The paper deals with weak congruences of algebras having at least two constants in the similarity type. The presence of constants is a necessary condition for complementedness of the weak congruence lattices of non-trivial algebras. Some sufficient conditions for the same property are also given. In particular, so called 0,1-algebras have complemented weak congruence lattices if and only if their subalgebra lattices are complemented. In this context we also investigate relations among algebras with balanced congruences, balanced weak congruences, consistent and strongly consistent algebras. We prove that an algebra has balanced weak congruences if and only if it is strongly consistent and has balanced congruences on all subalgebras. For a variety, strong consistency of algebras is equivalent with having balanced weak congruences. Finally, we prove that for a class of algebras which additionally are Hamiltonian, there is a homomorphism from the congruence lattice onto the subalgebra lattice.