Rationality requires belief in God
It is a common perception in the (post?) Christian world that 'faith' or 'belief' is irrational; that a matter of faith is one for which there is little or no evidence. To the Muslim this seems quite absurd. In Islam people who disbelieve in Allah and in the prophethood of Muhammad can only do so out of ignorance or out of wilful denial of what is evident to them as clear as seeing:
Say thou: "This is my way: I do invite unto Allah on evidence clear as the seeing with one's eyes, I and whoever follows me: Glory to Allah! and never will I join gods with Allah!"
Qur'an Surah 12 Verse 108
For the Muslim, belief in Allah forms a foundational requirement of rationality and belief in Muhammad as the prophet of Allah follows from applying the rational mind to the evidence of his mission.
There are many ways in which we may describe rational thinking. From the world view of Islam there is good thinking and bad thinking and 'rational' thinking is nothing more than morally good thinking. Someone who strives to do that which is right in the way he thinks, who is sincere and avoids self deception, someone who seeks the truth, avoids biases and prejudice in his opinions, who seeks knowledge from all available sources, someone who reflects on what he sees and always looks for explanations and doesn't accept that things are as they are simply without any reason, someone who tries to avoid any contradictions in his understandings, such a person is rational and such a person is one who seeks to think, learn and decide in a morally good way.
If someone tries to do what is morally right they are already acknowledging that there is a moral good to achieve. Such moral value only exists if existence as a whole achieves something morally good, i.e. if existence has a purpose.
If you think that existence as a whole has no purpose and achieves nothing, then it follows that whatever you do in life achieves exactly nothing as well - your life would contribute no value to existence.
There is no moral good in doing what your own desires find good or what your family or nation finds good. The only way what you do achieves a moral good is if it achieves something towards the purpose of existence. Every good that can be achieved is either instrumental to the purpose of existence and therefore a true moral good or it is false good and worth nothing.
The core of all value and purpose to existence is commonly known by the name Allah or God. He is the ultimate purpose to existence and the achievement of what is morally good equates to getting closer to Him. Allah is more than we can ever fully understand but in order to have morally good thinking it is essential to believe in Allah in some sense whatever name we use for Him.
It is also essential to reject as having any hold over us anything or anyone who asserts that a good can be achieved without it being instrumental to the purpose to existence. i.e. without it being for the sake of Allah.
A person who has amoral thinking or immoral thinking acknowledges that there is no inherent value to the way they think and will see no problem in changing the criteria by which they decide to accept or reject conclusions. Whatever definition of rationality could be used there is, for such people, no reason why they should accept it. If being rational by some definition leads to conclusions they don't like then they can just change the definition. In extreme cases there can never be enough evidence and strong arguments for them to accept a statement as true, they just continually move the criteria of what constitutes for them convincing evidence and argument.
Being rational means being a moral thinker.
Being a moral thinker means believing in Allah.
Only the irrational disbelieve.
To understand more about moral thinking please read the section on The Sin of Disbelief in the online book Islam for the Western Mind