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I know that both can mean "correct". I frequently hear things like "合ってる？" to mean :"is it right?" I don't know much about the use of 正しい, but I know that it's also a common word for "correct".
What would be the difference between the two? Is there a preference for one over the other in certain situations and formality levels?
I understand your question is "what is the difference between 合う and 正しい？" as is said by Aeon Akechi.
合う means basically A is equal to B or A and B are equal. So if it is used like "あなたの答｛こた｝えは合っている", the phrase means "your answer is equal to the correct answer".
On the other hand 正しい means basically "right" along with the meaning of "correct".
So, 正 in 正しい is used in both ways.
For example, 正解{せいかい} means a correct answer, while 正義｛せいぎ｝ means justice, right or righteousness.
合う is a verb, 正しい is an adjective.
I've seen 合う's primary use as an intransitive verb with the meaning of "to fit" or "to match".
Examples: その洋服は彼女の赤い髪に合う。(That dress matches her red hair.) このジャケットはぴったり合う。(This jacket sets well.)
正しい's use as an i-adjective is a little different. 私の意見は大抵において正しい。(My opinion is generally correct.) 厳密に言うと、君は正しくない。(Strictly speaking, you're not correct.)
