By equivalent sides it is meant the sides opposite the same angle.
Consider 2 ∇s ABC EFG where ∠CAB = ∠GEF, ∠ABC = ∠EFG, ∠BCA = ∠FGE
Let ∇ABC be the lesser of the two triangles.
Place ∇ABC on ∇EFG so that point A coincides with point E and AB lies on EF and AC lies on EG. This can be seen on the bottom right in the accompanying diagram.
∠GFE = ∠CBE
∴ FG parallel to BC ... (corr. ang.s equal)
∴ EB:FB = EC:EG ... line drawn through ∇ parallel to side divides other sides proportionally
∴ EB:FB = EC:CG and as EB = AB and EC = AC
AB:FB = AC:CG
∴ AB:EF = AC:EG and AB:AC = EF:EG
... (Quantities in proportion are also in proportion by composition )
Similarly by placing ABC on the other corners so that B coincides with F and then C coincides with G, it can be shown
BC:FG = BA:FE and BC:BA = FG:FE
CA:GE = CB:GF and CA:CB = GE:GF