These are a couple of angles with a common vertex and a common side. The other two sides comprise a continuation of one another and form a straight line. Thus, supplementary angles together form a flat angle. Therefore, the amount of supplementary angles is equal to 180 degrees.

Angle is the geometric figure formed by two rays (the sides of the angle), emerging from one point (called the vertex of the angle).

The plane containing the two sides of the angle is divided into two areas by the angle. Each of these areas, united with the sides of the angle is called a flat angle (or just an angle). One of the plane angles (usually the smaller of the two) is sometimes imprecisely called internal, and the other one is external. Points of a plane angle that don’t belong to its sides form the inner area of the flat angle.

According to the other definition, flat angle is the part of the plane, which is the union of all rays emanating from a given point (the vertex) and crossing some line lying in the plane (which is called a line subtending the plane angle).

Often, for the sake of brevity, angle is referred to as an angular measure, that is the number that defines the angle.

Besides the most common flat angles, more common objects can be also considered as the angles – figures formed by intersecting arches, half-planes, and other figures in Euclidean and other types of geometry in metric spaces of different dimensions.

The angles can have various names, it all depends on the angles’ degrees:

- Zero angle (0 degrees). Sides of a zero angle match and its interior area is an empty set.
- Acute angle (from 0 to 90 degrees, not including boundary values).
- Right angle (90 degrees). Sides of the right angle are perpendicular to each other.
- Obtuse angle (from 90 to 180 degrees, not including boundary values).
- Oblique angle (any angle that is not equal to 0, 90, 180 or 270 degrees).
- Deployed angle (180 degrees). Sides of the deployed angle are antiparallel and form a straight line (or rather, two straight rays directed in opposite directions).
- Convex angle (from 0 to 180 degrees inclusively).
- Nonconvex angle (from 180 to 360 degrees, not including the boundary values).
- Full angle (360 degrees).

The term flat angle is used as a synonym for the angle defined at the beginning of the article, to distinguish it from the term of the angle used in stereometry (including dihedral, trihedral or polyhedral angle).

The properties of flat angles are often considered as angle ratios (supplementary angles, additional angles, adjacent, vertical, etc.) in case where angles are in the same plane.

Vertical corners represent two angles that are formed by the intersection of two lines. These angles do not have common sides. In other words, the two angles are called vertical if the sides of one angle are extensions of the sides of the other angle. The main feature of the vertical angles is that they are equal.

Adjacent corners, in their turn, are two corners with a common vertex and one side. These angles do not intersect with their interior area lying in the same plane. The angle formed by the external (not common) sides of adjacent angles is equal to the sum of the values of the adjacent angles themselves.

Adjacent Corners: Special Cases

- If adjacent angles are equal, then their common side is bisector.
- Additional angles are two angles with a common vertex, one of the sides of which is common, while the remaining sides are making a right angle. The amount of additional angles is 90 degrees. Sine, tangent, and secant of the angle are respectively equal to the cosine, cotangent, and cosecant of an additional angle.
- Supplementary angles are two angles with a common vertex, one of the sides of which is common, while the remaining sides are collinear (not coincident). The amount of the supplementary angles is equal to 180 degrees.
- Conjugate angles are two angles with a common top and two sides, while their inner areas are different. The bonding of these corners is a whole plane, but as complementary angles together they form a full angle. The sum of the conjugate angles is 360 degrees.

These are two angles with a common vertex and a common side, while the other two sides of the angles lie on one line. The total of supplementary angles is 180 degrees. For example, for an angle of 115 degrees, an angle of 65 degrees will be an adjacent corner.

A couple of supplementary angles are angles with one common side, while the other sides are on the same line. At the intersection of two lines, four pairs of supplementary angles are obtained.

Properties of Supplementary Angles

- The sinuses of supplementary angles are equal.
- Cosine and tangent are also equal, however different with the signs (one of the cosine (or tangents) is positive «+», while the other one is negative «-»).

Sum of adjacent corners is 180 degrees.

The proof of the theorem:

Let’s assume that we have supplementary angles of A1B and A2B. The ray B passes between the sides A1 and A2 of the adjacent angle. Thus, the sum of angles of A1B and A2B is equal to the adjacent corner, which is equal to 180 degrees.

Deductions of the theorem:

- If the two angles are equal, then the angles that are adjacent to them, are also equal.
- If the angle is not deployed, it is not equal to 180 degrees.
- The angle, which is adjacent to the right angle (90 degrees), is also right angle.
- The angle that is adjacent to an acute angle (the value of which is less than 90 degrees), is obtuse (the value of which is greater than 90 degrees), and the angle that is adjacent to the obtuse angle is a sharp corner.

When there is a conversation about supplementary angles, usually vertical angles are mentioned as well. Vertical angles are two angles that are formed by the intersection of two straights, but are not adjacent.

Theorem of vertical angles:

At the intersection of two lines at one point, four angles are formed. Those of the angles that are not supplementary angles are vertical angles. Angles adjacent to them, which form a straight line, are supplementary angles. Each pair of vertical angles is adjacent to them, thus the vertical angles are equal.

Measuring Angles

The angles are measured:

- in radians;
- in degrees, minutes, seconds;
- in revolutions;
- in gradian.

Radians are the ratio of the length of the arc S to its radius R (systemic). Radian measurements are used in mathematical analysis (for example, a numeric argument of trigonometric functions and in determining the amounts (tabular and graphic) of back arcus functions). In plane geometry and mechanics when considering rotation at a point or axis, and other processes described by using trigonometric functions, - vibration waves, and so on.

The most common measurement of angles is degrees, minutes, seconds. Degree measurement is used in elementary geometry (angle measurement on drawings with a protractor), in surveying on the map, and on the ground (for measuring angles on the ground, a very precise instrument - wagon/theodolite is used).

Revolution is the ratio of the length of the arc S subtending an angle (i.e., fully inside the angle’s arc whose ends lie on the sides of the angle, and the center of curvature coincides with the apex angle) to the length L of the circumference containing the arc.

Gradian system of corners measurement has been proposed to use historically and now is almost never used because it has not supplanted a more common sexagesimal-degree measurement system.

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