The Teaching of Geometry

by David Eugene Smith

Theorem. If two sides of a quadrilateral are equal and parallel, then the other two sides are equal and parallel, and the figure is a parallelogram.

This was Euclid's first proposition on parallelograms, and Proclus speaks of it as the connecting link between the theory of parallels and that of parallelograms. The ancients, writing for mature students, did not add the words "and the figure is a parallelogram," because that follows at once from the first part and from the definition of "parallelogram," but it is helpful to younger students because it emphasizes the fact that here is a test for this kind of figure.

Theorem. The diagonals of a parallelogram bisect each other.

This proposition was not given in Euclid, but it is usually required in American syllabi. There is often given in connection with it the exercise in which it is proved that the diagonals of a rectangle are equal. When this is taken, it is well to state to the class that carpenters and builders find this one of the best checks in laying out floors and other rectangles. It is frequently applied also in laying out tennis courts. If the class is doing any work in mensuration, such as finding the area of the school grounds, it is a good plan to check a few rectangles by this method.

An interesting outdoor application of the theory of parallelograms is the following:

Suppose you are on the side of this stream opposite to XY, and wish to measure the length of XY. Run a line AB along the bank. Then take a carpenter's square, or even a large book, and walk along AB until you reach P, a point from which you can just see X and B along two sides of the square. Do the same for Y, thus fixing P and Q. Using the tape, bisect PQ at M. Then walk along YM produced until you reach a point Y' that is exactly in line with M and Y, and also with P and X. Then walk along XM produced until you reach a point X' that is exactly in line with M and X, and also with Q and Y. Then measure Y'X' and you have the length of XY. For since YX' is ⊥ to PQ, and XY' is also ⊥ to PQ, YX' is || to XY'. And since PM = MQ, therefore XM = MX' and Y'M = MY. Therefore Y'X'YX is a parallelogram.

The properties of the parallelogram are often applied to proving figures of various kinds congruent, or to constructing them so that they will be congruent.

For example, if we draw A'B' equal and parallel to AB, B'C' equal and parallel to BC, and so on, it is easily proved that ABCD and A'B'C'D' are congruent. This may be done by ordinary superposition, or by sliding ABCD along the dotted parallels.