Outside angle set

The exterior angle sentence ( English Exterior Angle theorem ) is a theorem of geometry , which means that each external angle of a triangle is as large as the two non-abutting internal angle together. It was first used in the 3rd century BC. Proved as Theorem 32 in Book 1 of the Elements of Euclid .

Formulation of the sentence

Internal angles α, β, γ and external angles α ′, β ′, γ ′ of a triangle in the Euclidean plane

The exterior angle theorem of Euclidean geometry says that the exterior angle at one corner of a triangle is always equal to the sum of the interior angles at the other two corners; in a triangle, for example, is the sum of the interior angles at the corners and equal to the outer angle at the corner . ${\ displaystyle ABC}$ ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle C}$

proof

The theorem of external angles is a simple consequence of the theorem of the sum of angles , because for the (internal) angles denoted by , and the applies , and thus also ; as for the outer angle at the corner it is true that it has an amount of as a complementary angle to the inner angle . Which promptly gives you the set of external angles: ${\ displaystyle \ alpha}$ ${\ displaystyle \ beta}$ ${\ displaystyle \ gamma}$ ${\ displaystyle \ triangle {ABC}}$ ${\ displaystyle \ alpha + \ beta + \ gamma = 180 ^ {\ circ}}$ ${\ displaystyle 180 ^ {\ circ} - \ alpha = \ beta + \ gamma}$ ${\ displaystyle \ alpha '}$ ${\ displaystyle A}$ ${\ displaystyle \ alpha}$ ${\ displaystyle 180 ^ {\ circ} - \ alpha}$

{\ displaystyle \ alpha '= \ beta + \ gamma}

.

Analogously one proves and . ${\ displaystyle \ beta '= \ alpha + \ gamma}$ ${\ displaystyle \ gamma '= \ alpha + \ beta}$

Weak outside angle set

The weak exterior angle set , also known as the set of exterior angles , says:

Every exterior angle of any triangle is always strictly larger than each of the two non-adjacent interior angles .

In formulas:

{\ displaystyle \ alpha '> \ beta, \ alpha'> \ gamma, \ beta '> \ alpha, \ beta'> \ gamma, \ gamma '> \ alpha, \ gamma'> \ beta}

.

It follows that each interior angle is always strictly smaller than each of the two non-adjacent exterior angles.

Relationship between exterior angle set and weak exterior angle set

The weak set of exterior angles obviously follows from the set of exterior angles. But it can also be proved without using the external angle theorem by means of the cosine theorem and Cauchy-Schwarz inequality and then derive the external angle theorem from the weak external angle theorem with the addition of the parallels axiom .

From the axiom of parallels and the weak theorem of exterior angles it follows that the sum of interior angles in the triangle is 180 °, from which, with the proof given above, the exterior angle theorem in its strong form results.

Inferences

The law of exterior angles - even in its weak form , which is valid without assuming the axiom of parallels - leads to a number of consequences, of which the following are often mentioned:

In every triangle, the larger side is always opposite the larger angle and, conversely, the larger angle is always the larger side.
In any triangle, the sum of the lengths of two sides is strictly greater than the length of the third side. ( Triangle inequality )
At least two of the three interior angles of any triangle are acute angles.
The angle sum of two interior angles of any triangle is always smaller than a straight angle.

From the external angle theorem (assuming the axiom of parallels), if one takes the usual angle measurement in degrees as a basis and at the same time always considers only one of the two secondary angles of the associated internal angle as the external angle:

The sum of the outer angles of a triangle is . ${\ displaystyle 360 ^ {\ circ}}$

This conclusion can still be generalized. Because in the Euclidean plane the corresponding statement is even valid for all convex polygons - regardless of the number of vertices:

In the Euclidean plane the sum of the outer angles of a convex polygon is always any number of corners . ${\ displaystyle 360 ^ {\ circ}}$

The latter result is also sometimes referred to as the external angle set.

The set of external angles in absolute geometry

The proof of the weak exterior angle theorem is not based on the axiom of parallels and thus belongs to the theorems of absolute elementary geometry . (Those parts of Euclidean geometry that do not need the axiom of parallels and that are therefore also valid in non-Euclidean geometries such as hyperbolic geometry are referred to as absolute geometry .)

