Learn Russian: Почему ты не поверил твоей умной сестре? - Why didn't you believe your smart sister?

They are describing the sister, so they have to match the noun сестре. The verb поверить (to believe) requires the person you believe to be in the Dative case. So, сестра becomes сестре, and the words for 'your' and 'smart' also take their feminine Dative singular forms: твоей and умной.

Is it a coincidence that both твоей and умной end in a similar way?

Not a coincidence! In the feminine Dative case, possessive pronouns and adjectives share very similar endings. Most adjectives take -ой or -ей (like умной), and possessive pronouns take -ей (like твоей and моей).

In English we just say 'believe your sister.' Why does Russian use the Dative case here?

Think of the Dative case as the 'to' or 'giving' case. In Russian, you do not believe someone directly. Instead, you 'give your belief TO someone.' That is why verbs like верить and its completed form поверить always need the Dative case for the person being believed.

Why is the verb поверил masculine if the sentence is about a smart sister?