Angle sum of a triangle

Statement

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

Remarks

Mathlib proves the formula under the weaker hypothesis that $B \neq A$ and $C \neq A$ , with its convention for the angle at a repeated point. An equilateral triangle has three equal angles, so each one is $\pi / 3$ . 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 $\pi$ , and in hyperbolic geometry, where it is less.

History

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

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Where this leads

3 pages build on this one.

Tag	`05MDKQBEAF`
URI	`m2://05MDKQBEAF`
Aliases	`triangle-angle-sum-theorem`
Kind	statement, theorem
Writing	W0 stub
MSC	51M04
Status	This copy of the site has no build data. Run `m2 status 05MDKQBEAF`.
Uses	Triangle, Angle at a point
Same as	Wikidata Q7840113 Wikipedia Sum of angles of a triangle

Angle sum of a triangle

Statement

Remarks

History

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

Used by