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.

Let triangle $\mathrm{△}ABC$ have angles $\mathrm{\angle }B$ and $\mathrm{\angle }C$ congruent.

Since we have

• $\mathrm{\angle }B\cong \mathrm{\angle }C$

• $\overline{BC}\cong \overline{CB}$ by the reflexive property (http://planetmath.org/Reflexive) of $\cong$ (note that $\overline{BC}$ and $\overline{CB}$ denote the same line segment)

• $\mathrm{\angle }C\cong \mathrm{\angle }B$ by the symmetric property (http://planetmath.org/Symmetric) of $\cong$

we can use ASA to conclude that $\mathrm{△}ABC\cong \mathrm{△}ACB$. Since corresponding parts of congruent triangles are congruent, we have that $\overline{AB}\cong \overline{AC}$. It follows that $\mathrm{△}ABC$ 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.