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It follows from Euclid's parallel postulate that if the two lines are parallel, then the angles of a pair of consecutive interior angles of a transversal are supplementary (Proposition 1.29 of Euclid's Elements). If one pair of consecutive interior angles is supplementary, the other pair is also supplementary.
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° [8] 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.
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 ...
The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute geometry because its proof does not depend upon the parallel postulate.
Saccheri–Legendre theorem. In absolute geometry, the Saccheri–Legendre theorem states that the sum of the angles in a triangle is at most 180°. [1] Absolute geometry is the geometry obtained from assuming all the axioms that lead to Euclidean geometry with the exception of the axiom that is equivalent to the parallel postulate of Euclid. [a]
Counting these parts, there are 32 axioms in this system. Amongst the postulates can be found the point-line-plane postulate, the Triangle inequality postulate, postulates for distance, angle measurement, corresponding angles, area and volume, and the Reflection postulate. The reflection postulate is used as a replacement for the SAS postulate ...
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie [1][2][3][4] (tr. The Foundations of Geometry) as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff.
ASA (angle-side-angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. The ASA postulate is attributed to Thales of Miletus. In most systems of axioms, the three criteria – SAS, SSS and ASA – are established as theorems.