Supplementary Angles Definition Geometry

An angle is the area of the plane between two straight lines with a common origin. The straight lines are called sides and the common origin is called the vertex.

Measuring angles

To measure angles you use the arc of a circle The arc of a circle is the amplitude of the angle that results from dividing the circumference into equal parts.

Radians with the unit radian

The radian (rad) is the measure of the central angle of a circle whose arc length is equal to the length of its radius. Our best available math tutors 5 (35 ratings) 1st lesson free! 5 (65 ratings) 1st lesson free! 5 (73 ratings) 1st lesson free! 5 (47 ratings) 1st lesson for free! 5 (36 ratings) 1st lesson for free! 5 (25 ratings) 1st lesson for free! 5 (36 ratings) 1st lesson for free! 5 (20 ratings) 1st lesson for free! 5 (35 ratings) 1st lesson for free! 5 (65 ratings) 1st lesson for free! 5 (73 ratings) 1st lesson for free! 5 (47 ratings) 1st lesson for free! 5 (36 ratings) 1st lesson for free! 5 (25 ratings) 1st lesson for free! 5 (36 ratings) 1st lesson for free! 5 (20 ratings) 1st lesson for free! Let’s go

Classification of angles according to their measure

acute angle

It measures less than .

Right angle

It measures exactly .

Blunt angle

It measures more than .

Flat angle

It measures .

Overshot angle

It measures less than a flat angle.

Sublime angle

It measures more than a flat angle.

Zero angle

It measures . The lines forming the angle lie directly on top of each other.

Solid angle

It measures .

Negative angle

It measures less than . Negative angle rotate clockwise clockwisei.e. in the direction in which the hands of a clock move. A negative angle can be converted into a positive angle by adding adding.

Angle greater than 360

It measures more than one revolution. An angle of corresponds – if you represent it – to an angle of . An angle of corresponds – if you represent it – to an angle of . If you want to assign an angle to the first turn, divide the angle by The quotient is the number of turns that results. The remainder is the resulting angle corresponding to the first turn.

Classification of angles according to their position

Secondary angle

Two angles that are adjacent to each other at a straight line intersection are called adjacent angles.

Adjacent angles

These are angles that have the vertex and one side in common. The other sides lie in a polygon of the other. Together they form a elongated angle.

Alternate angle

They are angles that have the vertex in common. The sides of one are an extension of the sides of the other. The angles and are equal. The angles and are equal.

Classification of angles according to their sum

Complementary angle

Two angles that add up to are called Complementary angle.

Supplementary angle

Two angles that complement each other are called Supplementary angle.

Step angle

Supplementary angles

The angles and are equal.

Alternate angles inside

The angles and are equal.

Alternate angle outside

The angles and are equal.

Angle at circle

Center angle

The angle whose vertex lies at the center of the circle and whose legs intersect the boundary points of the circular arc is called the center angle. The measure of an arc is the measure of the corresponding center angle.

Circle angle

The circle angle has its vertex on the circumference and its sides are secants to it. It measures half of the arc it spans.

Half angle

The half angle lies on the circle, one side is secant and the other is tangent to it. It measures half of the arc it spans.

Interior angle

Its vertex is inside the circle and its sides are secants to it. It measures half the sum of the measures of the arcs that span its sides and the extensions of its sides.

Exterior angle

Its vertex is a point outside the circle, and the sides of its angles are: either a secant to it, or a tangent and a secant, or tangent to it. It measures half the difference between the measures of the arcs of the circle that span its sides on the circumference.

Angle of a regular polygon

Center angle of a regular polygon

It is the one formed by two consecutive radians. Example If is the number of sides of a polygon: midpoint angle = Center angle of regular polygon =

Interior angle of a regular polygon

It is the one formed by two consecutive sides. Interior angle Center angle interior angle of a regular polygon

Exterior angle of a regular polygon

It is the one formed by one side and the extension of a successive side. The exterior and interior angles are complementary, i.e. they add up to . exterior angle = center angle exterior angle of regular polygon

Sum

The sum of two angles is another angle whose amplitude is the sum of the amplitudes of the two initial angles.

Subtraction

The subtraction of two angles is another angle whose amplitude is the difference between the amplitude of the larger angle and the amplitude of the smaller angle.

Multiplication of a number by an angle

Multiplying a number by an angle is another angle whose amplitude is the sum of as many angles as the number specifies.

