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The generalization of the Morse theory presented by Goresky and MacPherson isa landmark that divided completely the topological and geo\-me\-tri\-cal studyof singular spaces. Let \{$X_t\}_t$ be a suitable family of germs at $0$ ofcomplete intersection varieties in $\mathbb{C}^n$ and $\{f_t\}_t, \{g_t\}_t$families of non-constant polynomial functions on $X_t$. If the germs $X_t$,$X_t \cap f_t^{-1}(0)$ and $X_t\cap f_t^{-1}(0) \cap g_t^{-1}(0)$ arenon-degenerate, locally tame, complete intersection varieties, for each $t,$ weprove that the difference of the Brasselet numbers, ${\rm B}_{f_t,X_t}(0)$ and${\rm B}_{f_t,X_t\cap g_t^{-1}(0)}(0)$, is related with the number of Morsecritical points {on the regular part of the Milnor fiber} of $f_t$ appearing ina morsefication of $g_t$, even in the case where $g_t$ has a critical locuswith arbitrary dimension. This result connects topological and geometricproperties and allows us to determine some interesting formulae, mainly interms of the combinatorial information from Newton polyhedra.

Stratified Morse critical points and Brasselet number on non-degenerate locally tame singularities