a street that bisects yours

While a nit-picker might quibble about the use of bisect, yes, that is basically it, but helen's response missed my point. In the case of cross street vs. cross the street I meant in an equivalent context, as in "Don't cross (the) street without looking both ways."

Normally we would say that the is used when a specific thing is involved, e.g., "that is a book but it isn't the book I was talking about." In the street example however we are talking about a general piece of advice that holds for all streets (no quibbles about one way streets here, please; you can get killed just as dead by someone driving the wrong way on a one way street as you can by someone driving the right way) so why don't we say, "Don't cross a street without looking both ways", or just "Don't cross street without looking both ways"? We could certainly say, "Don't cross streets without looking both ways."