Angle

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The angle is the arc that is formed from the intersection of two rays, segments or straight lines, and can be measured in degrees (with the sexagesimal system) or in radians.

Another way to define the angle is as the region that is formed from the union of the intersection or union of two lines that share a common vertex or point.

The angles can then be formed from intersecting lines or rays.At this point, it is important to remember that the line is a one-dimensional element that is constituted by a succession of points that extends indefinitely, that is, it has no beginning or end. Likewise, the ray is the portion of the line that starts from a point on it and extends to infinity, that is, it has an origin, but not an end.

So, angles can be formed in a plane when we draw lines or rays, as we see below.

Angle formed by the crossing of two lines Angle formed by the union of two rays

On the other hand, angles are also formed by the union of segments that share a vertex. We must remember that a segment is a portion of a line that is bounded by two points, has an origin and an end.

The angles that are formed from segments can be observed in the polygons, as in the figure below where α, β and γ are the internal angles of the triangle.

It should also be clarified that an angle can be formed between two vectors that are segments of lines that follow a certain direction.

Types of angles

According to their measure, the angles can be:

  • Sharp: Measures less than 90º or π / 2 radians.

  • Blunt: Measures more than 90º or π / 2 radians and less than 180º or π radians.
  • Straight: It is equal to 90º or π / 2 radians.
  • Flat: Its measure is 180º or π radians.
  • Oblique or concave: It measures more than 180º or π radians and less than 360º or 2π radians (it should be noted that a convex angle is one that measures less than 180º).
  • Complete or perigonal: Measures exactly 360º or 2π radians

According to how they are located relative to each other, the angles can be:

  • Consecutive: They are one contiguous to the other. In the image below, α and β are consecutive angles.
  • Adjacent: They are part of the same line and add a straight angle, that is, they add 180º, as α and β in the following graph:
  • Opposed by the vertex: They share the same vertex and one is constituted by the extension of the sides that form the other angle. In the image below, α and δ are vertex opposites, as are β and γ.

Finally, according to the result of their summation, the angles can be:

  • Complementary: They add up to 90º.
  • Supplementary: Its sum is 180º.

In the image below, α and β are complementary. Meanwhile, δ and ε are supplementary.

Angle measurement

For the measurement of an angle you can use mainly two methods:

  • Sexagesimal system: It is one that, taking as a reference the angle observed on a flat surface (which we call the flat angle, as we already explained), is divided into 180 equal parts called degrees. Likewise, each degree subdivides into 60 minutes, and each minute into 60 seconds.
  • Radian system: The complete or perigonal angle, which represents a circumference, can be calculated by dividing the length of the arc (which is equal to 2πr as explained in the circumference article) by the radius of the figure:

α = L / r = 2πr / r = 2r

From this data, we deduce that the straight angle is π, for example, and that the right angle is π / 2.

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