In Plane and solid geometry by Fletcher Durell, he mentions in page 34 that
91. Homologous angles of two mutually equiangular triangles are corresponding angles in those triangles.
Homologous sides of two mutually equiangular triangles are sides opposite homologous angles in those triangles.
I hope to see that, length of an opposite side of the given angle doesn't have to be influenced by it's opposite angle. So why does the author say that "homologous sides are sides opposite to homologous angles"?
I'm asking this question because in some proofs like in Proposition XVI : it says triangle $F'BH$ = triangle $BHC$, so $F'H$ = $CH$ (homologous sides of equal triangles). What is the author trying to say by this statement?
It is a definition, a name to refer to these sides. It does not imply that their length are pairwise equal in general (although the ratio of the lengths is the same for all three pairs).
In the example, it is important to note that it is not "homologous sides of mutually equiangular triangles" but "homologous sides of equal triangles". Equal triangles, in this context, seems to mean isometric.