Try to rewrite each isometry as the composition of a certain number of reflections.
See solution.
Practice makes perfect
We are given two categories and four isometries classified under these two categories.
Odd Isometries
Even Isometries
Reflections
Glide Reflections
Translations
Rotations
One main difference we can point our between these categories is that the odd isometries reverse the orientation of a figure while the even isometries don't.
However, this reason does not have much to do with the name of the categories and so, let's go beyond. Recall that we learned that we can express all the isometries as compositions of reflections.
Isometry
Composition of Reflections across
Translation
Two parallel lines
Rotation
Two intersecting lines
Since a glide reflection involves a translation (two reflections) and one reflection, it's composed of three reflections.
Isometry
Composition of Reflections across
Glide Reflection
Three lines
Finally, in the case of a reflection, it can be written as the composition of an odd number of reflections across the same line. With all this information, let's build the following table.
Isometry
Number of Reflections Involved
Reflections
One
Glide Reflections
Three
Translation
Two
Rotation
Two
From the table above, we can see why the reflections and glide reflections are called odd isometries while the translations and rotations are called even isometries. This is the reason behind the names of the two given categories.
