Congruence and Similarity via Transformations
Define congruence formally as a rigid motion mapping one figure to another, and similarity as a composition of a dilation with rigid motions. Connect these transformation-based definitions to the SSS/SAS/AA criteria.
A lesson of Geometry, in Transformations, about 10 minutes.
By the end of this lesson, you will be able to define congruence as a sequence of rigid motions, define similarity as a sequence of rigid motions and a dilation, and use transformation language to describe geometric relationships.
The Big Idea
Transformations provide a unified definition of congruence and similarity:
- Congruence: two figures are congruent if one can be mapped to the other by a sequence of rigid motions (translations, rotations, reflections).
- Similarity: two figures are similar if one can be mapped to the other by a sequence of rigid motions and a dilation.
This approach — which is central to the Common Core framework — puts transformations at the foundation of all geometry, rather than treating them as an add-on.
Congruence via Rigid Motions
Definition: Two figures are congruent if and only if there exists a sequence of rigid motions (translations, rotations, reflections) mapping one onto the other.
This is equivalent to the classical definition: same shape, same size, six pairs of congruent parts.
Example: $\triangle ABC \cong \triangle DEF$ means there is a composition of rigid motions taking $A \to D$, $B \to E$, $C \to F$.
Similarity via Rigid Motions + Dilation
Definition: Two figures are similar if and only if there exists a sequence of rigid motions and a dilation mapping one onto the other.
Similarity preserves shape (angles) but not necessarily size.
Example: $\triangle ABC \sim \triangle DEF$ means there is a composition (rigid motions + dilation with scale factor $k$) mapping $\triangle ABC$ to $\triangle DEF$.
Example 1 — Congruence Transformation
Describe the rigid motion(s) mapping $\triangle ABC$ to $\triangle A'B'C'$ if $A(0,0)$, $B(4,0)$, $C(2,3)$ and $A'(6,0)$, $B'(10,0)$, $C'(8,3)$.
Every point shifted by $\langle 6, 0 \rangle$ → translation by $\langle 6, 0 \rangle$.
Example 2 — Similarity Transformation
$\triangle ABC$: $A(0,0)$, $B(2,0)$, $C(0,2)$. $\triangle DEF$: $D(0,0)$, $E(6,0)$, $F(0,6)$. Describe the transformation.
All coordinates tripled → dilation about the origin with $k = 3$. The triangles are similar (not congruent).