---
tag: 0BRBYRK6PA
kind: statement
role: theorem
name: Side-side-side congruence
aliases: [sss-congruence]
msc: [51M04]
uses: [093EDENVA1, 0NPPGHQNXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Congruence_(geometry)
    locator: Determining congruence
---

## Statement

Let $ABC$ and $A'B'C'$ be triangles in the Euclidean space $\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].
Then $ABC$ and $A'B'C'$ are [[congruent|093EDENVA1]] if and only if
$$d(A, B) = d(A', B'), \quad d(B, C) = d(B', C') \quad \text{and} \quad d(C, A) = d(C', A').$$

## Remarks

Other criteria for congruence are side-angle-side, angle-side-angle and angle-angle-side.
Side-side-angle is not a criterion: two sides and an angle that is not between them can fit two different triangles.
Angle-angle-angle is not a criterion either. Triangles with equal angles are [[similar|0YF6QXXWB5]], but they can have different sizes.


## Explanations

### L1 human

If you know the three side lengths of a triangle, you know the whole triangle. Three sticks of fixed lengths make one rigid shape, up to a turn or a flip. This is why triangles make frames stiff.