The monotonicity inequalities for the \(r\)-area of a complete oriented properly immersed \(r\)-minimal hypersurface in Euclidean space under appropriate quasi-positivity assumptions on certain invariants of the immersion are established. The proofs are based on the corresponding first variational formula. As an application, the degeneracy theorem is derived for an entire \(r\)-minimal graph whose defining function \(f\) has first and second derivatives decaying fast enough at infinity: its Hessian operator \(D^2f\) has at least \(n-r\) null eigenvalues everywhere.