parallel

Two lines $l$ and $m$ are parallel if there is no point $P$ such that $P$ is incident to both $l$ and $m$. Symbolically, $l\parallel m$.

The Euclidean parallel postulate states:

For every line $l$ and every point $P$ not on $l$, there is exactly one line $m$ incident to $P$ such that $l\parallel m$.

See also: Euclidean geometry

The Elliptic parallel postulate states:

For every line $l$ and every point $P$ not on $l$, there is no line $m$ such that $m$ is incident to $P$ and $l\parallel m$.

The Hyperbolic parallel postulate states:

For every line $l$ and every point $P$ not on $l$, there are at least two lines $m$ and $n$ both incident to $P$ such that $l\parallel m$ and $l\parallel n$.