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line, plane, lines and parallel

PERPENDICULAR, (from the Latin perpendicularis,) in geometry, a line falling directly on another line, so as to make equal angles on each side ; called also a normal line.

From the very notion of a perpendicular, it follows : 1. That the perpendicularity is mutual ; i. e., if a line, as o, be perpendicular to another, II ; that other is also perpendicular to the first.

2. That only one perpendicular can be drawn from one point in the same plane.

3. That if a perpendicular be continued through the line to which it was drawn perpendicular, the continuation will also be perpendicular to it.

4. That if there be two points of a right line, each of which is at an equal perpendicular distance from two points of another right line, the two lines are parallel to each other.

5. That two right lines perpendicular to one and the same line, are parallel to each other.

6. That a line, which is perpendicular to another, is also perpendicular to all the parallels of the other.

7. That perpendiculars to one of two parallel lines, ter minated by those lines, are equal to each other.

S. That a perpendicular line is the shortest of all those which can be drawn from the same point to the same right line.

'Hence the distance of a point from a line, is a right line drawn from the point perpendicular to the line or plane ; and hence the altitude of a figure is a perpendicular let fall from the vertex to the base.

Perpendiculars are best described in practice by means of a square ; one of whose legs is applied along that line, to or from which the perpendicular is to be let fall or raised.

A line is said to be perpendicular to a plane, when it is perpendicular to all right lines, that can be drawn in that plane, from the point on which it insists.

A plane is said to be perpendicular to another plane, when all right lines drawn in the one, perpendicular to the common section, are perpendicular to the other.

If a right line he perpendicular to two other right lines, intersecting each other at the common section, it will be per pendicular to the plane passing by those two lines.

Two right lines perpendicular to the same plane are parallel to each other.

It; of two parallel right lines, the one is perpendicular to any plane, the other must also be perpendicular to such plane.

If a right line be perpendicular to a plane, any plane pass ing by that line will be perpendicular to the same plane.

Planes, to which one and the same right line is perpen dicular, are parallel to each other : hence all right lines per pendicular to one of two parallel planes, are also perpendicular to the other.

If two planes, cutting each other, be both perpendicular to a third plane, their common section will also be perpendicular to the same plane.