Prop. 7. If two straight lines be parallel, the straight line drawn from any point in one to any point in the other is in the same plane with the parallels. From the definition of a plane further follows
Prop. 3. If two planes cut one another, their common section is a straight line.
§ 74. Whilst these propositions are virtually contained in the definition of a plane, the next gives us a new and fundamental property of space, showing at the same time that it is possible to have a straight line perpendicular to a plane, according to Def. 3. It states—
Prop. 4. If a straight line is perpendicular to two straight lines in a plane which it meets, then it is perpendicular to all lines in the plane which it meets, and hence it is perpendicular to the plane.
Def. 3 may be stated thus: If a straight line is perpendicular to a plane, then it is perpendicular to every line in the plane which it meets. The converse to this would be
All straight lines which meet a given straight line in the same point, and are perpendicular to it, lie in a plane which is perpendicular to that line.
This Euclid states thus:
Prop. 5. If three straight lines meet all at one point, and a straight line stands at right angles to each of them at that point, the three straight lines shall be in one and the same plane.
§ 75. There follow theorems relating to the theory of parallel lines in space, viz.:—
Prop. 6. Any two lines which are perpendicular to the same plane are parallel to each other; and conversely