An indexing 'head' (you said plate-?) often called a (super) spacer, divides 360 degrees into common and equal divisions (2, 3, 4, 6, 8, 12, 16, 24 might be common). Those are the only choices you get. It will make those divisions quickly and hold them rigidly. They are often used for production work, but can certainly be used for one off jobs. You will need the masking plates to be able to avoid indexing where you do not want to go.
A dividing head is used mostly for making larger numbers of divisions on things like gears, sprockets, and such, and quite accurately. In general, the setups are not as rigid and take more time to prepare, but there are exceptions. They are fairly slow to set up, but can be relatively quick to change out parts. It is possible to set a dividing head up for just doing one interval, but I would be looking for another way to get there.
Neither of those options is very useful for laying out single odd angles, like 139 degrees. A rotary table does that job much more quickly and easily. Note that there exist rotary tables with included or optional dividing plates, which can do a lot of useful things in a home shop, but are not so great for most production work unless you are clever and make quick change work setups for swapping out parts.