Angle
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.
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Angles smaller than a right angle (less than 90°) are called acute angles ("acute" meaning "sharp").
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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.
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Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called obtuse angles ("obtuse" meaning "blunt").
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An angle equal to 1/2 turn (180° or π radians) is called a straight angle.
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Angles larger than a straight angle but less than 1 turn (between 180° and 360°) are called reflex angles.
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An angle equal to 1 turn (360° or 2π radians) is called a full angle, complete angle, or a perigon.
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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.
Video: Alternate Interior Angles, Alternate Exterior Angles, Vertical Angles, Corresponding 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.
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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 ,,,,,,
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