Right angle

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A right angle is equal to 90 degrees. Right angle.svg
A right angle is equal to 90 degrees.
A line segment (AB) drawn so that it forms right angles with a line (CD). Perpendicular-coloured.svg
A line segment (AB) drawn so that it forms right angles with a line (CD).

In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), [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.

Contents

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.

Etymology

The meaning of "right" in "right angle" possibly refers to the latin adjective rectus, which can be translated into erect, straight, upright or perpendicular. A Greek equivalent is orthos, which means straight or 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

Right triangle, with the right angle shown via a small square. Rtriangle.svg
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. Triangle 30-60-90 rotated.png
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+221FRIGHT ANGLE (HTML ∟ ·∟). It should not be confused with the similarly shaped symbol U+231EBOTTOM LEFT CORNER (HTML ⌞ ·⌞, ⌞). Related symbols are U+22BERIGHT ANGLE WITH ARC (HTML ⊾ ·⊾), U+299CRIGHT ANGLE VARIANT WITH SQUARE (HTML ⦜ ·⦜), and U+299DMEASURED RIGHT ANGLE WITH DOT (HTML ⦝ ·⦝). [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:

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 (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

01-Rechter Winkel mittels Thaleskreis.gif
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
01-Rechter Winkel mittels Thaleskreis-II.gif
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

Related Research Articles

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Euclidean geometry Mathematical system attributed to Euclid

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.

Pythagorean triple Three positive integers, the squares of two of which sum to the square of the third

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and 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 primitive Pythagorean triple is one in which a, b and c are coprime. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.

Triangle Shape with three sides

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted .

Right triangle

A right triangle or right-angled triangle is a triangle in which one angle is a right angle. The relation between the sides and angles of a right triangle is the basis for trigonometry.

Triangle inequality property of geometry, also used to generalize the notion of "distance" in metric spaces

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If x, y, and z are the lengths of the sides of the triangle, with no side being greater than z, then the triangle inequality states that

Perpendicular Relationship between two lines that meet at a right angle (90 degrees)

In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects.

Hyperbolic geometry Non-Euclidean geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

Hypotenuse Longest side of a right-angled triangle, the side opposite of the right angle

In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 and the other has a length of 4, then their squares add up to 25. The length of the hypotenuse is the square root of 25, that is, 5.

Thaless theorem Angle formed by a point on a circle and the 2 ends of a diameter is a right angle

In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, who is said to have offered an ox, probably to the god Apollo, as a sacrifice of thanksgiving for the discovery, but it is sometimes attributed to Pythagoras.

Semicircle geometric shape

In mathematics, a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180°. It has only one line of symmetry. In non-technical usage, the term "semicircle" is sometimes used to refer to a half-disk, which is a two-dimensional geometric shape that also includes the diameter segment from one end of the arc to the other as well as all the interior points.

Ultraparallel theorem

In hyperbolic geometry, two lines may intersect, be ultraparallel, or be limiting parallel.

Special right triangle right triangle with a feature making calculations on the triangle easier

A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.

Sum of angles of a triangle

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Transversal (geometry)

In geometry, a transversal is a line that passes through two lines in the same plane at two distinct points. Transversals play a role in establishing whether two other lines in the Euclidean plane are parallel. The intersections of a transversal with two lines create various types of pairs of angles: consecutive interior angles, corresponding angles, and alternate angles. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive interior angles are supplementary, corresponding angles are equal, and alternate angles are equal.

Pythagorean theorem Equation relating the side lengths of a right triangle

In mathematics, the Pythagorean theorem, also known as Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":

Parallel postulate Geometric axiom

In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

Playfairs axiom Modern formulation of Euclids parallel postulate

In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid :

In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.

Geometric mean theorem Relates the altitude on the hypotenuse in a right triangle and the 2 line segments created

The right triangle altitude theorem or geometric mean theorem is a result in elementary geometry that describes a relation between the lengths of the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that the geometric mean of the two segments equals the altitude.

In geometry, the inverse Pythagorean theorem is as follows:

References

  1. "Right Angle". Math Open Reference. Retrieved 26 April 2017.
  2. Wentworth p. 11
  3. Wentworth p. 8
  4. Wentworth p. 40
  5. Unicode 5.2 Character Code Charts Mathematical Operators, Miscellaneous Mathematical Symbols-B
  6. Müller-Philipp, Susanne; Gorski, Hans-Joachim (2011). Leitfaden Geometrie [Handbook Geometry] (in German). Springer. ISBN   9783834886163.
  7. Heath p. 181
  8. Heath p. 181
  9. Heath p. 181
  10. Wentworth p. 9
  11. Heath pp. 200-201 for the paragraph