angles of an isosceles triangle

The following theorem holds in any geometryMathworldPlanetmath 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 triangleMathworldPlanetmath are congruent.


Let triangleMathworldPlanetmath ABC be isosceles such that the legs AB¯ and AC¯ are congruent.


Since we have

  • AB¯AC¯

  • AA by the reflexive property ( of

  • AC¯AB¯ by the symmetricMathworldPlanetmathPlanetmath property ( of

we can use SAS to conclude that ABCACB. Since corresponding parts of congruent triangles are congruent, it follows that BC. ∎

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