From my experience in the USA, the deadlines for the homeworks were possibly in the syllabus / in the online course management system, and the professor was not obliged to announce them at the end of the class. Your experience, however, might be different.
In any way, I would simply write a short note explaining that you failed to submit the homework on time, and asking for an extension / for the possibility to submit the homework. If technically possible, most professors will agree to do that.
Do not: blame the professor for not having announced the assignment. That won't take you far.