In geometry, an angle is formed by two lines that meet at a point. Each line is called a side of the angle, and the point they share is called the vertex of the angle. The term angle is used to denote both geometric figures and their size or magnitude as associated quantity. Angular measure or measure of angle are sometimes used to distinguish between the measure of the quantity and figure itself. The measurement of angles is intrinsically linked with circles and rotation, and this is often visualized or defined using the arc of a circle centered at the vertex and lying between the sides.
Fundamentals There is no universally agreed definition of an angle. Angles can be conceived of and used in a variety of ways and while valid definitions may be given for specific contexts, it is difficult to give a single formal definition that is completely satisfactory in capturing all aspects of the general concept of angle. One standard definition is that an angle is a figure consisting of two rays which lie in a plane and share a common endpoint. Alternatively, given such a figure, an angle might be defined as: the opening between the rays; the area of the plane that lies between the rays; or the amount of rotation about the vertex of one ray to the other. More generally, angles are also formed wherever two line segments come together, such as at the corners of triangles and other polygons, or at the intersection of two planes or curves, in which case the rays lying tangent to each curve at the point of intersection define the angle. It is common to consider that the sides of the angle divide the plane into two regions called the interior of the angle and the exterior of the angle. The interior of the angle is also referred to as an angular sector.
Notation and measurement
An angle symbol ( ∠ {\displaystyle \angle } or ^ {\displaystyle {\widehat {\quad }}} , read as "angle") together with one or three defining points is used to identify angles in geometric figures. For example, the angle with vertex A formed by the rays AB → {\displaystyle {\vec {\text{AB}}}} and AC → {\displaystyle {\vec {\text{AC}}}} is denoted as ∠ A {\displaystyle \angle {\text{A}}} (using the vertex alone) or ∠ BAC {\displaystyle \angle {\text{BAC}}} (with the vertex always named in the middle). The size or measure of the angle is denoted m ∠ A {\displaystyle m\angle {\text{A}}} or m ∠ BAC {\displaystyle m\angle {\text{BAC}}} . In geometric figures and mathematical expressions, it is also common to use Greek letters (α, β, γ, θ, φ, ...) or lower case Roman letters (a, b, c, ...) as variables to represent the size of an angle. Angular measure is commonly a scalar quantity, although in physics and some fields of mathematics, signed angles are used by convention to indicate a direction of rotation: positive for anti-clockwise; negative for clockwise.
Units of measurement Angles are measured in various units, the most common being the degree (denoted by the symbol °), radian (denoted by the symbol rad) and turn. These units differ in the way they divide up a full angle, an angle where one ray, initially congruent to the other, performs a compete rotation about the vertex to return back to its starting position. Degrees and turns are defined directly with reference to a full angle, which measures 1 turn or 360°. A measure in turns gives an angle's size as a proportion of a full angle and a degree can be considered as a subdivision of a turn. Radians are not defined directly in relation to a full angle (see § Measuring angles), but in such a way that its measure is 2π rad, approximately 6.28 rad. Historically the degree unit was chosen such as the straight angle or half the full angle was attributed the value of 180.
Addition and subtraction
The angle addition postulate states that if D is a point lying in the interior of ∠ BAC {\displaystyle \angle {\text{BAC}}} then: m ∠ BAC = m ∠ BAD + m ∠ DAC . {\displaystyle m\angle {\text{BAC}}=m\angle {\text{BAD}}+m\angle {\text{DAC}}.} This relationship defines what it means to add any two angles: their vertices are placed together while sharing a side to create a new larger angle. The measure of the new larger angle is the sum of the measures of the two angles. Subtraction follows from rearrangement of the formula.
Types
Common angles
An angle equal to 0° or not turned is called a zero angle. An angle smaller than a right angle (less than 90°) is called an acute angle. An angle equal to 1/4 turn (90° or π/2 rad) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular. An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an obtuse angle ("obtuse" meaning "blunt"). An angle equal to 1/2 turn (180° or π rad) is called a straight angle. An angle larger than a straight angle but less than 1 turn (between 180° and 360°) is called a reflex angle. An angle equal to 1 turn (360° or 2π rad) is called a full angle, complete angle, round angle or perigon. An angle that is not a multiple of a right angle is called an oblique angle.
Adjacent and vertical angles
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