Theorem. Two parallelograms are congruent if two sides and the included angle of the one are equal respectively to two sides and the included angle of the other.
This proposition is discussed in connection with the one that follows.
Theorem. If three or more parallels intercept equal segments on one transversal, they intercept equal segments on every transversal.
These two propositions are not given in Euclid, although generally required by American syllabi of the present time. The last one is particularly useful in subsequent work. Neither one offers any difficulty, and neither has any interesting history. There are, however, numerous interesting applications to the last one. One that is used in mechanical drawing is here illustrated.
If it is desired to divide a line AB into five equal parts, we may take a piece of ruled tracing paper and lay it over the given line so that line 0 passes through A, and line 5 through B. We may then prick through the paper and thus determine the points on AB. Similarly, we may divide AB into any other number of equal parts.
Among the applications of these propositions is an interesting one due to the Arab Al-Nairīzī (ca. 900 A.D.). The problem is to divide a line into any number of equal parts, and he begins with the case of trisecting AB. It may be given as a case of practical drawing even before the problems are reached, particularly if some preliminary work with the compasses and straightedge has been given.
Make BQ and AQ' perpendicular to AB, and make BP = PQ = AP' = P'Q'. Then ⧍XYZ is congruent to ⧍YBP, and also to ⧍XAP'. Therefore AX = XY = YB. In the same way we might continue to produce BQ until it is made up of n - 1 lengths BP, and so for AQ', and by properly joining points we could divide AB into n equal parts. In particular, if we join P and P', we bisect the line AB.
