Circle angle

The terms and statements explained below can be used for many questions in elementary geometry that involve angles on circles .

Terms

If you connect the different end points A and B of an arc with its center point M and a point P on the arc, the following angles exist:

• Circumferential angle or peripheral angle (ϕ) is called an angle whose apex P lies on the arc that complements the given arc via [AB] to form a complete circle (the circumference of the triangle ABP). ${\displaystyle \angle {\rm {APB}}}$
• Center angle (μ): If M is the center of the given circular arc, then the angle is called the associated center angle ( central angle ). ${\displaystyle \angle {\rm {AMB}}}$
• A chord tangent angle (τ) to the given arc is limited by the chord [AB] and the circle tangent at point A and B.

Many authors of geometry textbooks do not refer to a given arc of a circle, but to a given chord [AB] in the case of circumferential angles, central angles and chord tangent angles. If one takes such a definition as a basis, one has to distinguish between two types of circumferential angles, namely acute and obtuse circumferential angles . In this case, the center angle is defined as the smaller of the two angles enclosed by the circle radii [MA] and [MB]. The formulation of the sentences in the next section needs to be varied slightly when using this definition.

Circumferential, midpoint, and tendon tangent angles

Circular angle set (central angle set)

Sketch for the circle angle set

The central angle (central angle) of a circular arc is twice as large as one of the associated circumferential angles (peripheral angle).

The proof of this statement is particularly easy in the special case sketched on the left. The two angles at B and P are the same as the base angle in the isosceles triangle MBP. The third angle of the triangle MBP (with the vertex M) has the size . The theorem about the sum of angles consequently and further, as claimed, yields . ${\displaystyle 180^{\circ }-\mu }$ ${\displaystyle \phi +\phi +(180^{\circ }-\mu )=180^{\circ }}$ ${\displaystyle 2\phi \,=\,\mu }$

In the general case, M does not lie on one leg of the circumferential angle. The straight line PM then divides the circumferential angle and the central angle into two angles ( and or and ), for each of which the statement applies, since the requirements of the proven special case are fulfilled. Therefore, the statement also applies to the entire circumferential angle and the entire central angle . In addition, the validity of the peripheral angle theorem (see below) enables the general case to be converted into the special case without restricting the generality of the proof already provided for the special case. ${\displaystyle \phi _{1}}$ ${\displaystyle \phi _{2}}$ ${\displaystyle \mu _{1}}$ ${\displaystyle \mu _{2}}$ ${\displaystyle \phi _{1}+\phi _{2}}$ ${\displaystyle \mu _{1}+\mu _{2}}$

More evidence in the Wikibooks evidence archive

Supplementary illustration to the above picture General case . As can be seen therein: resp. ${\displaystyle \mu _{1}=2\phi _{1}}$ ${\displaystyle \mu _{2}=2\phi _{2}.}$

Thales theorem

A particularly important special case is when the given arc is a semicircle: In this case, the central angle is 180 ° (a straight angle), while the circumferential angles are 90 °, i.e. right angles . The Thales theorem thus proves to be a special case of the circular angle theorem.

Circumferential angle set (peripheral angle set)

All circumferential angles (peripheral angles) over an arc are the same. This arc is then called the barrel arc .

The set of circumferential angles is a direct consequence of the set of circular angles: According to the set of circular angles, each circumferential angle is half as large as the central angle (central angle). So all circumferential angles must be the same.

However, it may be necessary to prove the peripheral angle theorem in another way, since otherwise it cannot be used as a condition in the demonstration of the circular angle theorem.

Tendon tangent angle set

The two chord tangent angles of a circular arc are as large as the associated circumferential angle (peripheral angle) and half as large as the associated central angle (central angle).

${\displaystyle \delta =\gamma }$

Chord tangent angle theorem:
Since is isosceles, the following applies: Together with : ${\displaystyle \triangle ABM}$
${\displaystyle \alpha _{2}={\tfrac {180^{\circ }-2\gamma }{2}}=90^{\circ }-\gamma }$
${\displaystyle \alpha _{2}+\delta =90^{\circ }}$
${\displaystyle \delta =90^{\circ }-\alpha _{2}=90^{\circ }-(90^{\circ }-\gamma )=\gamma }$

Use in construction tasks

Circumferential angle set

In particular, the set of circumferential angles can often be used for geometric constructions. In many cases one looks for the set (the geometrical location ) of all points P from which a given segment (here [AB]) appears at a certain angle. The desired set of points generally consists of two circular arcs, the so-called barrel arcs (Fig. 1).

Image 1: Sketch for the pair of barrel arcs

Circle angle set

The circular angle set is also suitable as a construction component for solving z. B. the following tasks:

• Draw a quadrangle with the side length given. ${\displaystyle a}$

For this, the circumference of a decagon is first constructed with only one side length and then the circle angle set is applied twice in succession. ${\displaystyle a}$

• The trisection of the angle with the help of the hyperbola ; already in the 4th century Pappos used the properties of this sentence for their solution (Fig. 2).

Figure 2: Theorem of circular angles
Approach for the trisection of any angle. The right branch of the hyperbola later runs through the point . ${\displaystyle C}$

• A polygon is to be constructed from a given side length that has twice the number of corners of a polygon with the same side length (Fig. 3). ${\displaystyle a}$
• A polygon is to be constructed from a given side length that has half the number of corners of a polygon with the same side length (Fig. 4). ${\displaystyle a}$

Figure 3: Set of circular angles
Construction of a polygon with a given side length that has twice the number of corners of a polygon with the same side length. Example: The side length of the twenty-one (blue) you are looking for is the same as that of the given decagon. ${\displaystyle a}$

${\displaystyle a}$

Figure 4: Set of circular angles
Construction of a polygon with a given side length that has half the number of corners of a polygon with the same side length. This shows the vertical line from Example: The length of the side of the decagon you are looking for (blue) is the same as that of the specified 20. ${\displaystyle a}$ ${\displaystyle Ms}$ ${\displaystyle {\overline {AM}}.}$

${\displaystyle a}$

Web links

Wikibooks: Circle angle set  - learning and teaching materials

Wikibooks: tendon tangent angle set  - learning and teaching materials

• Alternative proof of the circumferential angle theorem, State Education Server Baden-Württemberg The proof presented here is impressive because of its simplicity and leads in a natural way to the relationships between the circumferential angle and the central angle as well as to the peculiarity of tendon quadrangles.

literature

• Max Koecher , Aloys Krieg : level geometry. 3rd, revised and expanded edition. Springer, Berlin a. a. 2007, ISBN 978-3-540-49327-3 , pp. 161-162
• Schülerduden - Mathematics I . Bibliographisches Institut & FA Brockhaus, 8th edition, Mannheim 2008, ISBN 978-3-411-04208-1 , pp. 415-417
• Günter Aumann : Circular Geometry: An Elementary Introduction . Springer, 2015, ISBN 978-3-662-45306-3 , pp. 15-18