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In Euclidean geometry, the triangle postulate states that the sum of the angles of a triangle is two right angles. This postulate is equivalent to the parallel postulate. [1] In the presence of the other axioms of Euclidean geometry, the following statements are equivalent: [2] Triangle postulate: The sum of the angles of a triangle is two ...
In a hyperbolic triangle the sum of the angles A, B, C (respectively opposite to the side with the corresponding letter) is strictly less than a straight angle. The difference between the measure of a straight angle and the sum of the measures of a triangle's angles is called the defect of the triangle.
The reason is that the value of sine for the angle of the triangle does not uniquely determine this angle. For example, if sin β = 0.5 , the angle β can equal either 30° or 150°. Using the law of cosines avoids this problem: within the interval from 0° to 180° the cosine value unambiguously determines its angle.
The same geometric strategy also works for triangles, as in the figure on the right: [4] if the large triangle has area 1, then the largest black triangle has area 1 / 4 , and so on. The figure as a whole has a self-similarity between the large triangle and its upper sub-triangle.
The sum of the angles in every triangle is 180° (triangle postulate). There exists a triangle whose angles add up to 180°. The sum of the angles is the same for every triangle. There exists a pair of similar, but not congruent, triangles. Every triangle can be circumscribed.
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.
The triangle group is the infinite symmetry group of a certain tessellation (or tiling) of the Euclidean plane by triangles whose angles add up to π (or 180°). Up to permutations, the triple (l, m, n) is one of the triples (2,3,6), (2,4,4), (3,3,3). The corresponding triangle groups are instances of wallpaper groups.
The angles of proper spherical triangles are (by convention) less than π, so that < + + < (Todhunter, [1] Art.22,32). In particular, the sum of the angles of a spherical triangle is strictly greater than the sum of the angles of a triangle defined on the Euclidean plane, which is always exactly π radians.
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