ANGLE

ANGLE, Ἀγκύλος, in Geometry, the Aperture or mutual Inclination of two Lines, which meet, and form an Angle in their Point of Intersection. See LINE. Such is the Angle ABC, (Tab. Geometry, Fig. 91.) formed by the Lines AB, and AC, meeting in the Point A—The Lines AB and AC, are called the Legs of the Angle; and the Point of Intersection, the Vertex. See LEG and VERTEX. Angles are sometimes denoted by a single Letter affixed to the Vertex, or angular Point, as A; and sometimes by three Letters, that of the Vertex being in the middle, as BAC. The Measure of an Angle, whereby its Quantity is expressed, is an Arc, DE, described from its Vertex A, with any Radius at pleasure, between its Legs, AC and BC. See ARC. Hence Angles are distinguished by the Ratio of the Arches which they thus subtend, to the Circumference of the whole Circle. See CIRCLE and CIRCUMFERENCE.—And thus, an Angle is said to be of so many Degrees, as are the Degrees of the Arch DE, See DEGREE. Hence also, since similar Arches, AB and DE, fig. 87. have the same Ratio to their respective Circumferences; and the Circumferences contain each the same Number of Degrees; the Arches AB, and DE, which are the Measures of the two Angles ACB, and ADE, are equal; and therefore the Angles themselves are so too. Hence, again, as the Quantity of an Angle is estimated by the Ratio of the Arch, subtended by it, to the Periphery; it does not matter what Radius that Arch is described withal: But the Measures of equal Angles are always either equal Arches or similar ones; and contrarily. It follows, therefore, that the Quantity of the Angle remains still the same, though the Legs be either produced or diminished. And thus similar Angles, and in similar Figures, the Homologous or Corresponding Angles are also equal. See SIMILAR, FIGURE, &c.



To measure, or find the Quantity of an ANGLE.
1°. On Paper—Apply the Centre of a Protractor on the Vertex of the Angle O; (Tab. Surveying, fig. 29.) so as the Radius Op lie on one of the Legs: The Degree shown in the Arch, by the other Leg of the Angle, will give the Angle required. See PROTRACTOR.
To do the same with a Line of Chords, see CHORD.
2°. On the Ground—Place a surveying Instrument, E.g., a Semi-circle, fig. 16.
in such manner as that a Radius thereof CG may lie over one Leg of the Angle, and the Center C over the Vertex——The first is obtained by looking through the Sights F and G, towards a Mark fixed at the End of the Leg; and the latter, by letting fall a Plummet from the Centre of the Instrument.
Then, the movable Index HI being turned this way and that, till through its Sights, you discover a Mark placed at the extreme of the other Leg of the Angle: The Degree it cuts in the Limb of the Instrument, shows the Quantity of the Angle.
See SEMICIRCLE, To take an Angle with a Quadrant, Theodolite, Plain Table, Circumferentor, Compass, &c., see Quadrant, Theodolite, Plain Table, Circumferentor, Compass, &c.

To plot or lay down any given Angle; i.e., the Quantity of an Angle being known to describe it on paper, see PLOTTING and PROTACTING.
To bissect a given Angle, as HIK, fig. 92, from the Centre L, with any Radius at pleasure, describe an Arch LM.
From L and M, with an Aperture greater than LM, strike two Arches, mutually intersecting each other in N.
Then, drawing the right line IN, we have HIN=NIK.

ANGLES are of various Kinds, and Denominations—With regard to the Form of their Legs, they are divided into Rectilinear, Curvilinear, and Mixed.

Rectilinear, or right-lined ANGLE, is that whose Legs are both right Lines; as ABC (Tab. Geometry, fig. 91.) See RECTILINEAR.

Curvilinear ANGLE, is that whose Legs are both of them Curves. See CURVE and CURVILINEAR.

Mixed, or Mixtilinear ANGLE, is that, one of whose Sides is a right Line, and the other a Curve. See MIXED.

With regard to their Quantity, Angles are again divided into Right, Acute, Obtuse, and Oblique.

Right ANGLE, is that formed by a Line falling perpendicularly on another; or that which subtends an Arch of 90 Degrees—Such is the Angle KLM, Fig. 93. See PERPENDICULAR, &c.

The Measure of a right Angle, therefore, is a Quadrant of a Circle; and consequently all right Angles are equal to each other. See QUADRANT.

Acute ANGLE is that which is less than a right Angle, or than 90° as AEC, fig. 86. See ACUTE.

Obtuse ANGLE, is that greater than a right Angle, or whose Measure exceeds 90° as AED. See OBTUSE.

Oblique ANGLE, is a common Name both for Acute and Obtuse Angles. See OBLIQUE.

With regard to their Situation in respect of each other, Angles are divided into Contiguous, Adjacent, Vertical, Alternate, and Opposite.

Contiguous ANGLES, are such as have the same Vertex, and one Leg common to both. Such are FGH, and HGI, fig. 94. See CONTIGUOUS.

Adjacent ANGLE, is that made by producing one of the Legs of another Angle—Such is the Angle AEC, fig. 86, made by producing a Leg ED, of the Angle AED, to C. See ADJACENT.

Two adjacent Angles, α and β, or any other Number of Angles made on the same Point E, over the same right Line CD, are together equal to two right ones; and consequently, to 180°. And hence, one of two contiguous Angles being given, the other is likewise given: as being the Complement of the former to 180°. See COMPLEMENT.

Hence, also, to measure an inaccessible Angle in the Field; taking an adjacent accessible Angle, and subtracting the Quantity thereof from 180°, the Remainder is the Angle required.

Again, all the Angles α, β, γ, δ, etc., made around a given Point E, are equal to four right ones; and therefore all make 360°.

Vertical ANGLES, are those whose Legs are Continuations of each other. Such are the Angles α and β, fig. 86. See VERTICAL.

If a right Line AB, cut another, CD, in E, the vertical Angles α and γ, as also β and δ, are equal.—And hence, if it be required to measure in a Field, or any other Place, an inaccessible Angle, α; and the other vertical Angle, γ, be accessible: This latter may be taken in lieu of the former. See SURVEYING.

Alternate ANGLES. See ALTERNATE—Such are the Angles α and β; fig. 36.

The alternate Angles β and α, are equal. See OPPOSITE ANGLES.

OPPOSITE ANGLES. See OPPOSITE.—Such are α and β, and also γ and δ.

EXTERNAL ANGLES, are the angles of any right-lined Figure made without it, by producing all the Sides severally.

All the external angles of any Figure taken together, are equal to four right angles: And the external angle of a Triangle is equal to both the internal and opposite ones, as is demonstrated by Euclid, Lib. 1. Prop. 32.

Internal ANGLES, are the angles made by the Sides of any right-lined Figure within. The Sum of all the internal angles of any right-lined Figure, is equal to twice as many right angles as the Figure hath Sides, excepting four. This is easily demonstrated from Euclid, Prop. 32. Lib. 1. The external Angle is demonstrated to be equal to the internal opposite one; and the two internal opposite ones, are equal to two right ones.

Homologous ANGLES, are such Angles in two Figures, as retain the same Order from the first, in both Figures.
See FIGURE.

ANGLE