Often I find a lay person dispensing hadith with no context or reference to back up its authenticity.
How can I check a hadith for it's pretty authenticity?
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Well, at first we should know that the hadith explains text of Quran, as the way it has layers of meaning and everybody has its own Interpretation about them or even couldn't get the meaning, so it needs expounder, which we call their Interpretation “hadith".
So first of all we should check them with Quran, then we should check the source of it (the person who said it) also it should come from an acceptable or valuable reference (the source which you get the hadith from)
Furthermore in your mind it should seem logical.