Basic Concepts

As part of the issue of measuring angles, in this section we will consider several concepts related to initial geometric information:

• corner;
• unfolded and undeveloped angle;
• degree, minute and second;
• degree measure of angle;
• right, acute and obtuse angles.

An angle is a geometric figure that consists of a point (vertex) and two rays (sides) emanating from it. An angle is called developed if both rays lie on the same straight line.

Thanks to the degree measure of angle, angles can be measured. Measuring angles is carried out similarly to measuring segments. Just like when measuring segments, when measuring angles, a special unit of measurement is used. Most often it is a degree.

Definition 1

A degree is a unit of measurement. In geometry, it represents the angle to which other angles are compared. The degree is equal to $\frac(1)(180)$ from the straight angle.

Now we can define the degree measure of an angle.

Definition 2

The degree measure of an angle is a positive number that indicates how many times a degree is placed in a given angle.

A protractor is used to measure angles.

An example of writing a degree measure: $\angle ABC = 150^(\circ)$. In the figure, this entry means the following:

Orally they say this: “Angle ABC is 150 degrees.”

Some parts of the degree have their own special names. A minute is a $\frac(1)(60)$ part of a degree, denoted by the sign $"$. A second is a $\frac(1)(60)$ part of a minute, denoted by $""$. An example of writing an angle in 75 degrees, 45 minutes and 28 seconds: $75^(\circ)45"28""$.

Those angles whose degree measures are equal are called equal. Accordingly, angles can be compared by saying that one angle is less than another or one angle is greater than another.

The definition of a rotated angle was given above. Using the concept of a degree measure, we can describe the difference between a developed and non-developed angle. The reversed angle is always $180^(\circ)$. An undeveloped angle is any angle less than $180^(\circ)$.

There are right, acute and obtuse angles. A right angle is equal to $90^(\circ)$, an acute angle is less than $90^(\circ)$, an obtuse angle is more than $90^(\circ)$ and less than $180^(\circ)$.

Figure 4. Right, acute and obtuse angles. Author24 - online exchange of student works

In everyday life there are examples of the need and importance of the ability to measure angles and understand degrees. Measuring angles is necessary in various studies, including in astronomy when determining the position of celestial bodies.

For practice, try to draw at least three unraveled angles and one unfolded one in different ways, measure the angles using a protractor and write down these results. You can set random numbers and practice the accuracy of drawing angles using a protractor, dividing them using a bisector (a bisector is a ray emanating from the vertex of a given angle and dividing the angle in half).

Sample problems

Example 1

Task. There is a drawing:

The rays $DE$ and $DF$ are the bisectors of the corresponding angles $ADB$ and $BDC$. You need to find the angle $ADC$ if $\angle EDF = 75^(\circ)$.

Solution. Since angle $EDF$ contains half of each angle $ADB$ and $BDC$, we can conclude that $EDF$ is exactly half of angle $ADC$ itself. We get simple calculations: $\angle ADC=75\cdot 2=150^(\circ)$.

Answer: $150^(\circ)$.

Let's give another interesting example.

Example 2

Task. A drawing is given.

Angle $ABC$ is right. Angles $ABE$, $EBD$ and $DBC$ are equal. You need to find the angle formed by the bisectors $ABE$ and $DBC$.

Solution. Since $ABC$ is a right angle, it means it is equal to $90^(\circ)$. Angle $\angle EBD=90/3=30^(\circ)$. Since the angles $ABE$, $EBD$ and $DBC$ are equal, any of them will be equal to $30^(\circ)$. The bisector of any of these angles will divide any of these angles into two angles equal to $15^(\circ)$. Since the two halves of the angles $ABE$ and $DBC$ belong to the desired angle, we can say that the desired angle is equal to $30+15+15=60^(\circ)$.

Answer. $60^(\circ)$

In this article we have fully covered the issue of the degree measure of an angle and how to measure angles.

How to find the degree measure of an angle?

For many people at school, geometry is a real test. One of the basic geometric shapes is an angle. This concept means two rays that originate at the same point. To measure the value (magnitude) of an angle, degrees or radians are used. You will learn how to find the degree measure of an angle in our article.

Types of angles

Let's say we have an angle. If we expand it into a straight line, then its value will be equal to 180 degrees. Such an angle is called a turned angle, and 1/180 of its part is considered one degree.

In addition to a straight angle, there are also acute (less than 90 degrees), obtuse (more than 90 degrees) and right angles (equal to 90 degrees). These terms are used to characterize the degree measure of an angle.

Angle measurement

The angle is measured using a protractor. This is a special device on which the semicircle is already divided into 180 parts. Place the protractor on the corner so that one of the sides of the corner coincides with the bottom of the protractor. The second beam must intersect the arc of the protractor. If this does not happen, remove the protractor and use a ruler to lengthen the beam. If the angle “opens” to the right of the vertex, its value is read on the upper scale, if to the left - on the lower one.

In the SI system, it is customary to measure the magnitude of an angle in radians, rather than in degrees. Only 3.14 radians fit in the unfolded angle, so this value is inconvenient and is almost never used in practice. This is why you need to know how to convert radians to degrees. There is a formula for this:

• Degrees = radians/π x 180

For example, the angle is 1.6 radians. Convert to degrees: 1.6/3.14 * 180 = 92

Properties of corners

Now you know how to measure and recalculate degrees of angles. But to solve problems, you also need to know the properties of angles. To date, the following axioms have been formulated:

• Any angle can be expressed in degrees greater than zero. The size of the rotated angle is 360.
• If an angle consists of several angles, then its degree measure is equal to the sum of all angles.
• In a given half-plane, from any ray it is possible to construct an angle of a given value, less than 180 degrees, and only one.
• The values ​​of equal angles are the same.
• To add two angles, you need to add their values.

