determining from angles that a triangle is isosceles

The following theorem holds in any geometryMathworldPlanetmath in which ASA 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.

If a triangleMathworldPlanetmath has two congruent angles, then it is isosceles.


Let triangle ABC have angles B and C congruent.


Since we have

  • BC

  • BC¯CB¯ by the reflexive property ( of (note that BC¯ and CB¯ denote the same line segmentMathworldPlanetmath)

  • CB by the symmetricMathworldPlanetmathPlanetmath property ( of

we can use ASA to conclude that ABCACB. Since corresponding parts of congruent triangles are congruent, we have that AB¯AC¯. It follows that ABC is isosceles. ∎

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

Title determining from angles that a triangle is isosceles
Canonical name DeterminingFromAnglesThatATriangleIsIsosceles
Date of creation 2013-03-22 17:12:23
Last modified on 2013-03-22 17:12:23
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 7
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 51M04
Classification msc 51-00
Related topic AnglesOfAnIsoscelesTriangle