ANGLE, strictly, the degree of inclination to one another of two lines which meet, or of two or more surfaces which meet. The same term is incorrectly used for the point in which lines meet, and for the line in which two sur faces meet, but such point or such line is the vertex of the angle. The lines in the first case and the surfaces in the second case are known as the sides of the angle.
An angle made by two lines is called a plane angle; that made by two planes, a dihedral angle; that made by three or more planes, a solid angle, or a polyhedral angle. Angles may again be subdivided into rectilinear, curvilinear and mixed angles. A plane rectilinear angle is the inclination to each other of two straight lines which meet but are not in the same straight line. A curvilinear angle is the in clination to each other of two curved lines which meet in a point, and is equal to the in clination of the tangents to the curves at their intersection. A mixed angle is one formed by the meeting of a curve and a straight line, and is measured as the angle between the straight line and the tangent to the curve at the point of intersection.
Plane rectilinear angles are generally divided into right and oblique, or into right, obtuse and acute. When a straight line standing upon another straight line makes the two adjacent angles (those on the right and left of it) equal to one another, each of them is called a right angle. An oblique angle is one which is not a right angle. An obtuse angle is that which is greater than one right angle, but less than two.
An acute angle is that which is less than a right angle; both are • oblique. A spherical angle is one formed by the intersection or the meeting of two great circles of a sphere.
The inclination of any two lines which form the sides of an angle is measured by the length of the arc they cut out on the circumference of a circle struck from the vertex of the angle as a centre. The radius of this circle is ob viously a matter of convenience, as the arcs will be the same for any radius. Angles are expressed, therefore, in degrees, minutes and seconds and decimals of a second: as, for ex ample, an angle of or an angle of 3° 14' 17.298". In mathematical nomenclature an angle upon a diagram is usually designated by three letters, one on each side and one at the vertex, the vertex letter being the middle one of the triad. as, the angle BCD, in which C is the letter at the vertex. In some cases where no ambiguity could arise, the same angle might be named by its vertex letter: as, the angle C.
The complement of an angle is that length of arc by which it falls short of 90°: that is, the complement of an angle of 32° is 58°. The supplement of an angle is that length of arc by which it falls short of 180°: that is, the supplement of is 105°.
In astronomy angles are considered between lines which never meet. These angles are measured as between parallels to the original lines, so drawn or projected that they do meet.