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determining from angles that a triangle
is isosceles∗
Wkbj79†
2013-03-21 23:07:29
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 1. If a triangle has two congruent angles, then it is isosceles.
Proof. Let triangle 4ABC have angles ∠B and ∠C congruent.
A
B
C
Since we have
• ∠B ∼
= ∠C
• BC ∼
= CB by the reflexive property of ∼
= (note that BC and CB denote
the same line segment)
• ∠C ∼
= ∠B by the symmetric property of ∼
=
∗ hDeterminingFromAnglesThatATriangleIsIsoscelesi created: h2013-03-21i by: hWkbj79i
version: h39528i Privacy setting: h1i hTheoremi h51M04i h51-00i
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we can use ASA to conclude that 4ABC ∼
= 4ACB. Since corresponding
parts of congruent triangles are congruent, we have that AB ∼
= AC. It follows
that 4ABC is isosceles.
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.
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