PlanetMath: angles of an isosceles triangle

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angles of an isosceles triangle

(Theorem)

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 The angles opposite to the congruent sides of an isosceles triangle are congruent.

Proof. Let triangle $\triangle ABC$ be isosceles such that the legs $\overline{AB}$ and $\overline{AC}$ are congruent.

$\begin{pspicture}(-3,-2)(3,3) \pspolygon(-2,-2)(0,2)(2,-2) \psline(-1.2,0.1)(-0.... ...rput[b](0,2.2){$A$} \rput[r](-2.2,-2){$B$} \rput[l](2.2,-2){$C$} \end{pspicture}$

Since we have

$\overline{AB} \cong \overline{AC}$
$\angle A \cong \angle A$ by the reflexive property of $\cong$
$\overline{AC} \cong \overline{AB}$ by the symmetric property of $\cong$

we can use SAS to conclude that $\triangle ABC \cong \triangle ACB$ . Since corresponding parts of congruent triangles are congruent, it follows that $\angle B \cong \angle C$ . $\qedsymbol$

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.

"angles of an isosceles triangle" is owned by Wkbj79.

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See Also: determining from angles that a triangle is isosceles, pons asinorum

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Cross-references: determining from angles that a triangle is isosceles, converse theorem, ASA, corresponding parts of congruent triangles are congruent, legs, triangle, isosceles triangle, sides, congruent, angles, spherical geometry, neutral geometry, hyperbolic geometry, Euclidean geometry, SAS, geometry
There are 5 references to this entry.

This is version 7 of angles of an isosceles triangle, born on 2007-06-04, modified 2007-06-09.
Object id is 9521, canonical name is AnglesOfAnIsoscelesTriangle.
Accessed 2169 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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