Question

long definition of angle

Answer 1

Angle

Definition: A shape, formed by two lines or rays diverging from a common point (the vertex).

Try this Adjust the angle below by dragging the orange dot.

Attribute
The vertex is the common point at which the two lines or rays are joined. Point B is the figure above is the vertex of the angle∠ABC.LegsThe legs (sides) of an angle are the two lines that make it up. In the figure above, the line segments AB and BC are the legs of the angle ∠ABC.InteriorThe interior of an angle is the space in the 'jaws' of the angle extending out to infinity. See Interior of an AngleExteriorAll the space on the plane that is not the interior. See Interior of an Angle

Identifying an angle

An angle can be identified in two ways.

Like this: ∠ABC

The angle symbol, followed by three points that define the angle, with the middle letter being the vertex, and the other two on the legs. So in the figure above the angle would be ∠ABC or ∠CBA. So long as the vertex is the middle letter, the order is not important. As a shorthand we can use the 'angle' symbol. For example '∠ABC' would be read as 'the angle ABC'.

Or like this: ∠B

Just by the vertex, so long as it is not ambiguous. So in the figure above the angle could also be called simply  '∠B'

Measure of an angle

The size of an angle is measured in degrees (see Angle Measures). When we say 'the angle ABC' we mean the actual angle object. If we want to talk about the size, or measure, of the angle in degrees, we should say 'the measure of the angle ABC' - often written m∠ABC. 

However, many times we will see '∠ABC=34°'. Strictly speaking this is an error. It should say 'm∠ABC=34°'

Types of angle

Altogether, there are six types of angle as listed below. Click on an image for a full description of that type and a corresponding interactive applet. 

Acute angle
Less than 90°Right angle
Exactly 90°Obtuse angle
Between 90° and 180°Straight angle
Exactly 180°. Reflex angle
Between 180° and 360°. Complete angle
Exactly 360°

In Trigonometry

When used in trigonometry, angles have some extra properties: They can have a measure greater than 360°, can be positive and negative, and are positioned on a coordinate grid with x and y axes. They are usually measured in radians instead of d egrees. For more on this see Angle definition and properties (trigonometry).

Angle construction

In the Constructions chapter, there are animated demonstrations of various constructions of angles using only a compass and straightedge. 

Answer 2

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Angle is a combination of two rays with a common endpoint. The latter is known as the vertex of the angle and the rays as the sides, sometimes as the legs and sometimes the arms of the angle.