Right Interior Exterior
83-422: Adjacent Vertical Complementary Supplementary Dihedral In geometry and trigonometry , a right angle is an angle of exactly 90 degrees or π {\displaystyle \pi } / 2 radians 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. The term
166-430: A − 1 ) ( b − 1 ) 2 . {\displaystyle {\tfrac {(a-1)(b-1)}{2}}.} The area (by Pick's theorem equal to one less than the interior lattice count plus half the boundary lattice count) equals a b 2 {\displaystyle {\tfrac {ab}{2}}} . The first occurrence of two primitive Pythagorean triples sharing
249-460: A > 0 {\displaystyle c>b>a>0} to a rational n m {\displaystyle {\tfrac {n}{m}}} is achieved by studying the two sums a + c {\displaystyle a+c} and b + c {\displaystyle b+c} . One of these sums will be a square that can be equated to ( m + n ) 2 {\displaystyle (m+n)^{2}} and
332-491: A ) {\displaystyle {\tfrac {(c+a)}{b}}={\tfrac {b}{(c-a)}}} . Since ( c + a ) b {\displaystyle {\tfrac {(c+a)}{b}}} is rational, we set it equal to m n {\displaystyle {\tfrac {m}{n}}} in lowest terms. Thus ( c − a ) b = n m {\displaystyle {\tfrac {(c-a)}{b}}={\tfrac {n}{m}}} , being
415-520: A + b using elementary algebra and verifying that the result equals c . Since every Pythagorean triple can be divided through by some integer k to obtain a primitive triple, every triple can be generated uniquely by using the formula with m and n to generate its primitive counterpart and then multiplying through by k as in the last equation. Choosing m and n from certain integer sequences gives interesting results. For example, if m and n are consecutive Pell numbers ,
498-454: A and b are coprime, at least one of them is odd. If we suppose that a is odd, then b is even and c is odd (if both a and b were odd, c would be even, and c would be a multiple of 4, while a + b would be congruent to 2 modulo 4 , as an odd square is congruent to 1 modulo 4). From a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} assume
581-419: A and b will differ by 1. Many formulas for generating triples with particular properties have been developed since the time of Euclid. That satisfaction of Euclid's formula by a, b, c is sufficient for the triangle to be Pythagorean is apparent from the fact that for positive integers m and n , m > n , the a , b , and c given by the formula are all positive integers, and from
664-825: A is even) from a unique pair m > n > 0 of coprime odd integers. In the presentation above, it is said that all Pythagorean triples are uniquely obtained from Euclid's formula "after the exchange of a and b , if a is even". Euclid's formula and the variant above can be merged as follows to avoid this exchange, leading to the following result. Every primitive Pythagorean triple can be uniquely written where m and n are positive coprime integers, and ε = 1 2 {\displaystyle \varepsilon ={\frac {1}{2}}} if m and n are both odd, and ε = 1 {\displaystyle \varepsilon =1} otherwise. Equivalently, ε = 1 2 {\displaystyle \varepsilon ={\frac {1}{2}}} if
747-397: A is odd. We obtain c 2 − a 2 = b 2 {\displaystyle c^{2}-a^{2}=b^{2}} and hence ( c − a ) ( c + a ) = b 2 {\displaystyle (c-a)(c+a)=b^{2}} . Then ( c + a ) b = b ( c −
830-435: A + c ) = n / m is also the tangent of half of the angle that is opposite the triangle side of length b . There is a correspondence between points on the unit circle with rational coordinates and primitive Pythagorean triples. At this point, Euclid's formulae can be derived either by methods of trigonometry or equivalently by using the stereographic projection . For the stereographic approach, suppose that P ′
913-413: A is odd, and ε = 1 {\displaystyle \varepsilon =1} if a is even. The properties of a primitive Pythagorean triple ( a , b , c ) with a < b < c (without specifying which of a or b is even and which is odd) include: In addition, special Pythagorean triples with certain additional properties can be guaranteed to exist: Euclid's formula for
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#1733084998767996-485: 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). The solid angle subtended by an octant of a sphere (the spherical triangle with three right angles) equals π /2 sr . Adjacent angles In Euclidean geometry , an angle
1079-543: A , b , c ) can be drawn within a 2D lattice with vertices at coordinates (0, 0) , ( a , 0) and (0, b ) . The count of lattice points lying strictly within the bounds of the triangle is given by ( a − 1 ) ( b − 1 ) − gcd ( a , b ) + 1 2 ; {\displaystyle {\tfrac {(a-1)(b-1)-\gcd {(a,b)}+1}{2}};} for primitive Pythagorean triples this interior lattice count is (
1162-443: A , b , c ) , a well-known example is (3, 4, 5) . If ( a , b , c ) is a Pythagorean triple, then so is ( ka , kb , kc ) for any positive integer k . A triangle whose side lengths are a Pythagorean triple is a right triangle and called a Pythagorean triangle . A primitive Pythagorean triple is one in which a , b and c are coprime (that is, they have no common divisor larger than 1). For example, (3, 4, 5)
