Exterior angle

Inside angle (blue) and outside angle (green) of a triangle

The exterior angles of a convex polygon are the exterior angles between one side of the polygon and the extension of an adjacent side. Each exterior angle is the secondary angle of an interior angle and supplements this to 180 °. The sum of the outer angles of a polygon is independent of the number of its corners and always results in 360 °.

According to the set of exterior angles , each exterior angle in a triangle is equal to the sum of the two non-adjacent interior angles. The bisectors of the outer angles of a triangle intersect at the center points of the circle .

Designations

Designations of inside and outside angles on the square ABCD

If the sides of a convex polygon are extended beyond the corners , four angles of intersection arise , two of which are opposite each other congruent . The angle that lies inside the polygon is called the inside angle of the polygon. If the corners of the polygon are marked with , the interior angles are usually mentioned . The apex angle opposite the interior angle is the same size as this and is designated in the same way. The two remaining angles are also vertex angles and are called the outer angles of the polygon. The outer angles are usually referred to as (see illustration). Each exterior angle is the secondary angle of its associated interior angle, that is, it applies ${\displaystyle A,B,C,\ldots }$ ${\displaystyle \alpha ,\beta ,\gamma ,\ldots }$ ${\displaystyle \alpha ',\beta ',\gamma ',\ldots }$

${\displaystyle \alpha +\alpha '=\beta +\beta '=\gamma +\gamma '=\ldots =180^{\circ }}$.

Both the interior angles and the exterior angles of a polygon are clearly assigned to the corners of the polygon. The interior angle belonging to an exterior angle is called the adjacent interior angle , while the remaining interior angles of the polygon are called non-adjacent interior angles . Correspondingly, an outside angle that is assigned to an inside angle is referred to as the adjacent outside angle and the remaining outside angles of the polygon are referred to as non-adjacent outside angles .

Examples

For an equilateral triangle , the inside and outside angles apply ${\displaystyle \Delta ABC}$

${\displaystyle \alpha =\beta =\gamma =60^{\circ }~{\text{und}}~\alpha '=\beta '=\gamma '=120^{\circ }}$.

For a rectangle, the same applies to the inside and outside angles ${\displaystyle \Box ABCD}$

${\displaystyle \alpha =\beta =\gamma =\delta =90^{\circ }~{\text{und}}~\alpha '=\beta '=\gamma '=\delta '=90^{\circ }}$.

In the case of an equiangular polygon , for example a regular polygon , with corners, all external angles are of the same size and each measure . ${\displaystyle n}$ ${\displaystyle 360^{\circ }/n}$

properties

Illustration of the exterior angle set

Exterior angle sets

→ Main article : Exterior angle set

The exterior angle theorem of Euclidean geometry states that the exterior angle of a triangle is always equal to the sum of the two non-adjacent interior angles. It applies accordingly

${\displaystyle \alpha '=\beta +\gamma ,~\beta '=\alpha +\gamma ~{\text{und}}~\gamma '=\alpha +\beta }$

(see the figure for the third equation). The weak exterior angle theorem states that the exterior angle of a triangle is always strictly larger than either of the two non-adjacent interior angles, i.e.

${\displaystyle \alpha '>\beta ,~\alpha '>\gamma ,~\beta '>\alpha ,~\beta '>\gamma ,~\gamma '>\alpha ~{\text{und}}~\gamma '>\beta }$.

From the weak exterior angle theorem it also follows that each interior angle is always strictly smaller than each of the two non-adjacent exterior angles.

Angle sum

The inside angle sum is in a convex polygon with corners ${\displaystyle n}$

${\displaystyle \alpha +\beta +\gamma +\ldots =(n-2)\cdot 180^{\circ }}$.

Since adjacent inside and outside angles each add up to 180 °, the result for the outside angle sum is a convex polygon with corners ${\displaystyle n}$

${\displaystyle \alpha '+\beta '+\gamma '+\ldots =(180^{\circ }-\alpha )+(180^{\circ }-\beta )+(180^{\circ }-\gamma )+\ldots =n\cdot 180^{\circ }-(n-2)\cdot 180^{\circ }=360^{\circ }}$.

The sum of the outer angles of a convex polygon is therefore always 360 °, regardless of the number of corners. Here, two congruent exterior angles are counted only once.

Bisector

Triangle with outer angle bisector (green), inner angle bisector (red), inscribed circle (blue) and circles (orange)

In the corners of a convex polygon, the bisectors of the associated external and internal angles always intersect at a right angle .

In a triangle the bisector of the outer angles of various corners in the cut Ankreismittelpunkten , and the triangle. Each of these three points of intersection lies at the same time on the bisector of the respective non-adjacent interior angle. ${\displaystyle J_{A}}$ ${\displaystyle J_{B}}$ ${\displaystyle J_{C}}$

Furthermore , the outer angle bisector divides the extended opposite side of the triangle in a triangle in the ratio of the lengths of the two sides adjacent to the angle (see also circle of Apollonios ). The intersections of the outer angle bisector with the extended opposite sides, if they exist, all lie on a straight line .

Generalizations

Inside and outside angles at a re-entrant corner

Non-convex polygons

Inside and outside angles can also be defined for non-convex polygons. At a re-entrant corner, however, the outer angles are then inside the polygon. In such a case, the outer angle is assigned a negative angular dimension so that the sum of the angles from the outer angle and the associated inner angle is still 180 °. In this way, the sum of the outer angles of a non-convex polygon, as in the convex case, is 360 °.

More general geometries

The term outer angle can also be defined in more general geometries, such as absolute geometry and Riemannian geometry . The weak set of external angles also applies in absolute geometry, while the set of external angles no longer has to be correct in non-Euclidean geometries.

In spherical geometry , the sum of the outer angles of a polygon is always smaller than 360 °, while in hyperbolic geometry the sum of the outer angles is always greater than 360 °.

literature

• Ilka Agricola, Thomas Friedrich: Elementary Geometry: Expertise for Studies and Mathematics Lessons . Springer, 2010, ISBN 978-3-8348-9826-5 , pp. 15-18 .
• Susanne Müller-Philipp, Hans-Joachim Gorski: Guideline Geometry: For students of teaching posts . Springer, 2009, ISBN 978-3-8348-0097-8 , pp. 236-238 .

Web links

Wiktionary: external angle  - explanations of meanings, word origins, synonyms, translations

Commons : Exterior angles  - collection of images, videos, and audio files

• Eric W. Weisstein : Exterior Angle . In: MathWorld (English).
• Eric W. Weisstein : Exterior Angle Bisector . In: MathWorld (English).