Let A = \ 1, 2, 3, 4, 5\ and R be a relation on A defined as R = \...

An equivalence relation on a set partitions the set into disjoint equivalence classes.

The given relation is $R = \{(1,2), (2,3), (4,5)\}$.

For the enlarged relation to be an equivalence relation, it must be reflexive, symmetric, and transitive.

The presence of $(1,2)$ and $(2,3)$ implies that $1, 2,$ and $3$ must belong to the same equivalence class. The presence of $(4,5)$ implies that $4$ and $5$ must belong to the same equivalence class.

To minimize the number of ordered pairs in the equivalence relation, we should not merge these two classes. Thus, the equivalence classes are $\{1, 2, 3\}$ and $\{4, 5\}$.

The equivalence relation corresponding to this partition is the union of the Cartesian products of the classes with themselves:
$\{1, 2, 3\} \times \{1, 2, 3\} \cup \{4, 5\} \times \{4, 5\}$.

The total number of ordered pairs in this equivalence relation is $3^2 + 2^2 = 9 + 4 = 13$.

Since $R$ already contains $3$ ordered pairs, the minimum number of ordered pairs that must be added is $13 - 3 = 10$.

Answer: $10$