How are a priori synthetic judgements possible?

Lecture 4/8. Kant claims that, "our sense representation is not a representation of things in themselves, but of the way in which they appear to us. Hence it follows that the propositions of geometry... cannot be referred with the assurance to actual objects; but rather that they are necessarily valid of space... [and] space is nothing else than the form of all external appearances". [Prolegomena 286-287]