# Parallel Lines

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## Parallel Lines

In Euclidean geometry, parallel lines are coplanar lines that do not intersect. In absolute geometry, there is at least one line that passes through a point not lying on a given line and that does not intersect the given line. Only one such line exists in Euclidean geometry (Euclid’s fifth, or parallel postulate). In Lobachevskian geometry, a plane contains an infinite set

Figure 1

of lines that pass through a point C not lying on a given line AB and that do not intersect AB (see Figure 1). Only two of these lines are said to be parallel to AB. The line CE is said to be parallel to AB in the direction from A to B if (1) B and E are on the same side of AC, (2) CE does not intersect AB, and (3) every ray within the angle ACE intersects AB. The line CF parallel to AB and directed from B to A is analogously defined.

References in classic literature ?
In the dust of years that lay thick upon the floor--leading from the door by which they had entered, straight across the room to within a yard of Manton's crouching corpse--were three parallel lines of footprints--light but definite impressions of bare feet, the outer ones those of small children, the inner a woman's.
He acquiesced in the suggestion; and we immediately, therefore, began surveying the country lying before us, with a view of determining upon the best route for us to pursue; but it presented little choice, the whole interval being broken into steep ridges, divided by dark ravines, extending in parallel lines at right angles to our direct course.
The trappers proceed across the prairie in a long line; or sometimes three parallel lines, sufficiently distant from each other to prevent the packs from interfering.
He desired that ship to unite with his own in the search; by sailing over the sea some four or five miles apart, on parallel lines, and so sweeping a double horizon, as it were.
She is happy, she makes another person happy, and she's not broken down as I am, but most likely just as she always was, bright, clever, open to every impression," thought Darya Alexandrovna,--and a sly smile curved her lips, for, as she pondered on Anna's love affair, Darya Alexandrovna constructed on parallel lines an almost identical love affair for herself, with an imaginary composite figure, the ideal man who was in love with her.
I will here give a very brief sketch of the geology of the several parallel lines forming the Cordillera.
The black, in defence, aimed a kick at Jerry, who, leaping in instead of away--another inheritance from Terrence--avoided the bare foot and printed a further red series of parallel lines on the dark leg.
Saxon remembered her school geography, and with her mind's eye she saw a certain outline map of a continent with jiggly wavering parallel lines that denoted coast.
These words were, so to speak, the point of union of two scenes, which had, up to that time, been developed in parallel lines at the same moment, each on its particular theatre; one, that which the reader has just perused, in the Rat-Hole; the other, which he is about to read, on the ladder of the pillory.
Across their foreheads were tattooed three parallel lines of color, and on each breast three concentric circles.
All through April he was putting the perennials we had sown in the autumn into their permanent places, and all through April he went about with a long piece of string making parallel lines down the borders of beautiful exactitude and arranging the poor plants like soldiers at a review.
Hawkeye was not mistaken; for when the Hurons found their course was likely to throw them behind their chase they rendered it less direct, until, by gradually bearing more and more obliquely, the two canoes were, ere long, gliding on parallel lines, within two hundred yards of each other.

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