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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry.As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement.
For example, the first and fourth of Euclid's postulates, that there is a unique line between any two points and that all right angles are equal, hold in elliptic geometry. Postulate 3, that one can construct a circle with any given center and radius, fails if "any radius" is taken to mean "any real number", but holds if it is taken to mean ...
The curriculum that SSMCIS devised had influences from earlier reform work in Europe, [3] going back to the Bourbaki group's work in France in the 1930s and the Synopses for Modern Secondary School Mathematics published in Paris in 1961. [9] Indeed, most European secondary schools were teaching a more integrated approach. [14]
Investigations was developed between 1990 and 1998. It was just one of a number of reform mathematics curricula initially funded by a National Science Foundation grant. The goals of the project raised opposition to the curriculum from critics (both parents and mathematics teachers) who objected to the emphasis on conceptual learning instead of instruction in more recognized specific methods ...
The Principles and Standards for School Mathematics was developed by the NCTM. The NCTM's stated intent was to improve mathematics education. The contents were based on surveys of existing curriculum materials, curricula and policies from many countries, educational research publications, and government agencies such as the U.S. National Science Foundation. [3]
See the figures in this article for examples. The three defining points may also identify angles in geometric figures. For example, the angle with vertex A formed by the rays AB and AC (that is, the half-lines from point A through points B and C) is denoted ∠BAC or ^. Where there is no risk of confusion, the angle may sometimes be referred to ...
The line segments AB and CD are perpendicular to each other. In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity.Although many authors use the two terms perpendicular and orthogonal interchangeably, the term perpendicular is more specifically used for lines and planes that intersect to form a right angle, whereas orthogonal is used in generalizations ...
Special cases of Apollonius' problem are those in which at least one of the given circles is a point or line, i.e., is a circle of zero or infinite radius. The nine types of such limiting cases of Apollonius' problem are to construct the circles tangent to: three points (denoted PPP, generally 1 solution)
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