— GEOMETOR • elements

G.7 plane surface

A plane surface lies evenly with the lines on itself.

A circle ( A B ) is a plane figure contained by one arc defined by a center A and a point B establishing its defining radius / A B /, such that all segments / A B /, / A C /, / A D / falling upon the arc from A are equal.

G.22 plane angle

A plane angle is the inclination to one another of two lines or arcs in a plane that meet and do not lie on the same line.

G.52 parallel lines

Parallel lines are lines in the same plane which, extended indefinitely in both directions, do not meet.

G.319 similar plane numbers

Between two similar plane numbers there is one mean proportional number; and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side.

G.322 mean proportional number

If one mean proportional number fall between two numbers, the numbers is similar plane numbers.

G.502 part segment plane

A part of a segment cannot be in the plane of reference and a part in a plane more elevated.

G.503 segments cut plane

If two segments cut one another, they are in one plane, and every triangle is in one plane.

G.504 planes cut common

If two planes cut one another, their common section is a segment.

G.505 line at right angles to a plane

A:term:segment <G.4> is:term:**at right angles to a plane** <right-angles-to-plane>, when it makes:term:right angles <G.33> with all the segments which meet it and are in the:term:plane <G.7>.

G.506 segment set right

If a segment be set up at right angles to two segments which cut one another, at their common point of section, it is at right angles to the plane through them.

G.507 segment set right

If a segment be set up at right angles to three segments which meet one another, at their common point of section, the three segments are in one plane.

G.508 segments right angles

If two segments be at right angles to the same plane, the segments is parallel.

G.509 segments parallel points

If two segments be parallel and points be taken at random on each of them, the segment joining the points is in the same plane with the parallel lines.

G.510 segments parallel right

If two segments be parallel, and one of them be at right angles to any plane, the remaining one is at right angles to the same plane.

G.511 straight lines parallel

Straight lines which are parallel to the same segment and are not in the same plane with it are also parallel to one another.

G.512 segments meeting parallel

If two segments meeting one another be parallel to two segments meeting one another not in the same plane, they will contain equal angles.

G.513 elevated point draw

From a given elevated point to draw a segment perpendicular to a given plane.

G.514 set segment right

To set up a segment at right angles to a given plane from a given point in it.

G.515 point segments set

From the same point two segments cannot be set up at right angles to the same plane on the same side.

G.517 planes segment right

Planes to which the same segment is at right angles is parallel.

G.518 segments meeting parallel

If two segments meeting one another be parallel to two segments meeting one another, not being in the same plane, the planes through them are parallel.

G.519 parallel planes cut

If two parallel planes be cut by any plane, their common sections are parallel.

G.520 segments cut parallel

If two segments be cut by parallel planes, they is cut in the same ratios.

G.521 plane at right angles to a plane

A:term:**plane is at right angles to a plane** <plane-at-right-angles-to-plane> when the:term:segments <G.4> drawn, in one of the:term:planes <G.7>,:term:at right angles <G.33> to the common section of the planes are at right angles to the remaining plane.

G.522 segment right angles

If a segment be at right angles to any plane, all the planes through it is at right angles to the same plane.

G.523 planes cut right

If two planes which cut one another be at right angles to any plane, their common section is at right angles to the same plane.

G.524 solid angle

A:term:**solid angle** <solid-angle> is the inclination constituted by more than two:term:lines <G.2> which meet one another and are not in the same:term:surface <G.533>, towards all the lines. Otherwise: A:term:**solid angle** <solid-angle> is that which is contained by more than two:term:plane angles <G.22> which are not in the same:term:plane <G.7> and are constructed to one:term:point <G.1>.

G.525 solid angle contained

If a solid angle be contained by three plane angles, any two, taken together in any manner, are greater than the remaining one.

G.530 solid contained parallel

If a solid be contained by parallel planes, the opposite planes in it are equal and parallelogrammic.

G.531 parallelepipedal solid cut

If a parallelepipedal solid be cut by a plane which is parallel to the opposite planes, then, as the base is to the base, so will the solid be to the solid.

G.532 segment point construct

On a given segment, and at a given point on it, to construct a solid angle equal to a given solid angle.

G.537 prism

A:term:**prism** <prism> is a:term:solid figure <G.529> contained by:term:planes <G.7> two of which, namely those which are opposite, are equal, similar and:term:parallel <G.52>, while the rest are parallelograms.

G.538 parallelepipedal solid cut

If a parallelepipedal solid be cut by a plane through the diagonals of the opposite planes, the solid is bisected by the plane.

G.541 parallelepipedal solids equal

Parallelepipedal solids which are on equal bases and of the same height are equal.

G.542 parallelepipedal solids height

Parallelepipedal solids which are of the same height are to one another as their bases.

G.544 equal parallelepipedal solids

In equal parallelepipedal solids the bases are reciprocally proportional to the heights; and those parallelepipedal solids in which the bases are reciprocally proportional to the heights are equal.

G.545 there equal plane

If there be two equal plane angles, and on their vertices there be set up elevated segments containing equal angles with the original segments respectively, if on the elevated segments points be taken at random and perpendiculars be drawn from them to the planes in which the original angles are, and if from the points so arising in the planes segments be joined to the vertices of the original angles, they will contain, with the elevated segments, equal angles.

Posts tagged 'plane-surface'