By Eisenhart L. P.
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Additional info for Affine Geometries of Paths Possessing an Invariant Integral
AB A line. be a segment of the given line and let proceed as in Ex. 2. C Draw CA and CB and 95. Exs. j can be drawn parallel to I. It seems reasonable to suppose that no other line Zg can be drawn through P parallel to I, although this cannot be proved from the See § 60. assume the following preceding theorems. Hence we : PLANE GEOMETRY. 38 96. Axiom Through a VIII. one straight line can he drawn on a 'point not line only 'parallel to that line. Historical Note. This so-called axiom of parallels has attracted more attention than any other proposition in geometry.
106. angle viewed from the vertex has a right side and a left side. Theorem. 107- If two angles have right sides respectively parallel, their side to rigid side, and left side to left side, the angles are equal. ^ 1 and ^ 2 Given such that a II a' and & II &' To prove that Z 1 = Z 2. Produce a and b' till they meet, Proof forming Z 3. Complete the proof. : Wliy do a and V meet when produced ? Make a proof also by producing h and a' 108, Theorem. till they meet. -ude, re- aiid left side to left side, the angles are equal.
Tn the figure of § '^o is out of the plane in l\ABC lie? 2. Show how to 3. A is By Ofia = 4. test for congruent to the second A O/A' on Draw any here given ? measure the height of a Lay out a Su(iGF,s iiiiN. move wliicli tlie triangles Is it necessary in the figure by using the second which necessary to it triangle A ABC, test page triangle, tree congruence. i'l. whether 4. 'oustruct another tri- : BECTILINEAR FIGURES. 17 angle congruent to it. Use § 35 and also § 32. Use the protractor to construct the angles.