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)

 ABCD Theorem - translation

or a rotation (a turn). 

ABCD theorem - rotation


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.

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