Understanding these rules and knowing how to measure angles is the key to successfully learning geometry.

An angle is a figure that consists of a point - the vertex of the angle and two different half-lines emanating from this point - the sides of the angle (Fig. 14). If the sides of an angle are complementary half-lines, then the angle is called a developed angle.

An angle is designated either by indicating its vertex, or by indicating its sides, or by indicating three points: the vertex and two points on the sides of the angle. The word "angle" is sometimes replaced

The Angle symbol in Figure 14 can be designated in three ways:

A ray c is said to pass between the sides of an angle if it comes from its vertex and intersects some segment with ends on the sides of the angle.

In Figure 15, ray c passes between the sides of the angle as it intersects the segment

In the case of a straight angle, any ray emanating from its vertex and different from its sides passes between the sides of the angle.

Angles are measured in degrees. If you take a straight angle and divide it into 180 equal angles, the degree measure of each of these angles is called a degree.

The basic properties of angle measurement are expressed in the following axiom:

Each angle has a certain degree measure greater than zero. The rotated angle is 180°. The degree measure of an angle is equal to the sum of the degree measures of the angles into which it is divided by any ray passing between its sides.

This means that if a ray c passes between the sides of an angle, then the angle is equal to the sum of the angles

The degree measure of an angle is found using a protractor.

An angle equal to 90° is called a right angle. An angle less than 90° is called an acute angle. An angle greater than 90° and less than 180° is called obtuse.

Let us formulate the main property of setting aside corners.

From any half-line, into a given half-plane, you can put an angle with a given degree measure less than 180°, and only one.

Consider the half-line a. Let us extend it beyond the starting point A. The resulting straight line divides the plane into two half-planes. Figure 16 shows how, using a protractor, to plot an angle with a given degree measure of 60° from a half-line to the upper half-plane.

T. 1. 2. If two angles from a given half-line are put into one half-plane, then the side of the smaller angle, different from the given half-line, passes between the sides of the larger angle.

Let be the angles laid off from a given half-line a into one half-plane, and let the angle be less than the angle . Theorem 1. 2 states that the ray passes between the sides of the angle (Fig. 17).

The bisector of an angle is the ray that emanates from its vertex, passes between the sides and divides the angle in half. In Figure 18, the ray is the bisector of the angle

In geometry there is the concept of a plane angle. A plane angle is a part of a plane bounded by two different rays emanating from one point. These rays are called sides of the angle. There are two plane angles with given sides. They are called additional. In Figure 19, one of the plane angles with sides a and is shaded.

Lecture: The magnitude of the angle, the degree measure of the angle, the correspondence between the magnitude of the angle and the length of the arc of a circle

Angle measure is the amount by which a ray deviates relative to its original position.

The measure of an angle can be measured in two quantities: degrees and radians, hence the name of the units - degree and radian measure of angle.

Degree measure of angle

The degree measure makes it possible to estimate how many degrees, minutes or seconds are placed in a particular angle.

Angles in degrees are calculated from the point of view that the full rotation of the beam is 360°. Half a turn of 180° is a straight angle, a quarter - 90° is a right angle, etc.

Radian measure of angle

Now let's figure out what the radian measure of an angle is. As we know from physics, there are additional units. For example, to measure temperature, the main unit is Kelvin, and the additional unit is Celsius. To measure length we use meters, but the British use feet. This list goes on and on. The point is for you to understand that, in addition to the degree measure of angle, there is a radian measure, which also has a right to exist.

To determine the radian measure of an angle, a circle is used. It is believed that the radian measure is the length of the arc of a circle described by the central angle.

Recall that a central angle is an angle whose vertex is in the center of the circle, and the rays rest on some arc.

So, an angle of 1 rad has a degree measure of 57.3°. The radian measure of an angle is described either by natural numbers or using the number π ≈ 3.14.

For geometry it is more convenient to use the degree measure of angle, but for trigonometry they use the radian measure.

Mathematics, geometry - these sciences, as well as most other exact sciences, are extremely difficult for many. People find it difficult to understand formulas and strange terminology. What is hidden under this strange concept?

Definition

To begin with, you need to consider simply the measure of the angle. The image of a ray and a straight line will help with this. First you need to draw, for example, a horizontal straight line. Then a ray is drawn from its first point, not parallel to the straight line. Thus, a certain distance, a small angle, appears between the straight line and the ray. The measure of an angle is the size of this very beam rotation.

This concept denotes a certain digital value that will be greater than zero. It is expressed in degrees, as well as its components, that is, minutes and seconds. The number of degrees that fits into the angle between the ray and the straight line will be the degree measure.

Properties of corners

• Absolutely each angle will have a certain degree measure.
• If it is fully deployed, the number will be 180 degrees.
• To find the degree measure, the sum of all angles broken by the beam is considered.
• Using any ray, you can create a half-plane in which you can actually make an angle. It will have a degree measure, the value of which will be less than 180, and there can only be one such angle.

How to find out the measure of an angle?

As a rule, the minimum degree measure is 1 degree, which is 1/180 of the rotated angle. However, sometimes you cannot get such a clear figure. In these cases, seconds and minutes are used.

Once found, the value can be converted to degrees, thus getting a fraction of a degree. Sometimes fractional numbers are used, like 80.7 degrees.

It is also important to remember key quantities. A right angle will always be 90 degrees. If the measure is greater, then it will be considered obtuse, and if less, then sharp.