Congruence transformations are transformations that preserve angle measures and side lengths. They are also called rigid motions. We will consider them one at a time.

Translation

Let's start by considering translations.

As we can see above, translations preserve angle measures and side lengths. Therefore, a translation is a congruence transformation.

Reflection

Let's now consider reflections.

Reflections also preserve side lengths and angle measures. Therefore, a reflection is also a congruence transformation.

Glide Reflection

A glide reflection consists of a translation followed by a reflection. Let's perform a glide reflection on line segment $\overline{A B} .$ First, it is translated to $\overline{A^{'} B^{'}} .$

Next, it is reflected in the line of reflection to $\overline{A^{''} B^{''}} .$

A glide reflection could be, instead, considered a reflection followed by a translation. This is because the image created does not depend on the order of the transformations.

This transformation is a combination of congruence transformations. Therefore, as we see, it is a congruence transformation because it preserves side lengths and angle measures.

Rotation

Finally, let's consider rotations.

Rotations are congruence transformations as they preserve angle measures and side lengths.

Conclusion

Let's summarize what we have found.

Translations, reflections, glide reflections, and rotations are congruence transformations

Since the transformations we have seen here are congruence transformations, any combination of them is also a congruence transformation. Therefore, a composition of a rotation and a glide reflection is a congruence transformation.

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