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If two triangles have all their angles equal, then the lengths of one side divided by its equivalent on the other triangle is the same for all sides.
By equivalent sides it is meant the sides opposite the same angle.
Consider 2 ∇s ABC EFG whereConsider 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