Does rotation in plane involve Euclid's parallel postulate? It is given a line $l$, a point $P$ not on $l$, and $PQ$ perpendicular to $l$, i.e., $Q$ is on $l$. Let $R$ be a point on $l$ with $QR = PQ$. Therefore, $&lt;PQR = 90^{\rm o}$.

I can't figure out if Euclid's parallel postulate is involved when I rotate the object $PQR$ clockwise by $90^{\rm o}$. Any hint or reference is highly appreciated.
I'm quite positive rotation does not involve Euclid's fifth postulate. I'll try to present strict definition of rotation in synthetic setting.

Definition 1. An ordered pair $(A,B)$ of halflines having the same origin will be called a directed angle. Also denote $AB:=(A,B)$

Definition 2. We introduce a relation $\simeq$ among directed angles: We say $AB\simeq CD$ iff these angles are congruent (or equivalently have the same measure) and have the same orientation (in case $AB$ and $CD$ are not $0$ or $180$ degrees). Having the same orientation roughly speaking means that going from $A$ to $B$ and from $C$ to $D$ are both clockwise or both counterclockwise.

It can be proven that

Proposition 1. $\simeq$ is an equivalence relation.

Proposition 2. Given a halfline $A$ of origin $o$ and a directed angle $PQ$, there is a unique halfline $B$ of origin $o$ such that $AB\simeq PQ$.

Now if we fix a directed angle $PQ$ and a point $o$ on a plane, we can define a rotation around $o$ by a directed angle $PQ$ the following way: Obviously we map $o$ to $o$. Take $a\neq o$. By proposition 2 we can find a halfline $B$ of origin $o$ such that $AB\simeq PQ$, where $A$ is a halfline $\overrightarrow{oa}$. Next we find a unique point $b$ on $B$ such that $oa\equiv ob$. Finally we map $a$ to $b$.

What is the most important thing about rotations is that it can be proven in neutral geometry that they are isometries. It follows quite easily from lemma

Lemma. Let halflines $P,Q,P_1,Q_1$ all have the same origin. If $PP_1\simeq QQ_1$, then $PQ\simeq P_1Q_1$.

To prove that rotations are isometries you have to consider few cases but basically you apply the lemma and SAS congruence rule.
It seems that rotation indeed involves Euclids fifth postulate. Section 8 of this paper is all about that.