View source for Parallel straight lines

''in Euclidean geometry''

Lines that lie in one plane and that do not intersect. In [[Absolute geometry|absolute geometry]] there passes through a point not lying on a given line at least one line that does not intersect the given one. In [[Euclidean geometry|Euclidean geometry]] there is only one such line. This fact is equivalent to Euclid's [[Fifth postulate|fifth postulate]] (on parallels). In [[Lobachevskii geometry|Lobachevskii geometry]] in the plane there pass through a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p0713801.png" /> (see Fig.) outside a given line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p0713802.png" /> infinitely-many lines that do not intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p0713803.png" />. Of these only two are called parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p0713804.png" />. A line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p0713805.png" /> is said to be parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p0713806.png" /> in the direction from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p0713807.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p0713808.png" /> if: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p0713809.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p07138010.png" /> lie on the same side of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p07138011.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p07138012.png" /> does not intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p07138013.png" />; and 3) every ray within the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p07138014.png" /> intersects the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p07138015.png" />. The line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p07138016.png" /> parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p07138017.png" /> in the direction from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p07138018.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071380/p07138019.png" /> is defined similarly.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071380a.gif" />

Figure: p071380a



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Parallel lines"  ''Canad. Math. Bulletin'' , '''21''' :  4  (1978)  pp. 385–397</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Greenberg,   "Euclidean and non-Euclidean geometries" , Freeman  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Hilbert,   "Grundlagen der Geometrie" , Teubner, reprint  (1968)</TD></TR></table>