| Personable. Don't let them dismiss your letter as "just another appeal" - they get thousands every year. Universities want people with determination and integrity, but perhaps most importantly, character, not just another number to add to their statistics. Make sure you come across as both committed and purposeful, as well as a REAL person with real passion and an appetite for excellence. Use language fluently, adding sentences that convey YOU as a person. Don't let it sound as if a robot could've written it.
Logical. Tell them exactly why you're a perfect candidate for a position at their institution. Prove to them that you shouldn't have been dismissed by SHOWING them something. They've seen your application before and there was obviously a reason they rejected you. Don't just use the appeal as a re-application; they're not going to change their mind for nothing. SHOW them that you have changed, your application has changed, or that for some tangible, tenable reason you should be reconsidered.
Professional. Fairly self-explanatory. For a school like UCLA you want to make sure they view you as a mature, eloquent young person and that they feel they should take you and your plea seriously. Don't use any highfalutin language or pompous jargon or you'll just come off as pretentious. Show them that you're an intelligent, articulate and capable student by speaking clearly and effectively. Be persuasive but not forceful. Don't ramble on for pages. You want them to read the first few sentences of your appeal and actually reconsider your dismissal right there and then.
Respectful. Don't attack them for not letting you in. You're imploring them to reconsider your application, not trying to make them feel bad for not accepting you in the first place. It may sound elementary, but it's a mistake a lot of people make when attempting to put forth an argument for an appeal. Many don't mean to but end up coming off as disrespectful, if not downright rude. They have the power to reject you for any reason, just make sure it's not for p*ssing them off.
NB: This is coming from a student who has not yet applied, been accepted into, been rejected by or written an appeal to a US university. She just wanted to ramble for a while. Take all of the above with a grain of salt.
And good luck. | 
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