The rigid tubular appendages involved in the transfer of genes during bacterial conjugation are called pili (also known as fimbriae).

Here's a breakdown:

* Pili are hair-like structures on the surface of bacteria.

* Fimbriae are a specific type of pili that are shorter and more numerous than other pili.

* During conjugation, a sex pilus (a specialized type of pilus) extends from the donor bacterium to the recipient bacterium, forming a bridge between them.

* This bridge allows the transfer of genetic material, typically in the form of a plasmid, from the donor to the recipient.

So, the answer is pili, specifically sex pili.