3. Two parallel lines determine a plane. This requires a slight but informal proof to show that it properly follows as a corollary from the postulate, but a single sentence suffices.
While studying this book questions of the following nature may arise with an advanced class, or may be suggested to those who have had higher algebra:
How many straight lines are in general (that is, at the most) determined by n points in space? Two points determine 1 line, a third point adds (in general, in all these cases) 2 more, a fourth point adds 3 more, and an nth point n - 1 more. Hence the maximum is 1 + 2 + 3 + ... + (n - 1), or n(n-1)/2, which the pupil will understand if he has studied arithmetical progression. The maximum number of intersection points of n straight lines in the same plane is also n(n - 1)/2.
How many straight lines are in general determined by n planes? The answer is the same, n(n - 1)/2.
How many planes are in general determined by n points in space? Here the answer is 1 + 3 + 6 + 10 + ... + (n - 2)(n - 1)/2, or n(n - 1)(n - 2)/(1 × 2 × 3). The same number of points is determined by n planes.
Theorem. If two planes cut each other, their intersection is a straight line.
Among the simple illustrations are the back edges of the pages of a book, the corners of the room, and the simple test as to whether the edge of a card is straight by testing it on a plane. It is well to call attention to the fact that if two intersecting straight lines move parallel to their original position, and so that their intersection rests on a straight line not in the plane of those lines, the figure generated will be that of this proposition. In general, if we cut through any figure of solid geometry in some particular way, we are liable to get the figure of a proposition in plane geometry, as will frequently be seen.
Theorem. If a straight line is perpendicular to each of two other straight lines at their point of intersection, it is perpendicular to the plane of the two lines.
If students have trouble in visualizing the figure in three dimensions, some knitting needles through a piece of cardboard will make it clear. Teachers should call attention to the simple device for determining if a rod is perpendicular to a board (or a pipe to a floor, ceiling, or wall), by testing it twice, only, with a carpenter's square. Similarly, it may be asked of a class, How shall we test to see if the corner (line) of a room is perpendicular to the floor, or if the edge of a box is perpendicular to one of the sides?
In some elementary and in most higher geometries the perpendicular is called a normal to the plane.