PlanetMath: determining from angles that a triangle is isosceles

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determining from angles that a triangle is isosceles

(Theorem)

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

Proof. Let triangle $\triangle ABC$ have angles $\angle B$ and $\angle C$ congruent.

$\begin{pspicture}(-3,-2)(3,3) \pspolygon(-2,-2)(0,2)(2,-2) \psarc(-2,-2){0.4}{0}... ...rput[b](0,2.2){$A$} \rput[r](-2.2,-2){$B$} \rput[l](2.2,-2){$C$} \end{pspicture}$

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. $\qedsymbol$

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.

"determining from angles that a triangle is isosceles" is owned by Wkbj79.

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See Also: angles of an isosceles triangle

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Cross-references: angles of an isosceles triangle, converse theorem, SAS, corresponding parts of congruent triangles are congruent, line segment, isosceles, angles, congruent, triangle, spherical geometry, neutral geometry, hyperbolic geometry, Euclidean geometry, ASA, geometry
There are 6 references to this entry.

This is version 4 of determining from angles that a triangle is isosceles, born on 2007-06-04, modified 2007-06-09.
Object id is 9528, canonical name is DeterminingFromAnglesThatATriangleIsIsosceles.
Accessed 1738 times total.

Classification:

AMS MSC:	51-00 (Geometry :: General reference works )
	51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)

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