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Transcript
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 created: h2013-03-21i by: hWkbj79i version: h39521i Privacy setting: h1i hTheoremi h51-00i h51M04i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 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