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The last question in the image asks for an explanation of why \( \Delta GHI \) must be the same size and shape as \( \Delta ABC \) given that rigid body transformations map triangles in two steps.

Here's how to approach the question:

A rigid body transformation preserves the size and shape of geometric figures because such transformations include translations, rotations, and reflections, none of which alter the distances between points or angles. 

### Explanation:
- Since \( \Delta ABC \) is mapped to \( \Delta EFG \) by a rigid body transformation, this means \( \Delta EFG \) is congruent to \( \Delta ABC \). 
- A second rigid body transformation maps \( \Delta EFG \) to \( \Delta GHI \), so \( \Delta GHI \) must also be congruent to \( \Delta EFG \).
- Therefore, by the transitive property of congruence (if \( \Delta ABC \cong \Delta EFG \) and \( \Delta EFG \cong \Delta GHI \), then \( \Delta ABC \cong \Delta GHI \)), \( \Delta GHI \) must be the same size and shape as \( \Delta ABC \).

This maintains that the transformations do not alter the figure’s dimensions or angles, ensuring that the triangles remain congruent.

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### Follow-up Questions:
1. What are the different types of rigid transformations, and how do they work?
2. Can two figures be congruent if they are not oriented the same way? Explain.
3. How do translations affect the coordinates of points on a triangle?
4. What is the difference between congruence and similarity in triangles?
5. How would you prove that two triangles are congruent after a series of transformations?

### Tip:
When dealing with rigid transformations, always check if the transformation involves only translations, rotations, or reflections. These transformations preserve both the shape and size of geometric figures.

A transformation that is a rigid body motion maps ΔABC to ΔEFG. A second rigid body motion then maps ΔEFG to ΔGHI. Explain why ΔGHI must be the same size and shape as ΔABC.

Explore why ΔGHI must be congruent to ΔABC through two rigid body transformations. Learn how rigid motions preserve the size and shape of triangles through the transitive property of congruence. Suitable for students in Grades 8-10 studying geometric transformations.

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