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单词 SimilarityOfTriangles
释义

similarity of triangles


The following theorems are valid in Euclidean geometryMathworldPlanetmath:

Theorem AA.  If one triangle has a pair of angles that are congruent to a pair of angles in another triangle, then the two triangles are similarMathworldPlanetmath.

Theorem .  If a pair of sides of a triangle are proportional to a pair of sides in another triangle and if the angles included by the side-pairs are congruent, then the triangles are similar.

Theorem .  If the sides of a triangle are proportional to the sides of another triangle, then the triangles are similar.

The AA theoremMathworldPlanetmath may be regarded as the definition of the similarity of triangles. In some texts, the AA theorem is assumed as a postulate. The other two theorems may be proved by using the law of cosines for determining the the ratios other sides (for ) and the angles.

In hyperbolic geometry and spherical geometry, similar triangles are congruent. (See the AAA theoremMathworldPlanetmath for more details.) Thus, the theorem and theorem are invalid in these .

Titlesimilarity of triangles
Canonical nameSimilarityOfTriangles
Date of creation2013-03-22 17:49:41
Last modified on2013-03-22 17:49:41
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id12
Authorpahio (2872)
Entry typeTheorem
Classificationmsc 51F99
Classificationmsc 51M05
Classificationmsc 51-00
Synonymsimilar triangles
Related topicHarmonicMeanInTrapezoid
Related topicAreaOfSphericalCalotteByMeansOfChord
Related topicInterceptTheorem
DefinesAA
DefinesAA postulate
DefinesAA theorem
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