1245-493: A Pythagorean triple can be understood in terms of the geometry of rational points on the unit circle ( Trautman 1998 ). In fact, a point in the Cartesian plane with coordinates ( x , y ) belongs to the unit circle if x + y = 1 . The point is rational if x and y are rational numbers , that is, if there are coprime integers a , b , c such that By multiplying both members by c , one can see that
1328-1347: A constant η equal to 1 inverse radian (1 rad ) in a fashion similar to the introduction of the constant ε 0 . With this change the formula for the angle subtended at the center of a circle, s = rθ , is modified to become s = ηrθ , and the Taylor series for the sine of an angle θ becomes: Sin θ = sin x = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ = η θ − ( η θ ) 3 3 ! + ( η θ ) 5 5 ! − ( η θ ) 7 7 ! + ⋯ , {\displaystyle \operatorname {Sin} \theta =\sin \ x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\eta \theta -{\frac {(\eta \theta )^{3}}{3!}}+{\frac {(\eta \theta )^{5}}{5!}}-{\frac {(\eta \theta )^{7}}{7!}}+\cdots ,} where x = η θ = θ / rad {\displaystyle x=\eta \theta =\theta /{\text{rad}}}
1411-412: A deviation from a straight line ; the second, angle as quantity, by Carpus of Antioch , who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship. In mathematical expressions , it is common to use Greek letters ( α , β , γ , θ , φ , . . . ) as variables denoting the size of some angle (the symbol π
1494-451: A full turn are not equivalent. To measure an angle θ , a circular arc centered at the vertex of the angle is drawn, e.g., with a pair of compasses . The ratio of the length s of the arc by the radius r of the circle is the number of radians in the angle: θ = s r r a d . {\displaystyle \theta ={\frac {s}{r}}\,\mathrm {rad} .} Conventionally, in mathematics and
1577-420: A north-west orientation corresponds to a bearing of 315°. For an angular unit, it is definitional that the angle addition postulate holds. Some quantities related to angles where the angle addition postulate does not hold include: Pythagorean triple A Pythagorean triple consists of three positive integers a , b , and c , such that a + b = c . Such a triple is commonly written (
1660-419: A point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent. In other contexts, such as identifying a point on a spiral curve or describing an object's cumulative rotation in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of
1743-404: A primitive triple when m and n are coprime. Every primitive triple arises (after the exchange of a and b , if a is even) from a unique pair of coprime numbers m , n , one of which is even. It follows that there are infinitely many primitive Pythagorean triples. This relationship of a , b and c to m and n from Euclid's formula is referenced throughout
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#17330849987671826-526: 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. A right angle may be expressed in different units: Throughout history, carpenters and masons have known
1909-535: A quick way to confirm if an angle is a true right angle. It is based on the Pythagorean triple (3, 4, 5) and the rule of 3-4-5. From the angle in question, running a straight line along one side exactly three units in length, and along the second side exactly four units in length, will create a hypotenuse (the longer line opposite the right angle that connects the two measured endpoints) of exactly five units in length. Thales' theorem states that an angle inscribed in
1992-573: A right angle in a triangle is the defining factor for right triangles , 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 ). 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
2075-463: 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
2158-610: A right angle. 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. 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
2241-414: A straight line, they are supplementary. Therefore, if we assume that the measure of angle A equals x , the measure of angle C would be 180° − x . Similarly, the measure of angle D would be 180° − x . Both angle C and angle D have measures equal to 180° − x and are congruent. Since angle B is supplementary to both angles C and D , either of these angle measures may be used to determine
2324-559: A triangle is a right triangle . 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 ( ⦝ ). In diagrams,
2407-502: A triangle is supplementary to the third because the sum of the internal angles of a triangle is a straight angle. The difference between an angle and a complete angle is termed the explement of the angle or conjugate of an angle. The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles of the same size are said to be equal congruent or equal in measure . In some contexts, such as identifying
2490-421: A two-dimensional Cartesian coordinate system , an angle is typically defined by its two sides, with its vertex at the origin. The initial side is on the positive x-axis , while the other side or terminal side is defined by the measure from the initial side in radians, degrees, or turns, with positive angles representing rotations toward the positive y-axis and negative angles representing rotations toward
