Math 2.0

statement theorem

Angle sum of a triangle

Statement

Let ABCABC be a triangle in a Euclidean space Rn\mathbb{R}^n, and let ∠\angle denote the angle at a point. Then ∠BAC+∠CBA+∠ACB=π.\angle BAC + \angle CBA + \angle ACB = \pi.

Remarks

Mathlib proves the formula under the weaker hypothesis that B≠AB \neq A and C≠AC \neq A, with its convention for the angle at a repeated point. An equilateral triangle has three equal angles, so each one is π/3\pi / 3. In the axioms of plane geometry, the statement is equivalent to the parallel postulate. It is false in spherical geometry, where the sum is more than π\pi, and in hyperbolic geometry, where it is less.

History

Euclid's Elements proves the statement as part of Proposition 32 of Book I.

Explanations

L1 human

Cut a paper triangle and tear off its three corners. Put the corners side by side with their tips at one point. They always fill a straight line, which is half a turn, or π\pi radians.

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Angle at a point, concept, no evidence levelAngle at a pointconceptTriangle, concept, no evidence levelTriangleconceptAngle sum of a triangle, statement, no evidence levelAngle sum of a triang…statement
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Sources

  1. Web https://en.wikipedia.org/wiki/Sum_of_angles_of_a_triangle, Euclidean geometry

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