Question

Comment on Proposition 2.3.17 (Proposition 13 If a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles.) and Proposition 2.3.14 (If with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another.) in the context of the following geometries: (a) Spherical (b) Hyperbolic (c) Taxicab (d) Maxi

Answer 1

Gome toy Proposition 13 : _ lf a kłrtgeld-lire starts on-a fhis makes either ah ags Cingles-ulose-seunequalstr tuongit angles,, line standing on the Straiglt ins CDaks the anle CBA ABD le say thal either the c ,hoongltangles ORthir. But if nets dadw BE from the point B at zightaples angle CBE epuals the sumo tuo anples CBA А ВЕ , add th angte EBI to each, thet e the sum e the ns CBE EBD eqate sam of thoe angles cBA RBE, and EBD. THain meheLD A.LDBE-LEBA , adatte LAP С to eauh, heseto the sum the annes DBA4PBC eul He sur o. the thre argesP BE, EBAst ABC . the sum oh fhe same 3 angless&Phings whith equal to the same the ples ceeTEBDroefur zightang , : these mapthe angles feu Line stands on a stoagh e, then makes eithes 2 contd.。.

Date Prohosthon 14 nes t ng on the some side make the sum oi the adiavent anylbs eul to to rigltanes hen the tws seit lines are in a wth ore antey Let AB be any saaiy tt lie mot lying on the same siod make the sum the adeat o ngld anges We say that BD is in a Stoaight Uire with cB If BD is mot in a sta ne with BC, then pordune CBA&ABE epuals to the Sumof theargles CBA PAD Selectrout the angle CBA pomeaeb Ten the semzinin argle ABE eguas he desmei ring aneBD, the es erud oeater, whichis impossi stoaiglt lme with CB- E-D
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