In this activity we explore the composition of two isometries. Isometries preserve linearity, distance, and angle measures, and they map a preimage set of points to a congruent image set. [br][br]Isometries are one of the following five types: [br]Identity Transformation, Reflection, Translation, Rotation, or Glide-reflection.[br][br]This activity illustrates that the composition of two isometries is also another isometry. Composing the identity transformation with another isometry results in that non-identity isometry. The various checkboxes and controls above can be used to explore the various different possible compositions of two isometries. Through these explorations we can see exactly what single isometry is produced by composing two isometries, in all cases.[br][br]In sperate activities we have shown that every isometry can be decomposed into the composition of three or fewer reflections.[br][br][b]Definitions:  [/b]        [br]An [b][i]odd isometry[/i][/b] can be decomposed into a composition of an odd number of reflections.  [br]An [b][i]even isometry [/i][/b]can be decomposed into a composition of an even number of reflections.[br][br]Note that even though isometries are functions, odd isometries are typically not odd functions, and even isometries are typically not even functions.  In our explorations above, we have discovered the following results.[br][br][b]Proposition.[/b][br][br]A. Every isometry is either an even isometry or an odd isometry, but not both.[br]B. An isometry is an even isometry iff it is the identity transformation, a translation, or a rotation.[br]C.An isometry is an odd isometry iff it is a reflection or a glide-reflection.[br]D.The composition of two even isometries is an even isometry.[br]E. The composition of two odd isometries is an odd isometry.[br]F. The composition of an even and an odd isometry (in either order) is an odd isometry.[br][br][br][br][br][br]