2573-427: Is 1 / 256 of a turn. Plane angle may be defined as θ = s / r , where θ is the magnitude in radians of the subtended angle, s is circular arc length, and r is radius. One radian corresponds to the angle for which s = r , hence 1 radian = 1 m/m = 1. However, rad is only to be used to express angles, not to express ratios of lengths in general. A similar calculation using
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2656-611: Is irrational . Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322 , a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system. When searching for integer solutions, the equation a + b = c is a Diophantine equation . Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation. There are 16 primitive Pythagorean triples of numbers up to 100: Other small Pythagorean triples such as (6, 8, 10) are not listed because they are not primitive; for instance (6, 8, 10)
2739-484: Is "pedagogically unsatisfying". In 1993 the American Association of Physics Teachers Metric Committee specified that the radian should explicitly appear in quantities only when different numerical values would be obtained when other angle measures were used, such as in the quantities of angle measure (rad), angular speed (rad/s), angular acceleration (rad/s ), and torsional stiffness (N⋅m/rad), and not in
2822-403: 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
2905-490: Is a 1 to 1 mapping of rationals (in lowest terms) to primitive Pythagorean triples where n m {\displaystyle {\tfrac {n}{m}}} is in the interval ( 0 , 1 ) {\displaystyle (0,1)} and m + n {\displaystyle m+n} odd. The reverse mapping from a primitive triple ( a , b , c ) {\displaystyle (a,b,c)} where c > b >
2988-409: Is a multiple of (3, 4, 5). Each of these points (with their multiples) forms a radiating line in the scatter plot to the right. Additionally, these are the remaining primitive Pythagorean triples of numbers up to 300: Euclid's formula is a fundamental formula for generating Pythagorean triples given an arbitrary pair of integers m and n with m > n > 0 . The formula states that
3071-404: Is a point of the unit circle with x and y rational numbers. Then the point P ′ obtained by stereographic projection onto the x -axis has coordinates which is rational. In terms of algebraic geometry , the algebraic variety of rational points on the unit circle is birational to the affine line over the rational numbers. The unit circle is thus called a rational curve , and it
3154-434: Is a point on the x -axis with rational coordinates Then, it can be shown by basic algebra that the point P has coordinates This establishes that each rational point of the x -axis goes over to a rational point of the unit circle. The converse, that every rational point of the unit circle comes from such a point of the x -axis, follows by applying the inverse stereographic projection. Suppose that P ( x , y )
3237-407: Is a primitive Pythagorean triple whereas (6, 8, 10) is not. Every Pythagorean triple can be scaled to a unique primitive Pythagorean triple by dividing ( a , b , c ) by their greatest common divisor . Conversely, every Pythagorean triple can be obtained by multiplying the elements of a primitive Pythagorean triple by a positive integer (the same for the three elements). The name is derived from
3320-496: Is clear that the complete form is meant. Current SI can be considered relative to this framework as a natural unit system where the equation η = 1 is assumed to hold, or similarly, 1 rad = 1 . This radian convention allows the omission of η in mathematical formulas. It is frequently helpful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions or "sense" relative to some reference. In
3403-511: Is given in Maor (2007) and Sierpiński (2003). Another proof is given in Diophantine equation § Example of Pythagorean triples , as an instance of a general method that applies to every homogeneous Diophantine equation of degree two. Suppose the sides of a Pythagorean triangle have lengths m − n , 2 mn , and m + n , and suppose the angle between the leg of length m − n and
Right angle - Misplaced Pages Continue
3486-410: Is in the interior of angle AOC, then m ∠ A O C = m ∠ A O B + m ∠ B O C {\displaystyle m\angle \mathrm {AOC} =m\angle \mathrm {AOB} +m\angle \mathrm {BOC} } I.e., the measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC. Three special angle pairs involve
3569-511: Is independent of the size of the circle: if the length of the radius is changed, then the arc length changes in the same proportion, so the ratio s / r is unaltered. Throughout history, angles have been measured in various units . These are known as angular units , with the most contemporary units being the degree ( ° ), the radian (rad), and the gradian (grad), though many others have been used throughout history . Most units of angular measurement are defined such that one turn (i.e.,
3652-459: Is meant, and in these cases, no ambiguity arises. Otherwise, to avoid ambiguity, specific conventions may be adopted so that, for instance, ∠BAC always refers to the anticlockwise (positive) angle from B to C about A and ∠CAB the anticlockwise (positive) angle from C to B about A. There is some common terminology for angles, whose measure is always non-negative (see § Signed angles ): The names, intervals, and measuring units are shown in
