Perpendicularity was in the former attributed to lines considered in a surface. Therefore from thence is repeated this consectary of the perpendicle of a line with the surface it selfe.
If thou shalt conceive the right lines, ae, io, uy, to cut one another in the plaine beneath, in the common intersections: And the line rs, falling from above, to be to every one of them perpendicular in the common point s, thou hast an example of this consectary.
4 If three right lines cutting one another, be unto the same right line perpendicular in the common section, they are in the same plaine 5. p xj.
For by the perpendicle and common section is understood an equall state on all parts, and therefore the same plaine: as in the former example, as, ys, os, suppose them to be to sr, the same loftie line, perpendicular, they shall be in the same nearer plaine aiueoy.
5 If two right lines be perpendicular to the under-plaine, they are parallells: And if the one two
parallells be perpendicular to the under plaine, the other is also perpendicular to the same. 6. 8 p xj.
The cause is out of the first law or rule parallells. For if two right lines be perpendicular to the same under plaine, being joyned together by a right line, they shall make their inner corners equall to two right angles: And therefore they shall be parallells, by the [21. e v]. And if in two parallells knit together with a right line, one of the inner angles, be a right angle: the other also shall be a right angle. Because they are divided by a common perpendicular; As in the example. If the angles at a, and e, be right angles, ai, and eo, are parallells, and contrariwise, if ai, and eo be parallells, and the angle at a, be a right angle, the angle at e, also shall be a right angle.
6 If right lines in diverse plaines be unto the same right line parallel, they are also parallell betweene themselves. 9 p xj.
