Usually the bigger mystery is what is the adjusted basis?  (But maybe that's tomorrow's question.)  As Bob said, you can usually find the original purchase date online and you might even find the date of the gift/transfer.  If you're lucky you'll find the original purchase price (but that won't include any closing costs, seller incentives, subsequent improvements, accumulated depreciation, suspended PALs, etc.)

You're probably "stepping into the shoes" of the giver.  I'm sure they included a copy of the gift tax return that was filed reporting the adjusted basis of the gift, right?  As long as the FMV on date of gift was higher than the adjusted basis then you'll use adjusted basis as the starting point of "stepped into the shoes" basis for the gain/loss math (in which case I would use the original purchase date as date acquired on the 8949).

If the FMV on date of gift was lower than adjusted basis then you end up with what I call the "neutral zone".  Basis for loss is FMV on date of gift, basis for gain is adjusted basis.  Anything in between results in no gain/loss.

With real estate you're probably only looking at gains but we have no idea the timeline or numbers you're dealing with.  Either way I wouldn't stress too much about it, it's more important to get the correct tax treatment (LT/ST) than to figure out which specific date goes in the box.

Rick