PRINCIPIA · DEFINITION

Congruence

Two figures are congruent when there is a vertex correspondence under which every pair of corresponding sides is equal in length and every pair of corresponding angles is equal in measure—equivalently, one can be laid onto the other without changing shape or size, written ≅.

Dependencies: Distance, Angle. A derived definition: “exactly the same shape and size” means equal corresponding side lengths and equal corresponding angles.

Definition

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

Two congruent triangles with matching side and angle marks.

If every pair of corresponding sides has equal length and every pair of corresponding angles has equal measure, the figures are congruent, denoted by ≅\cong.

For triangles, △ABC≅△A′B′C′\triangle ABC \cong \triangle A'B'C' means that under the correspondence A ⁣↔ ⁣A′A\!\leftrightarrow\!A', B ⁣↔ ⁣B′B\!\leftrightarrow\!B', C ⁣↔ ⁣C′C\!\leftrightarrow\!C',

∣AB∣=∣A′B′∣,  ∣BC∣=∣B′C′∣,  ∣CA∣=∣C′A′∣,∠A=∠A′,  ∠B=∠B′,  ∠C=∠C′.|AB|=|A'B'|,\; |BC|=|B'C'|,\; |CA|=|C'A'|,\quad \angle A=\angle A',\; \angle B=\angle B',\; \angle C=\angle C'.

Intuitively, one figure can be moved or reflected to fit exactly over the other. This definition turns that superposition idea into six checkable equalities.

Remarks

  • Congruence is similarity with scale factor 11. The corresponding angles are equal and every corresponding side ratio is 11.
  • The correspondence is part of the definition. The order in △ABC≅△A′B′C′\triangle ABC \cong \triangle A'B'C' specifies which vertices match. A different order describes a different correspondence and may be false.
  • You need not check all six equalities. The congruence criteria (SAS congruence, SSS congruence, ASA congruence, AAS congruence, HL (hypotenuse-leg) congruence) show that three suitable independent measurements force the other three. This is the rigidity of a triangle.