"My mom is angry."
Translation:אמא שלי כועסת.
It's an Aramaic word and in Aramaic the definite article is at the end of the noun instead of the beginning. It would be redundant to add the Hebrew definite article. Aramaic was the primary Jewish language for centuries from roughly the Persian period to the end of the Talmudic period. Hebrew has always been the sacred language, but there are good reasons that there is so much Aramaic in the Bavli. Anyway, Hebrew nouns that end with א are usually Aramaic.
One of the solutions was "אמי כועסת". How does that work? Is "אמי" an informal way of indicating that the mother is your mother (possessed by you)? Is "אמא שלי" more formal?
It's actually the other way round - אמי is more formal. It's the possessive suffix, and can be used on any noun, but it is more formal than using "שלי". For example -
My house = הבית שלי/ביתי
My table = השולחן שלי/שולחני
My mother = אמא שלי/אמי
The reason you remove the 'א in "אמא", is because the word in Hebrew is actually אם. (the added 'א comes from Aramaic).