Lesson 3 — Isometries and Congruence
Big idea: Every isometry produces a congruent figure. Reflections and glide reflections produce mirror images; translations and rotations do not.
Words to know
| Word | Meaning |
|---|---|
| Congruence | Same size and shape. |
| Glide reflection | A reflection followed by a translation parallel to the reflecting line. |
| Mirror image | A figure that has been flipped, reversing orientation. |
Concepts to know
- Every isometry maps a figure to a congruent figure.
- Types of isometries:
- translation;
- rotation;
- reflection;
- glide reflection.
- A reflection or glide reflection changes a figure into its mirror image.
- A translation or rotation does not change it into a mirror image.
- Every isometry of the plane can be described as one, two, or three reflections.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Decide whether figures are congruent | Check whether an isometry maps one to the other. |
| Identify the isometry | Slide → translation; turn → rotation; flip → reflection; flip + slide → glide reflection. |
| Decide whether orientation changed | Reflection/glide reflection → yes; translation/rotation → no. |
| Explain why figures are congruent | Name the isometry that maps one to the other. |
| Use a composite | Break it into reflections if helpful. |
Worked example
A figure is flipped over a line and then slid along that line.
- The transformation is a glide reflection.
- It is an isometry, so the image is congruent to the original.
- Because it includes a reflection, the image is a mirror image (orientation reversed).
Watch out
- Congruent figures can be mirror images; orientation does not affect congruence.
- A glide reflection is not the same as a translation.
- A rotation preserves orientation; a reflection reverses it.
- Every isometry preserves distance and angle measure.