A preference is rational if it is complete and transitive. You need both properties satisfied for it to be rational. So an intratransitive and complete preference will still be irrational.
Indifference and strong preference on the other hand are relations derived from the preference relation itself.
The preference relation $\succcurlyeq$ itself says $x$ is at least as good as $y$, where $x$ and $y$ are two alternatives available to a decision maker. Formally $x \succcurlyeq y$ .
Strong preference relation $x \succ y$ (read as $x$ is strictly preferred to $y$), $\iff$ $x \succcurlyeq y$ and NOT $y \succcurlyeq x$.
Indifference relation $x \sim y$ (read as x is indifferent to y) $\iff$ $x \succcurlyeq y$ and $y \succcurlyeq x$