By David Gans
Booklet via Gans, David
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Extra info for An Introduction to Non-Euclidean Geometry
Sample text
Ill, 20 into a transversal subdivision St of that triangle. Then prove that Theorem 46 must hold for S inasmuch as it holds for St. * An angle of (180/7r)° at the center of a circle does not intercept an arc equal to the radius (see §14). Hence a standard unit should not be called a radian, as in Euclidean geometry. 10. QUADRILATERALS ASSOCIATED WITH A TRIANGLE 63 3. Why can we not claim to have proved that the smaller the area of a triangle, the smaller its defect? 4. If a triangle has an angle-sum of 120°, find its defect in (a) degrees, (b) right angle units, (c) standard units.
Since St is a transversal subdivision of triangle ABC, we have def ABC = atïCAD + def BCD + def AED + def BDE, (2) * D. Hubert, The Foundations of Geometry (transi, by E. J. Townsend), pp. , Chicago, Illinois, 1902). 62 III. PARALLELS WITH A COMMON PERPENDICULAR where def is the abbreviation of defect. Also, since triangle ABD is divided transversally by DE, def ABD = def AED + def BDE. (3) Substituting (3) in (2), we get def ABC = def CAD + def BCD + def ABD, which shows that the defect relation holds for the nontransversal subdivision S of triangle ABC.
Proof We consider only the case of alternate interior angles, for the case of corresponding angles then follows immediately. Let lines g, h be cut by a line t in points A, B so that £ 1 = £ 2 (Fig. Ill, 9). If these angles are Fig. Ill, 9 right angles, then t is a common perpendicular of g and h. If they are acute angles, let C be the midpoint of AB, and D the projection of C on g. On h take point E to the left of B so that BE = AD. Triangles CAD, CBE are congruent by side-angle-side. BCE= %ACD.