The Teaching of Geometry

by David Eugene Smith

If a dangerous shoal A is near a headland H, the angle HAX is measured and is put down upon the charts as the "vertical danger angle." Ships coming near the headland are careful to keep far enough away, say at S, so that the angle HSX shall be less than this danger angle. They are then sure that they will avoid the dangerous shoal.

Related to this proposition is the problem of supporting a tall iron smokestack by wire stays. Evidently three stays are needed, and they are preferably placed at the vertices of an equilateral triangle, the smokestack being in the center. The practical problem may be given of locating the vertices of the triangle and of finding the length of each stay.

Theorem. Two straight lines perpendicular to the same plane are parallel.

Here again we may cut through the figure by the plane of the two parallels, and we get the figure of plane geometry relating to lines that are perpendicular to the same line. The proposition shows that the opposite corners of a room are parallel, and that therefore they lie in the same plane, or are coplanar, as is said in higher geometry.

It is interesting to a class to have attention called to the corollary that if two straight lines are parallel to a third straight line, they are parallel to each other; and to have the question asked why it is necessary to prove this when the same thing was proved in plane geometry. In case the reason is not clear, let some student try to apply the proof used in plane geometry.

Theorem. Two planes perpendicular to the same straight line are parallel.

Besides calling attention to the corresponding proposition of plane geometry, it is well now to speak of the fact that in propositions involving planes and lines we may often interchange these words. For example, using "line" for "straight line," for brevity, we have:

One line does not determine a plane.	One plane does not determine a line.
Two intersecting lines determine a plane.	Two intersecting planes determine a line.
Two lines perpendicular to a plane are parallel.	Two planes perpendicular to a line are parallel.
If one of two parallel lines is perpendicular to a plane, the other is also perpendicular to the plane.	If one of two parallel planes is perpendicular to a line, the other is also perpendicular to the line.
If two lines are parallel, every plane containing one of the lines is parallel to the other line.	If two planes are parallel, every line in one of the planes is parallel to the other plane.

Theorem. The intersections of two parallel planes by a third plane are parallel lines.