms of their corresponding sides and angles. Given that point A points left and point X points down, perform a reflection if the angles and sides don’t match up after rotation.

4. **Final Alignment**:
Finally, ensure that the corresponding vertices align: vertex B should align with vertex Y and vertex C should align with vertex Z, keeping in mind that the lengths of AB and YX, as well as AC and XZ, are equal.

By following these steps, triangle ΔABC can be mapped onto triangle ΔXYZ. Thus, the complete process can be summarized as:

- **First, translate vertex A to vertex X.**
- **Next, rotate (and possibly reflect) ΔABC to align the sides and angles with ΔXYZ**.