Developed, obtuse, vertical and non-developed: types of geometry angles. Concept and types of angles Three unfolded angles

Students become familiar with the concept of angle in primary school. But as a geometric figure that has certain properties, they begin to study it from the 7th grade in geometry. Seems, quite a simple figure, what can be said about her. But, acquiring new knowledge, schoolchildren increasingly understand that they can learn quite interesting facts about it.

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When studied

The school geometry course is divided into two sections: planimetry and stereometry. In each of them there is considerable attention is given to the corners:

• In planimetry, their basic concept is given and an introduction is made to their types by size. The properties of each type of triangle are studied in more detail. New definitions are emerging for students - these are geometric figures formed by the intersection of two straight lines with each other and the intersection of several straight lines with transversals.
• In stereometry, spatial angles are studied - dihedral and trihedral.

Attention! This article discusses all types and properties of angles in planimetry.

Definition and measurement

When starting to study, first determine what is an angle in planimetry.

If we take a certain point on the plane and draw two arbitrary rays from it, we obtain a geometric figure - an angle, consisting of the following elements:

• vertex - the point from which the rays were drawn, denoted by a capital letter of the Latin alphabet;
• the sides are half-straight lines drawn from the vertex.

All the elements that form the figure we are considering divide the plane into two parts:

• internal - in planimetry does not exceed 180 degrees;
• external.

The principle of measuring angles in planimetry explained on an intuitive basis. To begin with, students are introduced to the concept of a rotated angle.

Important! An angle is said to be developed if the half-lines emerging from its vertex form a straight line. The undeveloped angle is all other cases.

If it is divided into 180 equal parts, then it is customary to consider the measure of one part to be equal to 10. In this case, they say that the measurement is made in degrees, and the degree measure of such a figure is 180 degrees.

Main types

Types of angles are divided according to criteria such as degrees, the nature of their formation, and the categories presented below.

By size

Taking into account the magnitude, angles are divided into:

• expanded;
• straight;
• blunt;
• spicy.

Which angle is called unfolded was presented above. Let's define the concept of direct.

It can be obtained by dividing the expanded into two equal parts. In this case, it is easy to answer the question: how many degrees is a right angle?

Divide 180 degrees of unfolded by 2 and we get that a right angle is 90 degrees. This is a wonderful figure, since many facts in geometry are connected with it.

It also has its own characteristics in the designation. To show a right angle in the figure, it is denoted not by an arc, but by a square.

Angles that are obtained by dividing a straight line by an arbitrary ray are called acute. Logically, it follows that an acute angle is less than a right angle, but its measure is different from 0 degrees. That is, it has a value from 0 to 90 degrees.

An obtuse angle is larger than a right angle, but smaller than a straight angle. Its degree measure varies from 90 to 180 degrees.

This element can be divided into different types of the figures in question, excluding the unfolded one.

Regardless of how a non-rotated angle is divided, the basic axiom of planimetry is always used - “the basic property of measurement.”

At dividing an angle with one beam or several, the degree measure of a given figure is equal to the sum of the measures of the angles into which it is divided.

At the 7th grade level, the types of angles according to their size end there. But to increase erudition, we can add that there are other varieties that have a degree measure greater than 180 degrees. They are called convex.

Figures at the intersection of lines

The next types of angles that students are introduced to are elements formed by the intersection of two straight lines. Figures that are placed opposite each other are called vertical. Their distinctive feature is that they are equal.

Elements that are adjacent to the same line are called adjacent. The theorem reflecting their property says that adjacent angles add up to 180 degrees.

Elements in a triangle

If we consider a figure as an element in a triangle, then the angles are divided into internal and external. A triangle is bounded by three segments and consists of three vertices. The angles located inside the triangle at each vertex are called internal.

If we take any internal element at any vertex and extend any side, then the angle that is formed and is adjacent to the internal one is called external. This pair of elements has the following property: their sum is equal to 180 degrees.

Intersection of two straight lines

Intersection of lines

When two straight lines intersect with a transversal, angles are also formed., which are usually distributed in pairs. Each pair of elements has its own name. It looks like this:

• internal crosswise lying: ∟4 and ∟6, ∟3 and ∟5;
• internal one-sided: ∟4 and ∟5, ∟3 and ∟6;
• corresponding: ∟1 and ∟5, ∟2 and ∟6, ∟4 and ∟8, ∟3 and ∟7.

In the case when a secant intersects two lines, all these pairs of angles have certain properties:

1. Internal crosswise lying and corresponding figures are equal to each other.
2. Internal one-way elements add up to 180 degrees.

We study angles in geometry, their properties

Types of angles in mathematics

Conclusion

This article presents all the main types of angles that are found in planimetry and are studied in the seventh grade. In all subsequent courses, the properties relating to all the elements considered are the basis for further study of geometry. For example, when studying, you will need to remember all the properties of the angles formed when two parallel lines intersect with a transversal. When studying the features of triangles, it is necessary to remember what adjacent angles are. Moving to stereometry, all volumetric figures will be studied and constructed based on planimetric figures.

