Angle

If you have an angle and want to copy that angle somewhere else, you can use the angle transfer method.

What is an angle

Defining an angle is not a completely straightforward matter and there are different versions. But for our purposes, this version might suffice: An angle is a part of a plane bounded by two half-lines that have a common origin. The important thing about this definition is that the angle is not just the two arms, but the whole area, the whole plane, that the two arms enclose. See the following figure:

But the previous picture wasn't quite true because we didn't specify what angle we mean. At first glance, we might imagine the surface mentioned as the angle ABC, but we could just as well imagine the other part of the plane:

Therefore, when we write angles, we should distinguish whether we mean a convex angle (one that is less than $180^{\circ}$) or a nonconvex angle (one that is greater than $180^{\circ}$). A nonconvex angle can also be called a concave angle. This can be written using an oriented angle, for example.

An angle can have either positive or negative orientation. In positive orientation we proceed counterclockwise and in negative orientation clockwise. So, when writing the angle ABC, the semi-direct AB is mentioned first and the semi-direct BC is mentioned second. In the positive direction, we look at the semi-colon AB and proceed counterclockwise. In our example above we would get the first image, a convex angle. If we were to specify a negative direction, we would get a non-convex angle, the second image. By default, the positive direction is assumed, and in this article the notations will be written in the positive sense.

Thus an angle consists of the two arms that bound the angle, the vertex of the angle from which the bisectors emanate, and the area bounded by the arms of the angle. If we have an angle ABC, the arms of the angle are AB and BC and the vertex is B.

The size of the angle

Just as we can measure the length of a line segment, we can also measure the magnitude of an angle. This is measured either in the classical degree measure, which I'm sure you are familiar with, or in the arc measure. The degree measure is simple, $90^{\circ}$ is a right angle (for example, the interior angles of a square are each $90^{\circ}$), $180^{\circ}$ is a right angle (such as being bisected by opposite half-lines), and $360^{\circ}$ is a solid angle (bisected by half-lines that lie on top of each other - such an angle is made up of the whole surrounding plane). A degree is further divided into minutes (denoted by ′) and seconds (denoted by ″). One degree has 60 minutes and one minute has sixty seconds. $1^{\circ} = 60′$ and 1′ = 60″.

Pictures of each angle:

The arc measure is already a bit harder on the imagination because it counts radians, which have no absolute value. However, I have already described the radian size and other properties for goniometric functions, so I won't repeat myself.

Angles can be further divided by size into acute angles, which have less than $90^{\circ}$, and obtuse angles, which have between $90^{\circ}$ and $180^{\circ}$. These are a sort of subset of the convex angle I showed above. More than $180^{\circ}$ has a non-convex angle that does not subdivide it.

A pair of angles

Some specific pairs of angles have various properties that are good to know about.

• Vertex angles are those angles that have a common vertex and their arms form opposite semi-major lines. Vertex angles are always congruent, having the same magnitude.

• Side angles are those angles that have one arm in common and the other arms are opposite semi-major lines. The sum of the side angles is always equal to $180^{\circ}$, a right angle.

• Congruent angles are angles whose first arms are parallel and the other lies on the same line. It must also be true that the angles have the same orientation. Congruent angles are congruent.

There are other types of angles, but these are just positions that can be derived from the three locations described above.

Operations with angles

We can perform certain elementary operations on angles. The most basic is addition. Numerically, there's not much to go wrong with this; you normally add the sizes of the angles. If an angle is more than 360°, it is usually converted to a simpler form, somewhere in the interval $0^{\circ} - 360^{\circ}$. In short, it is as if you were working in the three-hundred-sixteenths number system. Similarly, if you have minutes or seconds in the example, you have to calculate in the hexadecimal system (like a normal clock).

So, for the sake of argument, one example. Add the following angles: $\alpha = 183^{\circ} 51′$ and $\beta = 222^{\circ} 24′$. First count the minutes $51′ + 24′ = 75′ = 1^{\circ} 15′$ (the procedure is simple, you divide 75 by sixty. The result gives the number of degrees and the remainder the number of minutes remaining). Now we count the degrees: $183^{\circ} + 222^{\circ} = 405^{\circ}$. To this partial result we add the previous result $405^{\circ} + 1^{\circ} 15′ = 406^{\circ} 15′$. We will now decrease this angle by 360 until it is between 0 and 360. We decrease because if we describe an angle of size $360^{\circ}$, we are back at square one, the angle of size $361^{\circ}$ will be the same size on paper as the angle of size $1^{\circ}$. We come to this calculation: $406^{\circ} 15′ = 46^{\circ} 15′$.

• Wikipedia