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angles of an isosceles triangle∗
Wkbj79†
2013-03-21 23:06:53
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 4ABC be isosceles such that the legs AB and AC are
congruent.
A
B
C
Since we have
• AB ∼
= AC
• ∠A ∼
= ∠A by the reflexive property of ∼
=
• AC ∼
= AB by the symmetric property of ∼
=
we can use SAS to conclude that 4ABC ∼
= 4ACB. Since corresponding
parts of congruent triangles are congruent, it follows that ∠B ∼
= ∠C.
∗ hAnglesOfAnIsoscelesTrianglei
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1
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.
2