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.

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