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 $Z$?

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?