In ΔABC and ΔDEF, if AB = EF, BC = DE and CA = FD, then

Concept

Two triangles are congruent by the SSS (Side-Side-Side) criterion when all three sides of one triangle equal the three sides of the other. Writing the congruence as ΔABC ≅ ΔXYZ is a claim about vertex correspondence: it asserts AB = XY, BC = YZ and CA = ZX in that order — matching only the set of three side lengths is not enough; the vertices must line up in the stated order.

Application

The given equalities are AB = EF, BC = DE and CA = FD. Find the assignment of D, E, F to the positions X, Y, Z (so that ΔABC ≅ ΔXYZ) that makes AB = XY, BC = YZ, CA = ZX hold together:

1. AB = EF means XY must equal EF, so X and Y are F and E in some order; take X = F, Y = E, giving XY = FE = EF.

2. With Y = E fixed, BC = DE means YZ = EZ must equal DE, so Z = D.

3. Check the third equality with X = F, Z = D: CA = FD requires ZX = DF, and indeed DF = FD.

Cross-check

With X = F, Y = E, Z = D, all three given equalities hold simultaneously: AB = FE = EF, BC = ED = DE, CA = DF = FD. Since all three sides match under this single consistent vertex order, ΔABC ≅ ΔFED by SSS.