It is shown that if {yn} is a block of type I of a symmetric basis {xn} in a Banach spaceX, then {yn} is equivalent to {xn} if and only if the closed linear span [yn] of {yn} is complemented inX. The result is used to study the symmetric basic sequences of the dual space of a Lorentz sequence spaced(a, p). Let {xn,fn} be the unit vector basis ofd(a, p), for 1≤p<+∞. It is shown that every infinite-dimensional subspace ofd(a, p) (respectively, [fn] has a complemented subspace isomorphic tolp (respectively,lq, 1/p+1/q=1 when 1<p<+∞ andc0 whenp=1) and numerous other results on complemented subspaces ofd(a, p) and [fn] are obtained. We also obtain necessary and sufficient conditions such that [fn] have exactly two non-equivalent symmetric basic sequences. Finally, we exhibit a Banach spaceX with symmetric basis {xn} such that every symmetric block basic sequence of {xn} spans a complemented subspace inX butX is not isomorphic to eitherc0 orlp, 1≤p<+∞.