Similar · Principia

Two figures are similar when there is a vertex correspondence under which every pair of corresponding angles is equal and every pair of corresponding sides is in one common ratio—same shape, possibly different size, written ∼, that shared ratio being the ratio of similarity.

Dependencies: SAS similarity axiom, Distance, Angle. A derived definition: “same shape, possibly different size” means equal corresponding angles and proportional corresponding side lengths.

Definition

Suppose a one-to-one correspondence is specified between the vertices of two figures.

Two triangles of different sizes with matching angle marks are similar.

If every pair of corresponding angles is equal and every ratio of corresponding side lengths equals the same constant $k$ , then the figures are similar, denoted by $\sim$ . The constant $k$ is the scale factor or similarity ratio.

For triangles, $\triangle ABC \sim \triangle A'B'C'$ means

\angle A=\angle A',\;\angle B=\angle B',\;\angle C=\angle C',\qquad \frac{|A'B'|}{|AB|}=\frac{|B'C'|}{|BC|}=\frac{|C'A'|}{|CA|}=k.

Similarity is the precise language of scaling: shape remains unchanged while every length is multiplied by $k$ . The SAS similarity axiom axiom states that two proportional sides and their included equal angle are sufficient for triangle similarity.

Remarks

Scale factor $k=1$ gives congruence. The figures then have both the same shape and the same size.
For triangles, equal angles and proportional sides force one another. General polygons require both conditions, but two equal angle pairs already imply triangle similarity (AA similarity) and hence proportional sides.
Shape is preserved, size is not. Angles and length ratios remain unchanged; lengths scale by $k$ and areas by $k^2$ (Similar area ratio = k²).
Similarity is an equivalence relation. It is reflexive, symmetric, and transitive (Similarity is transitive), grouping figures by shape.