# Reflection and the ABCD Theorem

Copies of figures are common in mathematics as well as in real life. Teachers photocopy test papers, artists use figures make sculptures, people look copies of themselves when they look at the mirror.

In mathematics, congruent figures are exact copies of themselves. Two figures F and G are congruent if and only if G is the image of F under reflection or composite (combination of two or more) of reflections.

As you can see, a combination of two or more reflection (flip) is a translation (a slide)

or a rotation (a turn).

A transformation that is a reflection or composite of reflection is called congruence transformation or isometry. Isometry comes from the word isos which means “equal” and “metros” which means “measure.”

Every reflection preserves angle measure, betweenness, collinearity and distance. In Geometry, this is called the ABCD Theorem. We will not prove this theorem in this post, but we will discuss what it means.

• Angle Measure: If $\angle A^\prime B^\prime C^\prime$ is a reflection of  $\angle ABC$, then $\angle A^\prime B^\prime C^\prime \cong \angle ABC$.
• Betweenness: If $Q$ is between $P$ and $R$ and $P^\prime R^\prime$ is a reflection of  $PR$, then $Q^\prime$ is between $P^\prime$ and $R^\prime$.
• Collinearity: If $F$, $G$, and $H$ are collinear and $F^\prime$, $G^\prime$ and $H^\prime$ are reflections of $F$, $G$ and $H$, then $F^\prime$, $G^\prime$, and $H^\prime$ are collinear.
• Distance: If $P$ and $Q$ has distance $d$, and $P^\prime Q^\prime$ is a reflection of $PQ$, then $P^\prime Q^\prime$ has also distance $d$.