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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.
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.
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. If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle. There exists a quadrilateral in which all ...
Circular angles can be characterized geometrically by the property that if two chords P 0 P 1 and P 0 P 2 subtend angles L 1 and L 2 at the centre of a circle, their sum L 1 + L 2 is the angle subtended by a chord P 0 Q, where P 0 Q is required to be parallel to P 1 P 2. The same construction can also be applied to the hyperbola.
In computational geometry, a Delaunay triangulation or Delone triangulation of a set of points in the plane subdivides their convex hull [1] into triangles whose circumcircles do not contain any of the points. This maximizes the size of the smallest angle in any of the triangles, and tends to avoid sliver triangles.
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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