Division of an angle by a number

Dividing an angle by a number consists of finding another angle which, when multiplied by that number, gives the original angle.

Angle on a circle

For many problems in elementary geometry involving angles at circles the following terms and statements can be used.

Terms

If you connect the endpoints A and B of an arc, which are different from each other, with its center M and a point P on the arc, the following angles exist: Many authors of geometry textbooks do not refer to circumferential angles, midpoint angles, and chord tangent angles to a given circular arc, but to a given chord of a circle [AB]. If such a definition is used, two types of circumferential angles must be distinguished, namely acute and obtuse circumferential angles. In this case, the midpoint angle is defined as the smaller of the two angles enclosed by the circular radii [MA] and [MB]. The formulation of the propositions in the next section must be varied slightly when using this definition. Circumferential, central and chordal tangent angles

Circle angle theorem (center angle theorem)

Sketch of the Circle Angle Theorem The central angle (center angle) of a circular arc is twice as large as one of the corresponding circumferential angles (peripheral angle). The proof of this statement is particularly simple in the special case sketched on the left. The two angles at B and P are equal as base angles in the isosceles triangle MBP. The third angle of the triangle MBP (with vertex M) has the size . Consequently, the theorem about the sum of angles gives and further, as claimed, . In the general case, M does not lie on a leg of the circumferential angle. The straight line PM then divides circumferential angle and central angle into two angles ( and resp. and ), for each of which individually the statement holds, since the conditions of the proved special case are fulfilled. Therefore, the statement is also valid for the entire circumferential angle and the entire central angle . Furthermore, the validity of the Peripheral Angle Theorem (see below) makes it possible to transfer the general case into the special case without limiting the generality of the proof already given for the special case. Supplementary illustration to the above picture General case. As can be seen in it: resp. Theorem of Thales A particularly important special case exists if the given arc is a semicircle: In this case, the central angle is equal to 180° (a stretched angle), while the circumferential angles are equal to 90°, that is, right angles. Thus, Thales’ theorem proves to be a special case of the circle angle theorem.

Peripheral Angle Theorem (Peripheral Angle Theorem)

All circumferential angles (peripheral angles) over a circular arc are equal. This circular arc is then called a barrel arc. The circumferential angle theorem is a direct consequence of the circle angle theorem: According to the circle angle theorem, every circumferential angle is half as large as the center angle (central angle). So all circumferential angles must be equal. However, it may be necessary to prove the Peripheral Angle Theorem in another way, otherwise it cannot be used as a condition in the proof of the Circular Angle Theorem.

Tendon angle theorem

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

Application in construction tasks

Peripheral angle theorem

In particular, the Peripheral Angle Theorem can be used quite often for geometric constructions. In many cases, one is looking for the set (the geometric location) of all points P from which a given line (here [AB]) appears at a given angle. The set of points we are looking for generally consists of two circular arcs, called barrel arcs (Figure 1). Figure 1: Sketch of the pair of barrel arcs Alternative proof of the circumferential angle theorem, Landesbildungsserver Baden-Württemberg The proof presented here impresses by its simplicity and leads in a natural way to the connections between circumferential angle and central angle as well as to the peculiarity of chordal quadrilaterals.

Circle angle theorem

The circle angle theorem is also suitable as a construction building block for solving e.g. the following problems:

• Draw a quadrilateral for which the side length is given.

To do this, first construct the circumcircle of a decagon with only one side length and then apply the circle angle theorem twice in succession.

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

Figure 2: Circle angle theorem approach for the trisection of an arbitrary angle. Later the right branch of the hyperbola runs through the point. Figure 3: Circle angle theorem Construction of a polygon with a given side length, which has twice the number of corners of a polygon with the same side length. Example: The side length of the sought-for dodecagon (blue) is equal to that of the given decagon.

Literature

• Max Koecher, Aloys Krieg: Plane Geometry. 3rd, revised and extended edition. Springer, Berlin u.a. 2007, ISBN 978-3-540-49327-3.
• Günter Aumann: Circle geometry: an elementary introduction. Springer, 2015, ISBN 978-3-662-45306-3.

Based on an article in: Wikipedia.com. Page back © biancahoegel.de Date last modified: Jena, den: 08.11. 2020 Supplementary Angles Definition Geometry.

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