In David Hilbert's Fundamentals of Geometry , the weak set of exterior angles appears as the set of exterior angles . According to Hilbert, it is a “fundamental proposition which already plays an important role in Euclid and from which a number of important facts follow”.

The theorem of external angles is logically equivalent to the weak theorem of external angles if the axiom of parallels is used in addition to the axioms of absolute geometry . In the literature on absolute geometry, the weak set of external angles is sometimes referred to as the "set of external angles" (or "first set of external angles") and the set of external angles as the "strong set of external angles" (or "second set of external angles").

With the weak set of exterior angles, the size comparison of two angles plays an essential role. According to Hilbert, two angles are basically either the same , i.e. congruent , or unequal , in the latter case one of the two is strictly smaller than the other, which is then the strictly larger , or vice versa. In doing so, straight angles and obtuse angles are not used. Under these framework conditions, the size comparison of two angles is achieved by means of applications , with one angle being applied to one leg of the other in the vertex in such a way that the interior of the angle indicated overlaps with the interior of the other in the common leg and further points . The decision with regard to the question of size then depends on whether the free leg of the angle shown lies entirely inside the other angle or not. In the first case the indicated angle is the smaller, in the opposite case the larger one. If the interiors of both angles can even be brought into congruence in this way , both angles are the same; otherwise they are unequal.

The set of external angles in non-Euclidean geometries

Hyperbolic geometry

As in any geometry based on Euclid's axioms (without axiom of parallels), the weak law of external angles also applies in hyperbolic geometry . On the other hand, the law of external angles in its strong form does not apply in hyperbolic geometry; instead one has the so-called tightened external angle law in hyperbolic geometry :

In the hyperbolic plane, every outer angle of any triangle is strictly larger than the angle sum of the two non-adjacent inner angles.

This tightened set of exterior angles is also called the set of exterior angles of Lobachevsky geometry because it is based on the Lobachevsky axiom of parallels on which hyperbolic geometry is based.

Elliptical geometry

In elliptical geometry there is no set corresponding to the set of external angles. However, some of the conclusions presented above can be drawn in spherical geometry for Euler's spherical triangles , such as the triangle inequality given above.

literature

Ilka Agricola , Thomas Friedrich : Elementary Geometry . Expertise for studies and math lessons. 4th, revised edition. Springer Spectrum, Wiesbaden 2015, ISBN 978-3-658-06730-4 , doi : 10.1007 / 978-3-658-06731-1 .

Hermann Athens, Jörn Bruhn (ed.): Lexicon of school mathematics and related areas . tape 1 : A-E . Aulis Verlag Deubner, Cologne 1977, ISBN 3-7614-0242-2 , p. 404-405 .

Richard L. Faber : Foundations of Euclidean and Non-Euclidean Geometry. (= Monographs and Textbooks in Pure and Applied Mathematics . Volume 73 ). Marcel Dekker, New York / Basel 1983, ISBN 0-8247-1748-1 ( ams.org ).

Andreas Filler: Euclidean and non-Euclidean geometry (= mathematical texts . Volume 7 ). BI-Wissenschaftsverlag, Mannheim u. a. 1993, ISBN 3-411-16371-2 ( ams.org ).

Gerhard Hessenberg , Justus Diller: Fundamentals of geometry . 2nd Edition. Walter de Gruyter Verlag, Berlin 1967.

David Hilbert: Fundamentals of Geometry. With supplements by Paul Bernays (= Teubner Study Books: Mathematics ). 11th edition. Teubner Verlag, Stuttgart 1972, ISBN 3-519-12020-8 ( ams.org ).

Marvin Jay Greenberg : Euclidean and Non-Euclidean Geometries. Development and History . 3. Edition. WH Freeman and Company, San Francisco 1993, ISBN 0-7167-2446-4 ( ams.org ).

Hanfried Lenz : Non-Euclidean geometry (= BI university pocket books. 123 / 123a). Bibliographisches Institut, Mannheim 1967 ( ams.org ).

