Geometry › Congruence

Congruence & similarity

Two shapes look related — but how? Congruent means identical (just possibly moved); similar means scaled.

Congruent vs similar

Same size? Same shape? Tap a pair to compare.

Verdict

congruent

Translate, rotate, reflect — the shape is identical. Congruent.

Transformation: translate · reflect

Congruent

Two shapes are congruent if you can lay one perfectly on top of the other using translation, rotation and reflection only. Same shape, same size — sides and angles all match.

Triangle congruence tests

SSS — three sides equal.
SAS — two sides and the angle between them.
ASA — two angles and the side between them.
AAS — two angles and any non-included side.
RHS — right angle, hypotenuse, one side (right triangles only).

SSA is not enough

Two sides and a non-included angle do not guarantee congruence — there can be two different triangles that fit. (This is the "ambiguous case" in trig.)

Similar

Two shapes are similar if one is a scaled copy of the other — same angles, sides in the same ratio. Resize is allowed; resize plus translate / rotate / reflect makes them similar.

How sizes scale

Sides scale by the scale factor k.
Areas scale by k².
Volumes scale by k³.

Double the side of a square and the area quadruples. Triple the side of a cube and the volume goes up 27×. That's why elephants can't be the size of mice or vice versa — the math doesn't scale.