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  2. Saccheri–Legendre theorem - Wikipedia

    en.wikipedia.org/wiki/Saccheri–Legendre_theorem

    Similarly, the existence of at least one triangle with angle sum of less than 180 degrees implies the characteristic postulate of hyperbolic geometry. [ 3 ] One proof of the Saccheri–Legendre theorem uses the Archimedean axiom , in the form that repeatedly halving one of two given angles will eventually produce an angle sharper than the ...

  3. Sum of angles of a triangle - Wikipedia

    en.wikipedia.org/wiki/Sum_of_angles_of_a_triangle

    Exterior angles can be also defined, and the Euclidean triangle postulate can be formulated as the exterior angle theorem. One can also consider the sum of all three exterior angles, that equals to 360° [9] in the Euclidean case (as for any convex polygon), is less than 360° in the spherical case, and is greater than 360° in the hyperbolic case.

  4. AA postulate - Wikipedia

    en.wikipedia.org/wiki/AA_postulate

    In Euclidean geometry, the AA postulate states that two triangles are similar if they have two corresponding angles congruent. The AA postulate follows from the fact that the sum of the interior angles of a triangle is always equal to 180°. By knowing two angles, such as 32° and 64° degrees, we know that the next angle is 84°, because 180 ...

  5. Foundations of geometry - Wikipedia

    en.wikipedia.org/wiki/Foundations_of_geometry

    The 22 axioms of this system are given individual names for ease of reference. Amongst these are to be found: the Ruler Postulate, the Ruler Placement Postulate, the Plane Separation Postulate, the Angle Addition Postulate, the Side angle side (SAS) Postulate, the Parallel Postulate (in Playfair's form), and Cavalieri's principle. [51]

  6. Proofs of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/Proofs_of_trigonometric...

    The proofs given in this article use these definitions, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles , see Trigonometric functions . Other definitions, and therefore other proofs are based on the Taylor series of sine and cosine , or on the differential equation f ″ + f = 0 ...

  7. Synthetic geometry - Wikipedia

    en.wikipedia.org/wiki/Synthetic_geometry

    It relies on the axiomatic method for proving all results from a few basic properties initially called postulates, and at present called axioms. After the 17th-century introduction by René Descartes of the coordinate method, which was called analytic geometry , the term "synthetic geometry" was coined to refer to the older methods that were ...