You are here
Homedetermining from angles that a triangle is isosceles
Primary tabs
determining from angles that a triangle is isosceles
The following theorem holds in any geometry 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 triangle has two congruent angles, then it is isosceles.
Proof.
Since we have

$\angle B\cong\angle C$

$\overline{BC}\cong\overline{CB}$ by the reflexive property of $\cong$ (note that $\overline{BC}$ and $\overline{CB}$ denote the same line segment)

$\angle C\cong\angle B$ by the symmetric property of $\cong$
we can use ASA to conclude that $\triangle ABC\cong\triangle ACB$. Since corresponding parts of congruent triangles are congruent, we have that $\overline{AB}\cong\overline{AC}$. It follows that $\triangle ABC$ 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.
Mathematics Subject Classification
51M04 no label found5100 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections