A New Look at Geometry by Irving Adler

Author:Irving Adler
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2013-06-26T16:00:00+00:00

The second theorem shows how 11(d) changes as d changes:

II. If PΩ is parallel to QΩ, and QP is extended to R, then the exterior angle RPΩ is greater than angle Q.

Proof: Draw angle RPS equal to angle Q. Then either (1) PS lies inside angle QPΩ, or (2) PS coincides with PΩ, or (3) PS lies inside angle RPΩ. In case (1), PS intersects QΩ at some point T to form triangle PQT. Then exterior angle RPT is equal to the opposite interior angle Q. This is impossible, by proposition 16 of Euclid’s Book I. In case (2), let M be the midpoint of PQ. Draw MN perpendicular to PΩ, extended through P if necessary. On QΩ draw QL equal to PN so that N and L lie on opposite sides of PQ, and then draw ML. Since angle Q = angle RPS = angle NPM, and QM = PM, triangles NPM and LQM are congruent. Then angle NMP = angle LMQ, and consequently NML is a straight line. Moreover, since angle MLQ = angle MNP, NL is perpendicular to QΩ. Then angle LNΩ is an angle of parallelism. But this is impossible, since angle LNΩ is a right angle, and an angle of parallelism is acute. Therefore case (3) is the only one that may arise, and in this case angle RPΩ, being greater than angle RPS, is also greater than angle Q.

Let us apply this result to the diagram below, in which two lines are drawn parallel to AΩ from points P and Q on the line QPA which is perpendicular to AΩ. If the distances of P and Q from AΩ are d and d' respectively, and d' > d, then the theorem just proved shows that II(d') < II(d). If we picture the point P as moving toward Q, we see then that as the distance from AΩ increases, the corresponding angle of parallelism decreases. It can be shown that any acute angle, no matter how small, can be obtained as an angle of parallelism corresponding to some appropriate distance. Consequently, as the distance increases toward infinity, the corresponding angle of parallelism decreases toward zero. If P moves toward A, we see that as the distance from AΩ decreases toward zero the corresponding angle of parallelism increases toward 90°. The exact way in which II(d) varies with d is given explicitly by the following formula that was derived by Bolyai and Lobatschewsky:

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