3735-398: Is sometimes more convenient, as being more symmetric in m and n (same parity condition on m and n ). If m and n are two odd integers such that m > n , then are three integers that form a Pythagorean triple, which is primitive if and only if m and n are coprime. Conversely, every primitive Pythagorean triple arises (after the exchange of a and b , if
3818-462: Is the angle in radians. The capitalized function Sin is the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed, while sin is the traditional function on pure numbers which assumes its argument is a dimensionless number in radians. The capitalised symbol Sin {\displaystyle \operatorname {Sin} } can be denoted sin {\displaystyle \sin } if it
3901-412: Is the angle of the rays lying tangent to the respective curves at their point of intersection. The magnitude of an angle is called an angular measure or simply "angle". Angle of rotation is a measure conventionally defined as the ratio of a circular arc length to its radius , and may be a negative number . In the case of a geometric angle, the arc is centered at the vertex and delimited by
3984-400: 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 are also known as plane angles as they lie in the plane that contains the rays. Angles are also formed by the intersection of two planes; these are called dihedral angles . Two intersecting curves may also define an angle, which
4067-452: Is this fact which enables an explicit parameterization of the (rational number) points on it by means of rational functions. A 2D lattice is a regular array of isolated points where if any one point is chosen as the Cartesian origin (0, 0), then all the other points are at ( x , y ) where x and y range over all positive and negative integers. Any Pythagorean triangle with triple (
4150-404: Is typically determined by a normal vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie. In navigation , bearings or azimuth are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so
4233-447: Is typically not used for this purpose to avoid confusion with the constant denoted by that symbol ). Lower case Roman letters ( a , b , c , . . . ) are also used. In contexts where this is not confusing, an angle may be denoted by the upper case Roman letter denoting its vertex. See the figures in this article for examples. The three defining points may also identify angles in geometric figures. For example,
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#17330849987674316-563: The Proto-Indo-European root *ank- , meaning "to bend" or "bow". Euclid defines a plane angle as the inclination to each other, in a plane, of two lines that meet each other and do not lie straight with respect to each other. According to the Neoplatonic metaphysician Proclus , an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used by Eudemus of Rhodes , who regarded an angle as
4399-404: The Pythagorean theorem , stating that every right triangle has side lengths satisfying the formula a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} ; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance,
4482-678: The SI , the radian is treated as being equal to the dimensionless unit 1, thus being normally omitted. The angle expressed by another angular unit may then be obtained by multiplying the angle by a suitable conversion constant of the form k / 2 π , where k is the measure of a complete turn expressed in the chosen unit (for example, k = 360° for degrees or 400 grad for gradians ): θ = k 2 π ⋅ s r . {\displaystyle \theta ={\frac {k}{2\pi }}\cdot {\frac {s}{r}}.} The value of θ thus defined
4565-484: The area of a circle , π r . The other option is to introduce a dimensional constant. According to Quincey this approach is "logically rigorous" compared to SI, but requires "the modification of many familiar mathematical and physical equations". A dimensional constant for angle is "rather strange" and the difficulty of modifying equations to add the dimensional constant is likely to preclude widespread use. In particular, Quincey identifies Torrens' proposal to introduce
4648-446: The cotangent of its complement, and its secant equals the cosecant of its complement.) The prefix " co- " in the names of some trigonometric ratios refers to the word "complementary". If the two supplementary angles are adjacent (i.e., have a common vertex and share just one side), their non-shared sides form a straight line . Such angles are called a linear pair of angles . However, supplementary angles do not have to be on
4731-967: The hypotenuse of length m + n is denoted as β . Then tan β 2 = n m {\displaystyle \tan {\tfrac {\beta }{2}}={\tfrac {n}{m}}} and the full-angle trigonometric values are sin β = 2 m n m 2 + n 2 {\displaystyle \sin {\beta }={\tfrac {2mn}{m^{2}+n^{2}}}} , cos β = m 2 − n 2 m 2 + n 2 {\displaystyle \cos {\beta }={\tfrac {m^{2}-n^{2}}{m^{2}+n^{2}}}} , and tan β = 2 m n m 2 − n 2 {\displaystyle \tan {\beta }={\tfrac {2mn}{m^{2}-n^{2}}}} . The following variant of Euclid's formula
