I recently received my DNA match from Ancestry we share 757 centimorgans shared across 35 DNA segments. Could this be a half brother instead of 1st cousin like it predicts? I have no history of my father.
There are reference values for determining relationship grade based on the amount of shared DNA. Please refer to http://thednageek.com/the-limits-of-predicting-relationships-using-dna/ I am totally sure that it is improbable that this person is half-brother. It should be half-uncle/nephew or 1st cousin.
If you are sure that you share direct paternal ancestor, it is good idea to check your yDNA. It will push you to buy Bigy or Y elite test for both.
If you suspect that you share direct maternal line, there is only option to test mtDNA.
Also it is very interesting if this person is X-Match to you. Chromosome X has different pattern of inheritance than autosomal DNA and may give you additional clues about relationship