We had redid our cupboards a couple years ago and still have to properly cover the hole that was left for the vent on the previous vent system. It looks like they used staples of some type but the way it is just needing 1 and a half (or less) tiles it‘s awkward.
Most ceiling tiles have a tounge groove so each tile fits into another. When you are piecing in tiles it's impossible to get the tounges in the grooves without breaking them. I would suggest, cutting off the tounges with a razor knife and install with Liquid nails or some other construction adhesive. To keep the tiles in place while the glue is drying, cut a 2X4 long enough and put it under the tile while the glue sets up. You might want to put a towell between the tile and the 2X4 to keep from damaging the tile. For the 1/2 tile, score the backside of a tile with a razor knife then break it in half.
