Download Vorlesungen über nicht-Euklidische Geometrie by Felix Klein, Walter Rosemann PDF

By Felix Klein, Walter Rosemann

Als Felix Klein den Plan faBte, die wichtigsten seiner autogra phierten Vorlesungen im Druck erscheinen zu lassen, gedachte er, mit der Nichteuklidischen Geometrie zu beginnen und den alten textual content zu vor mit Hille eines jiingeren Geometers, des Herro Dr. Rosemann, in der Anlage und den Einzelheiten einer griindlichen Neubearbeitung zu unterziehen. Diese Arbeit erwies sich als langwieriger wie urspriing lich geschatzt. Klein selbst konnte ihren AbschluB nicht mehr erleben. Zwar hatte er in taglichen, durch mehr als ein J ahr fortgesetzten Be sprechungen den Stoff bis in die Einzelheiten hinein mit seinem Mit arbeiter durchdacht, gesichtet und geordnet; aber die eigentliche Aus arbeitung des Textes muBte er von vornherein Herro Rosemann uber lassen. Bei Kleins Tode lagen die Fahnenkorrekturen der ersten Ka pitel vor; es bedurfte jedoch noch jahrelanger opferwilliger Arbeit seitens Herro Rosemanns, urn auf Grund des urspriinglichen Programmes das Manuskript fertigzustellen und den Druck durchzufiihren. So ist bei diesem Werke eigener Antell und Verdienst, aber auch eigene Ver antwortung des Bearbeiters viel heher zu bewerten als sonst ublich.

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M∠RPM = 42° m∠MPS = ______ M R P S 42 Team-LRN – MEASURING ANGLES – 19. m∠GEO = 100° m∠GEM = 60° m∠MEO = ______ M G O E Use the following figure to answer problems 20–25. C B A D 35˚ P E 20. m∠BPC = ______ 21. m∠CPD = ______ 22. m∠EPB = ______ 23. m∠CPE = ______ 24. m∠APE = ______ 25. m∠DPE = ______ 43 Team-LRN – MEASURING ANGLES – Use the following figure for practice problems 26–29. S. Army uses a unit of angle measure called a mil. A mil is defined as ᎏ 6,4ᎏ 00 of a circle. The protractor is marked in mils.

Here are two examples of obtuse angles: 91° 170° 26 Team-LRN – TYPES OF ANGLES – Straight Angles A straight angle has a 180° measure. This is an example of a straight angle (∠ABC is a straight angle): 180˚ A B C Practice Use the following figure to answer practice problems 10–13. J K N O L M 10. Name three acute angles. 11. Name three obtuse angles. 12. Name two straight angles. 13. If ∠MON measures 27°, then ∠JOK measures ________ degrees. Complete each statement. 14. An angle with measure 90° is called a(n) __________ angle.

32 Team-LRN – WORKING WITH LINES – 11. Skew lines are __________ coplanar. 12. Skew lines __________ intersect. Angles Formed by Parallel Lines and a Transversal If a pair of parallel lines are cut by a transversal, then eight angles are formed. In the following figure, line l is parallel to line m and line t is a transversal forming angles 1–8. Angles 3, 4, 5, and 6 are inside the parallel lines and are called interior angles. Angles 1, 2, 7, and 8 are outside the parallel lines and are called exterior angles.

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