# similarity of triangles

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 . 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 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 ) and the angles.

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

Title | similarity of triangles |

Canonical name | SimilarityOfTriangles |

Date of creation | 2013-03-22 17:49:41 |

Last modified on | 2013-03-22 17:49:41 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 12 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 51F99 |

Classification | msc 51M05 |

Classification | msc 51-00 |

Synonym | similar triangles |

Related topic | HarmonicMeanInTrapezoid |

Related topic | AreaOfSphericalCalotteByMeansOfChord |

Related topic | InterceptTheorem |

Defines | AA |

Defines | AA postulate |

Defines | AA theorem |