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 $\mathrm{△}ABC$ be isosceles such that the legs $\overline{AB}$ and $\overline{AC}$ are congruent.

Since we have

• $\overline{AB}\cong \overline{AC}$

• $\mathrm{\angle }A\cong \mathrm{\angle }A$ by the reflexive property (http://planetmath.org/Reflexive) of $\cong$

• $\overline{AC}\cong \overline{AB}$ by the symmetric property (http://planetmath.org/Symmetric) of $\cong$

we can use SAS to conclude that $\mathrm{△}ABC\cong \mathrm{△}ACB$. 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.