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enter image description here How do I move this selected faces up along the Z axis and snap it to the unselected faces? They were originally connected together, but I cut them out initially thinking I didn't need them. The vertices that are supposed to be connected together are on the same Z axis. I tried different settings in snapping but the selected faces seem to just jump all over the place. I need to keep the sizes of the selected faces the same, so I can't just select vertices and snap them up. I need to move everything up together.

Thank you.

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What I use is the Bridge Edge Loops function. It's not perfect since it won't move all the faces but will connect the top edges cleanly.

Make sure you are either in Edge or Vertex Mode and select the two Edge loops you want to connect.

Select edges

Press W and select Bridge Edge Loops.

Bridge edge loops

In the operation settings, check Merge and set Merge Factor to 0. If 0 moves the wrong Edge loop then set the value to 1.

Enter values

Result

Lastly you'll have to move the rest of the faces manually. It's not perfect but works well enough for my own usage. I believe this is what user277143 is referring to but as I can't add comments yet I instead created a new answer.

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Face mode snapping isn't often useful since it deals with face middle points, you want to snap vertices here.

With faces selected switch to Vertex select mode. Hold Shift and right click RMB a vertex of your selection that you want to use for snapping to make it the active vertex. Set snap type to Vertex and from snap options dropdown select Active (closest would probably work too).

Now when you start moving this active vertex is going to be placed at a vertex you are pointing at. Of course you can still hit Z to restrict translation to Z-axis.

Afterwards you might want to merge the vertices. Easiest way is to select all vertices by hitting A and using CTRL + V (or W) vertex menu option Remove doubles.
Make sure the threshold is small enough to not to merge vertices elsewhere ie smaller than your smallest intended distance between vertices.

Finally you can verify the integrity of surface by selecting none in vertice mode (A) and hitting CTRL+ALT+SHIFT+M which selects any non-manifold vertices in the mesh.

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I think there is no magic bullet.

Either:

  1. You select a pair of vertices and go to W > Merge > [Select the one you want].

  2. You select the two edge loops in Edge selection mode and go Space > search for Bridge Gap (or Gaps, I am not sure off the wording.)

The first way merges the vertices, the second creates faces inbetween.

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