I've struggled with this since I began modeling. It seems that bridging edges never works perfectly. I've watched Acemastermind's video about the three ways to connect polygons, and I've tried them on a simple cube and had no trouble. However when I tried to bridge the cyan edges on this sphere, it just didn't come out right. Bridge boundary edges looks like this, and adding a new polygon or welding the points using the weld tool either inverts all the polygons or creates a double edge. Any help is appreciated

That's weird, something must be wrong with your polygon normals, just start over again.
Create a sphere
then duplicate the sphere(ctrl+d) and scale it down a little
Merge both spheres together
delete the top half of the sphere object
invert the polygon normals only on the inside portion
then select the border edge loops of both the inside and outside portion by pressing the i key then select one edge with alt+mmb for the first edge loop then shift+alt+mmb on the other
then right click and select Bridge Boundary Edges and you're done.

Demo vid:
Bridge edges

Yes! Thank you, it works perfectly now.