Welcome, then. to the crazy world of beading! Weaves, such as the right angle, can have the original thread 'woven' through what you have already created. If you make many twists and turns, threw woven beads, the thread will be secure and you will notice that you cannot see it pull tension from where you finished. Then, cut that thread very close to where you last existed and pull taught while cutting. Any small piece of thread left should shrink back into a bead if you pull taught when cutting.
Cut another comfortable length of thread and basically do the same thing, weave in and out, up and down threw woven beads till you can see it is taught then exit out where you ended.
Because many seed beads can only handle a few number of passes through its hole, I would suggest trying to keep the two woven trails separate from themselves. It's no fun to try and squeeze a needle through a bead that will not allow it and have the bead break!
This is only my way of cutting an old thread and starting new. Every beader has their own comfortable way to complete the same task, but they are all very secure.
Erin