Exterior angle theorem

Builds on Triangle Angle at a point 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. Let $D$ be a point such that $C$ lies strictly between $B$ and $D$ on the line $BC$ . Then the exterior angle at $C$ is the sum of the two interior angles at the other vertices: $\angle ACD = \angle BAC + \angle CBA.$

Remarks

The angles $\angle ACD$ and $\angle ACB$ add up to $\pi$ . Thus the statement follows from the angle sum of a triangle. A consequence is that the exterior angle is greater than each of the two remote interior angles. This weaker form does not need the parallel postulate.

History

In Book I of Euclid's Elements, the weaker form is Proposition 16. The form with the sum is part of Proposition 32 of the same book. Many school books call the form with the sum the exterior angle theorem.

Reading path

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Where this leads

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Saccheri–Legendre theorem

Tag	`0NTSE7EREA`
URI	`m2://0NTSE7EREA`
Kind	statement, theorem
Writing	W1 stated
MSC	51M04
Status	This copy of the site has no build data. Run `m2 status 0NTSE7EREA`.
Uses	Triangle, Angle at a point, Angle sum of a triangle
Same as	Wikidata Q5421989 Wikipedia Exterior angle theorem

Exterior angle theorem

Statement

Remarks

History

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

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