How do you confirm a 19th Century family relationship?

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This is always a tough question, and (depending on the local record keeping) requires some work to answer. I'm not sure if you have pursued this, but the best way to prove such relationships are from surviving legal documents. If there is a will (do you know where the father died?), administration of an estate (even a small one) or a land transaction such as a deed or purchase, you may find such a reference. Another source are church records, especially if you know what town or city they lived in (esp true for Catholic church records). There are also obituaries (esp in church newspapers) in that era that often contain family details. There are indices for this. Check with the LDS resources.
The later censuses do carry the exact relationship, but 1830s is a problem. Finally, if there are local archives, you may find a county history or other references that mention the family).
Good Luck  
Are you sure the census can't provide you with some clues? Like the number of males or females of a certain age in the household. You could use this data to look for brothers and/or sisters.
What county/township are they listed in? Often relations will live nearby. If not a few households away, then maybe in neighboring townships.
You might look into a book or website that offers advice on how to get the most out of census records - even info you may not realize is useful - and where to go next when you've exhausted those leads.
Here are a couple of books that offer alternative research resources: Hidden Sources: Family History in Unlikely Places
Locating Your Roots: Discover Your Ancestors Using Land Records
Also, have you checked surname websites or message boards (like's boards)for other people researching the same surnames. You might get lucky and find a 'distant cousin' with the missing piece to your puzzle. Or you mind find a volunteer who lives in the area who would be willing to help with your research.
Good luck.
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