I imported a PDF of an old map and made a layer of polygons for land use. This new layer, though, has no spatial reference. I cannot project the layer because there is no coordinate system to project it from.
How do I give this layer of polygons a spatial reference?
Importing an old pdf of a map will just give you a map referenced in pixel coordinate space. You have to ... georeference ... the file in order to provide it with real world coordinates. Just defining its coordinate system will make a bad situation worse.
The pdf of the old map was georeferenced and it sits nicely in place above the basemap.
That should not affect my new polygons layer. I did Define Projection to the same projection as the other layers in my map, but that has not helped.
you set the projection of the new polygons to that of the old map then, not your other files. If the new polygons are given a wrong coordinate system, they won't align with your old data. If you want the new file to be in a different coordinate system, you need to define the projection correctly first, then you use the Project tool to produce a new file with the desired coordinate system.
I wrote a visual guide as to what can happen if you mix and match the order of things... define projection vs project