Angle
An angle is a geometric figure created by two line segments that extend from a single point or two planes which extend from a single line. The size of an angle, measured in units of degrees or radians, is related to the amount of rotation required to superimpose one of its sides on the other. First used by ancient civilizations, angles continue to be an important tool to science and industry today.
The study of angles has been known since the time of the ancient Babylonians (4,000-300 b.c.). These people used angles for measurement in many areas such as construction, commerce, and astronomy. The ancient Greeks developed the idea of an angle further and were even able to use them to calculate the circumference of the Earth and the distance to the moon.
A geometric angle is formed by two lines (rays) that intersect at a common endpoint called the vertex. The two rays are known as the sides of the angle. An angle can be specified in various ways. If the vertex of an angle is at point P, then the angle could be denoted byP. It can be further described by using a point from each ray. For example, the angle OPQ would have the point O on one ray, a vertex at point P, and the point Q on the remaining ray. An angle can also be denoted by a single number or character which is placed on it. The most common character used is the Greek letter (theta).
Units of measurement of an angle
An angle is commonly given an arithmetic value which describes its size. To specify its this value, an angle is drawn in a standard position on a coordinate system, with its vertex at the center and one side, called the initial side, along the x axis. The value of the angle then represents the amount of rotation needed to get from the initial side to the other side, called the terminal side. The direction of rotation indicates the sign of the angle. Traditionally, a counterclockwise rotation gives a positive value and a clockwise rotation gives a negative value. The three terms which are typically used to express the value of an angle include revolutions, degrees, or radians.
The revolution is the most natural unit of measurement for an angle. It is defined as the amount of rotation required to go from the initial side of the angle all the way around back to the initial side. One way to visualize a revolution is to imagine spinning a wheel around one time. The distance traveled by any point on the wheel is equal to one revolution. An angle can then be given a value based on the fraction of the distance a point travels divided by the distance traveled in one rotation. For example, an angle represented by a quarter turn of the wheel is equal to .25 rotations.
A more common unit of measurement for an angle is the degree. This unit was used by the Babylonians as early as 1,000 b.c. At that time, they used a number system based on the number 60, so it was natural for mathematicians of the day to divide the angles of an equilateral triangle into 60 individual units. These units became known as degrees. Since six equilateral triangles can be evenly arranged in a circle, the number of degrees in one revolution became 6 x 60 = 360. The unit of degrees was subdivided into 60 smaller units called minutes and in turn, these minutes were subdivided into 60 smaller units called seconds. Consequently, the notation for an angle which has a value of 44 degrees, 15 minutes, and 25 seconds would be 44° 15' 25''.
An angle may be measured with a protractor, which is a flat instrument in the shape of a semi-circle. There are marks on its outer edges which subdivide it into 180 evenly spaced units, or degrees. Measurements are taken by placing the midpoint of the flat edge over the vertex of the angle and lining the 0° mark up with the initial side. The number of degrees can be read off at the point where the terminal side intersects the curve of the protractor.
Another unit of angle measurement, used extensively in trigonometry, is the radian. This unit relates a unique angle to each real number. Consider a circle with its center at the origin of a graph and its radius along the x-axis. One radian is defined as the angle created by a counterclockwise rotation of the radius around the circle such that the length of the arc traveled is equal to the length of the radius. Using the formula for the circumference of a circle, it can be shown that the total number of radians in a complete revolution of 360° is 2. Given this relationship, it is possible to convert between a degree and a radian measurement.
Geometric characteristics of angles
An angle is typically classified into four categories including acute, right, obtuse, and straight. An acute angle is one which has a degree measurement greater than 0° but less than 90°. A right angle has a 90° angle measurement. An obtuse angle has a measurement greater than 90° but less than 180°, and a straight angle, which looks like a straight line, has a 180° angle measurement.
Two angles are known as congruent angles if they have the same measurement. If their sum is 90°, then they are said to be complementary angles. If their sum is 180°, they are supplementary angles. Angles can be bisected (divided in half) or trisected (divided in thirds) by rays protruding from the vertex.
When two lines intersect, they form four angles. The angles directly across from each other are known as vertical angles and are congruent. The neighboring angles are called adjacent because they share a common side. If the lines intersect such that each angle measures 90°, the lines are then considered perpendicular or orthogonal.
In addition to size, angles also have trigonometric values associated with them such as sine, cosine, and tangent. These values relate the size of an angle to a given length of its sides. These values are particularly important in areas such as navigation, astronomy, and architecture.
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