How can one deal with aggressive and rude people who habitually get into arguments and pull others into it even when avoided?

People who ask questions are of two types:

1. Sincere people who are honestly misinformed - they need to be sat down with and spoken to with gentleness and wisdom. Allah says in the Qur’an, “Call to the way of Your Lord with Wisdom and gentle preaching, and debate with them in a way that is best.” [1]
2. Problem-makers who simply intend to argue and promote wrong ideas without listening to the opposing view. They should be given a brief answer, more for the benefit of others and their fight should not be entertained. The Qur'an teaches us not to engage in debate with foolish people. Allah says in the Qur’an, “The servants of The Most Merciful are those who walk on this earth with humility, and when the foolish try to argue with them, they respond with peace.” [2]

Allah knows best.

References:

[1] Surah an-Nahl, 16: 125

[2] Surah Al-Furqan, 25:63