PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (236)
Orphanage
Unclass'd (1)
Unproven (540)
Corrections (49)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Copyright
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] 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

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.
(view preamble | get metadata)

View style:

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


This object's parent.
Log in to rate this entry.
(view current ratings)

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, valid, SAS, geometry, theorem
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 5923 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
 51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)