# angles of an isosceles triangle

The following theorem holds in any geometry^{} in which SAS is valid. Specifically, it holds in both Euclidean geometry and hyperbolic geometry (and therefore in neutral geometry) as well as in spherical geometry.

###### Theorem 1.

The angles opposite to the congruent sides of an isosceles triangle^{} are congruent.

###### Proof.

Let triangle^{} $\mathrm{\u25b3}ABC$ be isosceles such that the legs $\overline{AB}$ and $\overline{AC}$ are congruent.

Since we have

we can use SAS to conclude that $\mathrm{\u25b3}ABC\cong \mathrm{\u25b3}ACB$. Since corresponding parts of congruent triangles are congruent, it follows that $\mathrm{\angle}B\cong \mathrm{\angle}C$. ∎

In geometries in which SAS and ASA are both valid, the converse theorem of this theorem is also true. This theorem is stated and proven in the entry determining from angles that a triangle is isosceles.

Title | angles of an isosceles triangle |
---|---|

Canonical name | AnglesOfAnIsoscelesTriangle |

Date of creation | 2013-03-22 17:12:06 |

Last modified on | 2013-03-22 17:12:06 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 10 |

Author | Wkbj79 (1863) |

Entry type | Theorem |

Classification | msc 51-00 |

Classification | msc 51M04 |

Related topic | DeterminingFromAnglesThatATriangleIsIsosceles |

Related topic | PonsAsinorum |