Congruent triangles

Definition

Let $n$ be a positive integer, and let $ABC$ and $A'B'C'$ be triangles in the Euclidean space $\mathbb{R}^n$ . The triangles are congruent, written $ABC \cong A'B'C'$ , if there is an isometry $f : \mathbb{R}^n \to \mathbb{R}^n$ with $f(A) = A'$ , $f(B) = B'$ and $f(C) = C'$ . An isometry is a map $f$ with $d(f(x), f(y)) = d(x, y)$ for all points $x$ and $y$ , where $d$ is the Euclidean distance.

Remarks

Congruence is an equivalence relation on triangles. Congruent triangles have equal corresponding sides and equal corresponding angles. The side-side-side criterion gives the converse. The definition depends on the order of the vertices. For example, $ABC \cong ACB$ holds only when $d(A, B) = d(A, C)$ . Mathlib's Congruent asks for equal distances between all pairs of corresponding points of two indexed families. In a Euclidean space this is equivalent, by the side-side-side criterion.

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Euclidean distance concept
Triangle concept
Congruent triangles definition

Tag	`0X3TEM69AA`
URI	`m2://0X3TEM69AA`
Kind	definition
Writing	W0 stub
MSC	51M04
Defines	Congruent triangles
Uses	Triangle, Euclidean distance

Congruent triangles

Definition

Remarks

Reading path

Further reading

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