4814-628: The triangle with sides a = b = 1 {\displaystyle a=b=1} and c = 2 {\displaystyle c={\sqrt {2}}} is a right triangle, but ( 1 , 1 , 2 ) {\displaystyle (1,1,{\sqrt {2}})} is not a Pythagorean triple because the square root of 2 is not an integer or ratio of integers . Moreover, 1 {\displaystyle 1} and 2 {\displaystyle {\sqrt {2}}} do not have an integer common multiple because 2 {\displaystyle {\sqrt {2}}}
4897-468: The adjacent angles, the vertical angles are equal in measure. According to a historical note, when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as: When two adjacent angles form
4980-558: The angle subtended by the circumference of a circle at its centre) is equal to n units, for some whole number n . Two exceptions are the radian (and its decimal submultiples) and the diameter part. In the International System of Quantities , an angle is defined as a dimensionless quantity, and in particular, the radian unit is dimensionless. This convention impacts how angles are treated in dimensional analysis . The following table lists some units used to represent angles. It
5063-457: The angle with vertex A formed by the rays AB and AC (that is, the half-lines from point A through points B and C) is denoted ∠BAC or B A C ^ {\displaystyle {\widehat {\rm {BAC}}}} . Where there is no risk of confusion, the angle may sometimes be referred to by a single vertex alone (in this case, "angle A"). In other ways, an angle denoted as, say, ∠BAC might refer to any of four angles:
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#17330849987675146-494: The area of a circular sector θ = 2 A / r gives 1 radian as 1 m /m = 1. The key fact is that the radian is a dimensionless unit equal to 1 . In SI 2019, the SI radian is defined accordingly as 1 rad = 1 . It is a long-established practice in mathematics and across all areas of science to make use of rad = 1 . Giacomo Prando writes "the current state of affairs leads inevitably to ghostly appearances and disappearances of
5229-407: The clockwise angle from B to C about A, the anticlockwise angle from B to C about A, the clockwise angle from C to B about A, or the anticlockwise angle from C to B about A, where the direction in which the angle is measured determines its sign (see § Signed angles ). However, in many geometrical situations, it is evident from the context that the positive angle less than or equal to 180 degrees
5312-523: The denominator, this would imply a to be even despite defining it as odd. Thus one of m and n is odd and the other is even, and the numerators of the two fractions with denominator 2 mn are odd. Thus these fractions are fully reduced (an odd prime dividing this denominator divides one of m and n but not the other; thus it does not divide m ± n ). One may thus equate numerators with numerators and denominators with denominators, giving Euclid's formula A longer but more commonplace proof
5395-413: The fact that A proof of the necessity that a, b, c be expressed by Euclid's formula for any primitive Pythagorean triple is as follows. All such primitive triples can be written as ( a , b , c ) where a + b = c and a , b , c are coprime . Thus a , b , c are pairwise coprime (if a prime number divided two of them, it would be forced also to divide the third one). As
5478-468: 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
5561-450: The final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor). In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined in terms of an orientation , which
5644-412: The integers form a Pythagorean triple. For example, given generate the primitive triple (3,4,5): The triple generated by Euclid 's formula is primitive if and only if m and n are coprime and exactly one of them is even. When both m and n are odd, then a , b , and c will be even, and the triple will not be primitive; however, dividing a , b , and c by 2 will yield
5727-565: The measure of Angle B . Using the measure of either angle C or angle D , we find the measure of angle B to be 180° − (180° − x ) = 180° − 180° + x = x . Therefore, both angle A and angle B have measures equal to x and are equal in measure. A transversal is a line that intersects a pair of (often parallel) lines and is associated with exterior angles , interior angles , alternate exterior angles , alternate interior angles , corresponding angles , and consecutive interior angles . The angle addition postulate states that if B
5810-467: The negative y -axis. When Cartesian coordinates are represented by standard position , defined by the x -axis rightward and the y -axis upward, positive rotations are anticlockwise , and negative cycles are clockwise . In many contexts, an angle of − θ is effectively equivalent to an angle of "one full turn minus θ ". For example, an orientation represented as −45° is effectively equal to an orientation defined as 360° − 45° or 315°. Although
5893-418: The numerator of m 2 − n 2 2 m n {\displaystyle {\tfrac {m^{2}-n^{2}}{2mn}}} would be a multiple of 4 (because an odd square is congruent to 1 modulo 4), and the denominator 2 mn would not be a multiple of 4. Since 4 would be the minimum possible even factor in the numerator and 2 would be the maximum possible even factor in
