In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or /2radians[1] corresponding to a quarter turn.[2] If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles.[3] The term is a calque of Latinangulus rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line.
A right angle is equal to 90 degrees.
A line segment (AB) drawn so that it forms right angles with a line (CD).
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,[4] making the right angle basic to trigonometry.
The meaning of right in right angle possibly refers to the Latin 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.
Right triangle, with the right angle shown via a small square.
Another option of diagrammatically indicating a right angle, using an angle curve and a small dot.
In Unicode, the symbol for a right angle is U+221F∟RIGHT ANGLE (∟). It should not be confused with the similarly shaped symbol U+231E⌞BOTTOM LEFT CORNER (⌞, ⌞). Related symbols are U+22BE⊾RIGHT ANGLE WITH ARC (⊾), U+299C⦜RIGHT ANGLE VARIANT WITH SQUARE (⦜), and U+299D⦝MEASURED RIGHT ANGLE WITH DOT (⦝).[5]
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.[6]
Euclid
Right angles are fundamental in Euclid's Elements. 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.[7] The straight lines which form right angles are called perpendicular.[8] 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).[9] Two angles are called complementary if their sum is a right angle.[10]
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.[11]
Conversion to other units
A right angle may be expressed in different units:
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(3, 4, 5) 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
Construction of the perpendicular to the half-line h from the point P (applicable not only at the end point A, M is freely selectable), animation at the end with pause 10 s
Alternative construction if P outside of the half-line h and the distance A to P' is small (B is freely selectable), animation at the end with pause 10 s
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
Wikimedia Commons has media related to Right angles.
Wentworth, G.A. (1895). A Text-Book of Geometry. Ginn & Co.
Euclid, commentary and trans. by T. L. HeathElements Vol. 1 (1908 Cambridge) Google Books
April 19, 2023
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For other uses see Right angle disambiguation In geometry and trigonometry a right angle is an angle of exactly 90 degrees or p displaystyle pi 2 radians 1 corresponding to a quarter turn 2 If a ray is placed so that its endpoint is on a line and the adjacent angles are equal then they are right angles 3 The term is a calque of Latin angulus rectus here rectus means upright referring to the vertical perpendicular to a horizontal base line A right angle is equal to 90 degrees A line segment AB drawn so that it forms right angles with a line CD 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 4 making the right angle basic to trigonometry Contents 1 Etymology 2 In elementary geometry 3 Symbols 4 Euclid 5 Conversion to other units 6 Rule of 3 4 5 7 Thales theorem 8 See also 9 ReferencesEtymology EditThe meaning of right in right angle possibly refers to the Latin adjective rectus erect straight upright perpendicular A Greek equivalent is orthos straight perpendicular see orthogonality In elementary geometry EditA 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 Edit Right triangle with the right angle shown via a small square Another option of diagrammatically indicating a right angle using an angle curve and a small dot In Unicode the symbol for a right angle is U 221F RIGHT ANGLE amp angrt It should not be confused with the similarly shaped symbol U 231E BOTTOM LEFT CORNER amp dlcorn amp llcorner Related symbols are U 22BE RIGHT ANGLE WITH ARC amp angrtvb U 299C RIGHT ANGLE VARIANT WITH SQUARE amp vangrt and U 299D MEASURED RIGHT ANGLE WITH DOT amp angrtvbd 5 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 6 Euclid EditRight angles are fundamental in Euclid s Elements 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 7 The straight lines which form right angles are called perpendicular 8 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 9 Two angles are called complementary if their sum is a right angle 10 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 11 Conversion to other units EditA right angle may be expressed in different units 1 4 turn 90 degrees p 2 radians 100 grad also called grade gradian or gon 8 points of a 32 point compass rose 6 hours astronomical hour angle Rule of 3 4 5 EditThroughout 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 3 4 5 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 Edit Construction of the perpendicular to the half line h from the point P applicable not only at the end point A M is freely selectable animation at the end with pause 10 s Alternative construction if P outside of the half line h and the distance A to P is small B is freely selectable animation at the end with pause 10 s Main article 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 Edit Wikimedia Commons has media related to Right angles Cartesian coordinate system Types of anglesReferences Edit Right Angle Math Open Reference Retrieved 26 April 2017 Wentworth p 11 Wentworth p 8 Wentworth p 40 Unicode 5 2 Character Code Charts Mathematical Operators Miscellaneous Mathematical Symbols B Muller Philipp Susanne Gorski Hans Joachim 2011 Leitfaden Geometrie Handbook Geometry in German Springer ISBN 9783834886163 Heath p 181 Heath p 181 Heath p 181 Wentworth p 9 Heath pp 200 201 for the paragraph Wentworth G A 1895 A Text Book of Geometry Ginn amp Co Euclid commentary and trans by T L Heath Elements Vol 1 1908 Cambridge Google Books Retrieved from https en wikipedia org w index php title Right angle amp oldid 1141839923, wikipedia, wiki, book, books, library,
