Define Congruence, Using Rigid Motions

Throughout this map, we say two objects are congruent if there exists a rigid motion taking one object to the other, or in other words, if they are "related by an rigid motion" (or "same/equivalent" up to rigid motions).

(For this reason, rigid motions are often called congruence transformations.)

Recall that

intuitively, you can think of rigid motions as "movements" that preserve the shapes (and sizes) of objects.

"Congruence" is how we formalize the concept of "same shape and size": intuitively, two objects are "congruent" if and only if they have the "same shape and size".

Warning: This differs from many textbook definitions of congruence. For example, some textbooks say "two triangles are congruent if and only if their sides and angles are the same." Matt and I decided that this is not a very enlightening definition of congruence (and it also doesn't apply to shapes other than triangles), so we tried to build an intuition-first/hands-on approach to geometry instead, including the definition of congruence in terms of rigid motions that you see above.

Define Congruence with Rigid Motions $\\$

Congruent figures are identical--they have the same size and shape. Here is an explanation going into further details defining congruence and similarity.

However, is there a way to understand congruence in terms of rigid motions?

Rigid motions are a series of transformations that does not change the size or shape of the figure.

Translation, rotation and reflection are all considered rigid motions.

**Two figures/shapes are congruent if they can transform into the other through rigid motions! **