The Teaching of Geometry

by David Eugene Smith

Surveyors make use of this proposition when they wish, without using a transit instrument, to run one line parallel to another.

For example, suppose two boys are laying out a tennis court and they wish to run a line through P parallel to AB. Take a 60-foot tape and swing it around P until the other end rests on AB, as at M. Put a stake at O, 30 feet from P and M. Then take any convenient point N on AB, and measure ON. Suppose it equals 20 feet. Then sight from N through O, and put a stake at Q just 20 feet from O. Then P and Q determine the parallel, according to the proposition just mentioned.

Theorem. If two parallel lines are cut by a transversal, the exterior-interior angles are equal.

This is also a part of Euclid I, 29. It is usually followed by several corollaries, covering the minor and obvious cases omitted by the older writers. While it would be possible to dispense with these corollaries, they are helpful for definite reference in later propositions.

Theorem. The sum of the three angles of a triangle is equal to two right angles.

Euclid stated this as follows: "In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles." This states more than is necessary for the basal fact of the proposition, which is the constancy of the sum of the angles.

The theorem is one of the three most important propositions in plane geometry, the other two being the so-called Pythagorean Theorem, and a proposition relating to the proportionality of the sides of two triangles. These three form the foundation of trigonometry and of the mensuration of plane figures.

The history of the proposition is extensive. Eutocius (ca. 510 A.D.), in his commentary on Apollonius, says that Geminus (first century B.C.) testified that "the ancients investigated the theorem of the two right angles in each individual species of triangle, first in the equilateral, again in the isosceles, and afterwards in the scalene triangle." This, indeed, was the ancient plan, to proceed from the particular to the general. It is the natural order, it is the world's order, and it is well to follow it in all cases of difficulty in the classroom.