Suppose that $X$ is a Whitney stratified algebraic variety with strata $\{S_i\}.$ Suppose that $Z$ is a hypersurface of $X$ which transversely intersects all strata of $X$, i.e. $S_i \cap Z$ is a smooth hypersurface of $S_i.$
Is it true that $\{S_i \cap Z\} $ is a Whitney stratification of $Z$? Is it true if $X$ is compact?
In order for transversality to make sense, I assume that X is contained in an ambient manifold? Assuming so, the following statement is on page 37 of Goresky and MacPherson's Stratified Morse Theory:
"The transversal intersection of two Whitney stratified spaces is again a Whitney stratified space, whose strata are the intersections of the strata of the two spaces [Ch]." And [Ch] is a reference to: Cheniot D: Sur les sections transversales d'un ensemble stratifie. C. R. Acad. Sci. Paris. ser. A275 A972), A915-A916
So assuming your hypersurface of $X$ means the intersection of $X$ with a hypersurface of the ambient manifold, then the answer appears to be "yes."
If you're not thinking of $X$ as being contained in an ambient manifold, can you say a bit more about what you mean by transverse intersections? (For that matter, I think $X$ needs to be contained in a manifold at least locally for the Whitney conditions to make sense, and in that case I think you can probably apply this statement locally to get the global result.)
