* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download PDF
Survey
Document related concepts
Multilateration wikipedia , lookup
Euler angles wikipedia , lookup
Line (geometry) wikipedia , lookup
Rational trigonometry wikipedia , lookup
Geometrization conjecture wikipedia , lookup
Trigonometric functions wikipedia , lookup
Noether's theorem wikipedia , lookup
Four color theorem wikipedia , lookup
Riemann–Roch theorem wikipedia , lookup
History of geometry wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Integer triangle wikipedia , lookup
History of trigonometry wikipedia , lookup
Transcript
similarity of triangles∗ pahio† 2013-03-22 0:24:56 The following theorems are valid in Euclidean geometry: 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 similar. Theorem SAS. 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 SSS. If the sides of a triangle are proportional to the sides of another triangle, then the triangles are similar. The AA theorem 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 SAS) and the angles. In hyperbolic geometry and spherical geometry, similar triangles are congruent. (See the AAA theorem for more details.) Thus, the SAS theorem and SSS theorem are invalid in these geometries. ∗ hSimilarityOfTrianglesi created: h2013-03-2i by: hpahioi version: h40292i Privacy setting: h1i hTheoremi h51F99i h51M05i h51-00i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1