Have Smart Guides and (if you want point snaps) Snap to Point ON, no other snaps.
You can draw a dummy straight line along the wanted route, stretch it to wanted length by holding Shift (=the angle stays) Convert it to guide by applying View > Guides > Make Guide. Actually a guide is not a must, the dummy line does the same, but it can cause harms if ...
So, you want to continue path AB from the endpoint B so that B has symmetric anchor handles, just the same handles as B had before you started the continuation process. You are NOT going to draw approximately right new handles.
There's a workaround:
Make a 180 degrees rotated copy (=blue) of the curve. Move it so that the copy starts from B. It should snap ...
You can only continue a path by clicking the end anchor with the Pen Tool.
If you want a handle on the continuation side, then you need to click-drag with the Pen Tool.
Merely selecting the end anchor with the Direct Selection Tool has never had anything to do with continuing the path in Illustrator. I can't speak to Fireworks, never used it.