My proofs are the following:
In Gof, Dumbledore says (in reference to the Goblet of Fire), "Only students who are of age---that is to say, seventeen years or older---will be allowed to put forward their names for participation." (American edition, page 188)
He also says, (in reference to the Age Line), "To insure that no underage students yield to temptation [...] Nobody under the age of seventeen will be able to cross this line." (American edition, page 256)
So, 16-and-unders are underage, while 17-and-overs are not. Though, I could be taking this the wrong way. Dumbledore may not be using the term "underage" in a legal sense---just to say that 17-and-overs are overage only in the Triwizard Tournament. But here's something else:
In OotP, Arthur refers to Fred and George, saying, "They are of age [...] they're legally adults now." (American edition, page 91) This is not in reference to any event; it's a general statement. Fred and George's ages are not mentioned in the section, but in GoF it is said that they turn 17 April, which is only a couple months before the aforementioned quote from Arthur occurs. They are 17 at the time.
I don't know whether or not what I'm stating is stupidly obvious, and therefore pointless. It's just that I've never seen anyone (in a story or not) acknowledge that the characters are legal adults once they turn 17.
Am I silly? Correct? Or, perhaps, does the term "legal adult" mean something in Britain that I am not grasping?