Prop. 8. If of two parallel straight lines one is perpendicular to a plane, the other is so also.
Prop. 7. If two straight lines are parallel, the straight line which joins any point in one to any point in the other is in the same plane as the parallels. (See above, § 73.)
Prop. 9. Two straight lines which are each of them parallel to the same straight line, and not in the same plane with it, are parallel to one another; where the words, “and not in the same plane with it,” may be omitted, for they exclude the case of three parallels in a plane, which has been proved before; and
Prop. 10. If two angles in different planes have the two limits of the one parallel to those of the other, then the angles are equal. That their planes are parallel is shown later on in Prop. 15.
This theorem is not necessarily true, for the angles in question may be supplementary; but then the one angle will be equal to that which is adjacent and supplementary to the other, and this latter angle will also have its limits parallel to those of the first.
From this theorem it follows that if we take any two straight lines in space which do not meet, and if we draw through any point P in space two lines parallel to them, then the angle included by these lines will always be the same, whatever the position of the point P may be. This angle has in modern times been called the angle between the given lines:—
By the angles between two not intersecting lines we understand the angles which two intersecting lines include that are parallel respectively to the two given lines.
§ 76. It is now possible to solve the following two problems:—
To draw a straight line perpendicular to a given plane from a given point which lies
1. Not in the plane (Prop. 11).