5976-437: The quantities of torque (N⋅m) and angular momentum (kg⋅m /s). At least a dozen scientists between 1936 and 2022 have made proposals to treat the radian as a base unit of measurement for a base quantity (and dimension) of "plane angle". Quincey's review of proposals outlines two classes of proposal. The first option changes the unit of a radius to meters per radian, but this is incompatible with dimensional analysis for
6059-404: The radian in the dimensional analysis of physical equations". For example, an object hanging by a string from a pulley will rise or drop by y = rθ centimetres, where r is the magnitude of the radius of the pulley in centimetres and θ is the magnitude of the angle through which the pulley turns in radians. When multiplying r by θ , the unit radian does not appear in the product, nor does
6142-461: The rational points on the circle are in one-to-one correspondence with the primitive Pythagorean triples. The unit circle may also be defined by a parametric equation Euclid's formula for Pythagorean triples and the inverse relationship t = y / ( x + 1) mean that, except for (−1, 0) , a point ( x , y ) on the circle is rational if and only if the corresponding value of t is a rational number. Note that t = y / ( x + 1) = b / (
6225-489: The reciprocal of ( c + a ) b {\displaystyle {\tfrac {(c+a)}{b}}} . Then solving for c b {\displaystyle {\tfrac {c}{b}}} and a b {\displaystyle {\tfrac {a}{b}}} gives As m n {\displaystyle {\tfrac {m}{n}}} is fully reduced, m and n are coprime, and they cannot both be even. If they were both odd,
6308-539: The rest of this article. Despite generating all primitive triples, Euclid's formula does not produce all triples—for example, (9, 12, 15) cannot be generated using integer m and n . This can be remedied by inserting an additional parameter k to the formula. The following will generate all Pythagorean triples uniquely: where m , n , and k are positive integers with m > n , and with m and n coprime and not both odd. That these formulas generate Pythagorean triples can be verified by expanding
6391-487: The same area occurs with triangles with sides (20, 21, 29), (12, 35, 37) and common area 210 (sequence A093536 in the OEIS ). The first occurrence of two primitive Pythagorean triples sharing the same interior lattice count occurs with (18108, 252685, 253333), (28077, 162964, 165365) and interior lattice count 2287674594 (sequence A225760 in the OEIS ). Three primitive Pythagorean triples have been found sharing
6474-655: The same area: (4485, 5852, 7373) , (3059, 8580, 9109) , (1380, 19019, 19069) with area 13123110. As yet, no set of three primitive Pythagorean triples have been found sharing the same interior lattice count. By Euclid's formula all primitive Pythagorean triples can be generated from integers m {\displaystyle m} and n {\displaystyle n} with m > n > 0 {\displaystyle m>n>0} , m + n {\displaystyle m+n} odd and gcd ( m , n ) = 1 {\displaystyle \gcd(m,n)=1} . Hence there
6557-582: The same line and can be separated in space. For example, adjacent angles of a parallelogram are supplementary, and opposite angles of a cyclic quadrilateral (one whose vertices all fall on a single circle) are supplementary. If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary. The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs. In Euclidean geometry, any sum of two angles in
6640-541: 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. The word angle comes from the Latin word angulus , meaning "corner". Cognate words include the Greek ἀγκύλος ([ankylοs] Error: {{Lang}}: Non-latn text/Latn script subtag mismatch ( help ) ) meaning "crooked, curved" and the English word " ankle ". Both are connected with
6723-976: The summation of angles: The adjective complementary is from the Latin complementum , associated with the verb complere , "to fill up". An acute angle is "filled up" by its complement to form a right angle. The difference between an angle and a right angle is termed the complement of the angle. If angles A and B are complementary, the following relationships hold: sin 2 A + sin 2 B = 1 cos 2 A + cos 2 B = 1 tan A = cot B sec A = csc B {\displaystyle {\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\[3pt]&\tan A=\cot B&&\sec A=\csc B\end{aligned}}} (The tangent of an angle equals
6806-422: The table below: 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. The equality of vertically opposite angles is called the vertical angle theorem . Eudemus of Rhodes attributed the proof to Thales of Miletus . The proposition showed that since both of a pair of vertical angles are supplementary to both of
6889-489: The unit centimetre—because both factors are magnitudes (numbers). Similarly in the formula for the angular velocity of a rolling wheel, ω = v / r , radians appear in the units of ω but not on the right hand side. Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics". Oberhofer says that the typical advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention and contextual knowledge
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