Arno Mitschka: Axiomatics in Geometry (= study books mathematics . Volume 7 ). Herder Verlag, Freiburg u. a. 1977, ISBN 3-451-16898-7 ( online ).

Fritz Reinhardt, Heinrich Soeder (Ed.): Dtv-Atlas for Mathematics. Boards and texts . 8th edition. Volume I: Fundamentals, Algebra and Geometry . Deutscher Taschenbuch Verlag, Munich 1990, ISBN 3-423-03007-0 .

Web links

Andreas Filler: Euclidean and non-Euclidean geometry . (PDF) HU Berlin
Michael Gieding: Excerpt from a script. PH Heidelberg

References and comments

^ Agricola-Friedrich (op.cit.), Sentence 7, p. 10.

^ Agricola-Friedrich (op.cit.), Sentence 8, p. 11.

↑ ^a ^b ^c D. Hilbert: Fundamentals of geometry . 1972, p. 24 .

^ ^A ^b G. Hessenberg, J. Diller: Fundamentals of geometry . 1967, p. 44 ff .

↑ ^a ^b A. Filler: Euclidean and non-Euclidean geometry . 1993, p. 105 ff .

↑ ^a ^b ^c H. Lenz: Non-Euclidean Geometry . 1967, p. 65 ff .

↑ ^a ^b ^c A. Mitschka: Axiomatics in Geometry . 1977, p. 115 ff .

↑ It should be noted that a sharp inequality is formulated here , which excludes the equality case . In contrast, the one pronounced in the theory of metric and pseudometric spaces - related! - Triangle inequality a fuzzy inequality in which the equality case is allowed.

↑ Beyond the simple triangle inequality, the following applies (cf. Hessenberg-Diller, p. 46): For three points A, B, Z the equation implies that Z lies on the line [AB] and thus between A and B. This results in Leibniz's minimal principle (named after Leibniz ) : The shortest line that connects two points is the line defined by the two points. ${\ displaystyle | AZ | + | ZB | = | AB |}$
↑ Lexicon of School Mathematics ... Volume 1 , p. 85 . ; the spelling here is “outside angle sentence”.
^ MJ Greenberg: Euclidean and Non-Euclidean Geometries . 1993, p. 118 ff .
^ MJ Greenberg: Euclidean and Non-Euclidean Geometries . 1993, p. 98 ff .
^ RL Faber: Foundations of Euclidean and Non-Euclidean Geometry . 1983, p. 113 ff .
↑ In Mitschka, the sentence is exactly called the "first sentence of the outer angle in the triangle" (p. 115), which, however, seems to be an inconsistent formulation and also does not correspond to the name of the later tightening (p. 129) in Mitschka omits the addition “in the triangle”.
↑ A. Mitschka: axioms in geometry . 1977, p. 129 .
↑ D. Hilbert: Fundamentals of geometry . 1972, p. 13-22 .
↑ A. Filler: Euclidean and non-Euclidean geometry . 1993, p. 168 .
↑ A. Filler: Euclidean and non-Euclidean geometry . 1993, p. 166-168 .
^ MJ Greenberg: Euclidean and Non-Euclidean Geometries . 1993, p. 90.120 .
↑ A. Filler: Euclidean and non-Euclidean geometry . 1993, p. 15th ff .

[1] Agricola-Friedrich (op.cit.), Sentence 7, p. 10.

[2] Agricola-Friedrich (op.cit.), Sentence 8, p. 11.

[Hilbert-3] D. Hilbert: Fundamentals of geometry . 1972, p. 24 .

[Hessenberg-Diller-4] A ^b G. Hessenberg, J. Diller: Fundamentals of geometry . 1967, p. 44 ff .

[Filler-5] A. Filler: Euclidean and non-Euclidean geometry . 1993, p. 105 ff .

[Lenz-6] H. Lenz: Non-Euclidean Geometry . 1967, p. 65 ff .

[Mitschka-7] A. Mitschka: Axiomatics in Geometry . 1977, p. 115 ff .

[8] It should be noted that a sharp inequality is formulated here , which excludes the equality case . In contrast, the one pronounced in the theory of metric and pseudometric spaces - related! - Triangle inequality a fuzzy inequality in which the equality case is allowed.