Let us assume A,B,C,D,E,F be 6 vertices in the graph G. Choose any vertex A in the graph G .Now, by pigeon hole principle we will get one of G or G' would contain at least 3 edges from A .
WLOG assume that the graph G contains at least 3 edges from A suppose the three edges be AB,AC,AD.
Now, if the edge BC belongs to G then the triangle ABC belongs to G.
if the edge CD belongs to G then the triangle ACD belongs to G
if the edge BD belongs to G then the triangle ABD belongs to G.
Now, if none of the edges BC , CD, BD belongs to G then the triangle BCD belongs to G.
In any way we have a triangle belongs to either G or G'
So, one of G or G' must contain a triangle.