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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
, a right angle is an
angle In Euclidean 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. Angles formed by two rays lie in the plane that contains the rays. Angles ...
of exactly 90 degrees or
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s corresponding to a quarter turn. If a
ray Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (gr ...
is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. The term is a
calque In linguistics, a calque () or loan translation is a word or phrase borrowed from another language by literal word-for-word or root-for-root translation. When used as a verb, "to calque" means to borrow a word or phrase from another language ...
of
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
''angulus rectus''; here ''rectus'' means "upright", referring to the vertical perpendicular to a horizontal base line. Closely related and important geometrical concepts are
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
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.


Etymology

The meaning of ''right'' in ''right angle'' possibly refers to the
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
adjective ''rectus'' 'erect, straight, upright, perpendicular'. A Greek equivalent is ''orthos'' 'straight; perpendicular' (see orthogonality).


In elementary geometry

A rectangle is a quadrilateral with four right angles. A square has four right angles, in addition to equal-length sides. The Pythagorean theorem states how to determine when a triangle is a right triangle.


Symbols

In
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...
, the symbol for a right angle is . It should not be confused with the similarly shaped symbol . Related symbols are , , and . In diagrams, the fact that an angle is a right angle is usually expressed by adding a small right angle that forms a square with the angle in the diagram, as seen in the diagram of a right triangle (in British English, a right-angled triangle) to the right. The symbol for a measured angle, an arc, with a dot, is used in some European countries, including German-speaking countries and Poland, as an alternative symbol for a right angle.


Euclid

Right angles are fundamental in
Euclid's Elements The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postu ...
. They are defined in Book 1, definition 10, which also defines perpendicular lines. Definition 10 does not use numerical degree measurements but rather touches at the very heart of what a right angle is, namely two straight lines intersecting to form two equal and adjacent angles. The straight lines which form right angles are called perpendicular. Euclid uses right angles in definitions 11 and 12 to define acute angles (those smaller than a right angle) and obtuse angles (those greater than a right angle). Two angles are called complementary if their sum is a right angle. Book 1 Postulate 4 states that all right angles are equal, which allows Euclid to use a right angle as a unit to measure other angles with. Euclid's commentator Proclus gave a proof of this postulate using the previous postulates, but it may be argued that this proof makes use of some hidden assumptions. Saccheri gave a proof as well but using a more explicit assumption. In Hilbert's axiomatization of geometry this statement is given as a theorem, but only after much groundwork. One may argue that, even if postulate 4 can be proven from the preceding ones, in the order that Euclid presents his material it is necessary to include it since without it postulate 5, which uses the right angle as a unit of measure, makes no sense.Heath pp. 200-201 for the paragraph


Conversion to other units

A right angle may be expressed in different units: * turn * 90° ( degrees) *
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s * 100 grad (also called ''grade'', ''gradian'', or ''gon'') * 8 points (of a 32-point
compass rose A compass rose, sometimes called a wind rose, rose of the winds or compass star, is a figure on a compass, map, nautical chart, or monument used to display the orientation of the cardinal directions (north, east, south, and west) and thei ...
) * 6 hours (astronomical
hour angle In astronomy and celestial navigation, the hour angle is the angle between two planes: one containing Earth's axis and the zenith (the '' meridian plane''), and the other containing Earth's axis and a given point of interest (the ''hour circle' ...
)


Rule of 3-4-5

Throughout history, carpenters and masons have known a quick way to confirm if an angle is a true "right angle". It is based on the most widely known Pythagorean triple and so called the "rule of 3-4-5". From the angle in question, running a straight line along one side exactly 3 units in length, and along the second side exactly 4 units in length, will create a hypotenuse (the longer line opposite the right angle that connects the two measured endpoints) of exactly 5 units in length. This measurement can be made quickly and without technical instruments. The geometric law behind the measurement is the Pythagorean theorem ("The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides").


Thales' theorem

Thales' theorem states that an angle inscribed in a semicircle (with a vertex on the semicircle and its defining rays going through the endpoints of the semicircle) is a right angle. Two application examples in which the right angle and the Thales' theorem are included (see animations).


See also

* Cartesian coordinate system * Types of angles


References

* *Euclid, commentary and trans. by T. L. Heath ''Elements'' Vol. 1 (1908 Cambridge
Google Books
{{Authority control Angle Orthogonality