In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.
Video: Types of angles
Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.
Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.
Types of angles
Individual angles
Right angle.
Reflex angle.
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
Angles smaller than a right angle (less than 90°) are called acute angles ("acute" meaning "sharp").
An angle equal to 1/4 turn (90° or π/2 radians) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular.
Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called obtuse angles ("obtuse" meaning "blunt").
An angle equal to 1/2 turn (180° or π radians) is called a straight angle.
Angles larger than a straight angle but less than 1 turn (between 180° and 360°) are called reflex angles.
An angle equal to 1 turn (360° or 2π radians) is called a full angle, complete angle, or a perigon.
Angles that are not right angles or a multiple of a right angle are called oblique angles.
The names, intervals, and measured units are shown in a table below:
Name
acute
right angle
obtuse
straight
reflex
perigon
Units
Interval
Turns
(0, 1/4)
1/4
(1/4, 1/2)
1/2
(1/2, 1)
1
Radians
(0, 1/2π)
1/2π
(1/2π, π)
π
(π, 2π)
2π
Degrees
(0, 90)°
90°
(90, 180)°
180°
(180, 360)°
360°
Gons
(0, 100)g
100g
(100, 200)g
200g
(200, 400)g
400g
Vertical and adjacent angle pairs
Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles.
When two straight lines intersect at a point, four angles are formed. Pairwise these angles are named according to their location relative to each other.
A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles or vertically opposite angles. They are abbreviated as ,,,,,,
In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), corresponding to a quarter turn If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles.
Video: Right angle
The term is a calque of Latin angulus rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line.
Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors. The presence of a right angle in a triangle is the defining factor for right triangles,making the right angle basic to trigonometry.
