| Ultimately, the answer was proven to be positive: in other spaces geometries this sum can be greater or lesser, but it then must depend on the triangle | The parallel postulate is equivalent to the Equidistance postulate, Playfair axiom, Proclus axiom, the Triangle postulate and the Pythagorean theorem |
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| In differential geometry [ ] In the , the question of a triangle's angular defect is understood as a special case of the where the curvature of a is not a function, but a with the in exactly three points — vertices of a triangle | Ratcliffe, John 2006 , , Graduate Texts in Mathematics, 149, Springer, p |
One can easily see how breaks Playfair's axiom, Proclus' axiom the parallelism, defined as non-intersection, is intransitive in an hyperbolic plane , the equidistance postulate the points on one side of, and equidistant from, a given line do not form a line , and Pythagoras' theorem.
21| In a , the of angles of a triangle equals the 180 , , two , or a half- | A circle cannot have arbitrarily small , so the three points property also fails |
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| A has three angles, one at each , bounded by a pair of adjacent | In the presence of the other axioms of Euclidean geometry, the following statements are equivalent:• The influence of this problem on mathematics was particularly strong during the 19th century |
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| Triangle postulate: The sum of the angles of a triangle is two right angles | Triangle area property: The of a triangle can be as large as we please |
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| So, the sum of three exterior angles added to the sum of three interior angles always gives three straight angles | This postulate is equivalent to the |
Cases [ ] Euclidean geometry [ ] In , the triangle postulate states that the sum of the angles of a triangle is two.
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