Angle

An angle is a geometric figure formed when two rays share a common endpoint. The point where they meet is called the vertex of the angle, and the two rays are called the sides of the angle. An angle expresses the degree of opening or closure between the two sides; it is not the same as the length of the sides.

Angles are measured in various units. The most common is the degree (°), where a full circle is divided into 360 degrees. Another unit used in advanced mathematics is the radian, where a full circle equals 2π radians. Therefore, 180 degrees equals π radians. Minutes and seconds of arc are also used (1° = 60′, 1′ = 60″), particularly in astronomy and land surveying.

Types of angles

Angles are classified by their measure. A right angle is 90 degrees. An acute angle is less than 90 degrees. An obtuse angle is greater than 90 degrees but less than 180 degrees. A straight angle is 180 degrees, while an angle greater than 180 degrees but less than 360 degrees is called a reflex angle. A full angle is 360 degrees.

Relationships between angles

Two angles whose measures sum to 90 degrees are called complementary; if their sum is 180 degrees, they are supplementary. When two straight lines intersect, the vertical angles (opposite angles) are equal. A line that crosses two parallel lines creates corresponding angles and interior angles that follow specific properties, which form the basis of many geometric proofs.

Angles and triangles

One of the fundamental theorems of Euclidean geometry is that the sum of the three angles in a triangle equals 180 degrees. The relationships between angles and the sides of a triangle are studied in trigonometry, which uses the functions sine, cosine, and tangent. If one angle and one side are known, the other sides can be calculated.

Applications

Angles are used in structural engineering, land surveying, navigation of ships and aircraft, and map projection. They are also important in physics, for example in measuring the angle of incidence and angle of reflection of light, and in measuring rotational motion.