Let's start by defining what an angle is. Firstly, it is Secondly, it is formed by two rays, which are called the sides of the angle. Thirdly, the latter emerge from one point, which is called the vertex of the angle. Based on these features, we can create a definition: an angle is a geometric figure that consists of two rays (sides) emerging from one point (vertex).

They are classified by degree value, by location relative to each other and relative to the circle. Let's start with the types of angles according to their magnitude.

There are several varieties of them. Let's take a closer look at each type.

There are only four main types of angles - straight, obtuse, acute and straight angles.

Straight

It looks like this:

Its degree measure is always 90 o, in other words, a right angle is an angle of 90 degrees. Only such quadrilaterals as square and rectangle have them.

Blunt

It looks like this:

The degree measure is always more than 90 o, but less than 180 o. It can be found in quadrilaterals such as a rhombus, an arbitrary parallelogram, and in polygons.

Spicy

It looks like this:

The degree measure of an acute angle is always less than 90°. It is found in all quadrilaterals except the square and any parallelogram.

Expanded

The unfolded angle looks like this:

It does not occur in polygons, but is no less important than all the others. A straight angle is a geometric figure whose degree measure is always 180º. You can build on it by drawing one or more rays from its top in any direction.

There are several other minor types of angles. They are not studied in schools, but it is necessary to at least know about their existence. There are only five secondary types of angles:

1. Zero

It looks like this:

The name of the angle itself already indicates its size. Its internal area is 0°, and the sides lie on top of each other as shown in the figure.

2. Oblique

An oblique angle can be a straight angle, an obtuse angle, an acute angle, or a straight angle. Its main condition is that it should not be equal to 0 o, 90 o, 180 o, 270 o.

3. Convex

Convex angles are zero, straight, obtuse, acute and straight angles. As you already understood, the degree measure of a convex angle is from 0° to 180°.

4. Non-convex

Angles with degree measures from 181° to 359° inclusive are non-convex.

5. Full

A complete angle is 360 degrees.

These are all types of angles according to their magnitude. Now let's look at their types according to their location on the plane relative to each other.

1. Additional

These are two acute angles forming one straight line, i.e. their sum is 90 o.

2. Adjacent

Adjacent angles are formed if a ray is passed through the unfolded angle, or rather through its vertex, in any direction. Their sum is 180 o.

3. Vertical

Vertical angles are formed when two straight lines intersect. Their degree measures are equal.

Now let's move on to the types of angles located relative to the circle. There are only two of them: central and inscribed.

1. Central

A central angle is an angle with its vertex at the center of the circle. Its degree measure is equal to the degree measure of the smaller arc subtended by the sides.

2. Inscribed

An inscribed angle is an angle whose vertex lies on a circle and whose sides intersect it. Its degree measure is equal to half the arc on which it rests.

That's it for the angles. Now you know that in addition to the most famous ones - acute, obtuse, straight and deployed - there are many other types of them in geometry.

This article will discuss one of the basic geometric shapes - an angle. After a general introduction to this concept, we will focus on a specific type of such a figure. Straight angle is an important concept in geometry, which will be the main topic of this article.

Introduction to Geometric Angle

In geometry there are a number of objects that form the basis of all science. The angle refers to them and is defined using the concept of a ray, so let's start with it.

Also, before you begin to determine the angle itself, you need to remember several equally important objects in geometry - this is a point, a straight line on a plane, and the plane itself. A straight line is the simplest geometric figure that has neither beginning nor end. A plane is a surface that has two dimensions. Well, a ray (or half-line) in geometry is a part of a line that has a beginning, but no end.

Using these concepts, we can make a statement that an angle is a geometric figure that lies entirely in a certain plane and consists of two divergent rays with a common origin. Such rays are called sides of an angle, and the common beginning of the sides is its vertex.

Types of angles and geometry

We know that angles can be completely different. Therefore, a little below will be a small classification that will help you better understand the types of angles and their main features. So, there are several types of angles in geometry:

1. Right angle. It is characterized by a value of 90 degrees, which means that its sides are always perpendicular to each other.
2. Sharp corner. These angles include all their representatives that are less than 90 degrees in size.
3. Obtuse angle. Here there can be all angles ranging from 90 to 180 degrees.
4. Unfolded corner. It has a size of strictly 180 degrees and externally its sides form one straight line.

The concept of a straight angle

Now let's look at the rotated angle in more detail. This is the case when both sides lie on the same straight line, which can be clearly seen in the figure a little lower. This means that we can say with confidence that in a reversed angle, one of its sides is essentially a continuation of the other.

It is worth remembering the fact that such an angle can always be divided using a ray that emerges from its apex. As a result, we get two angles, which in geometry are called adjacent.

Also, the unfolded angle has several features. In order to talk about the first of them, you need to remember the concept of “angle bisector”. Recall that this is a ray that divides any angle exactly in half. As for the unfolded angle, its bisector divides it in such a way that two right angles of 90 degrees are formed. This is very easy to calculate mathematically: 180˚ (degree of the rotated angle): 2 = 90˚.

If we divide a rotated angle with a completely arbitrary ray, then as a result we always get two angles, one of which will be acute and the other obtuse.

Properties of rotated corners

It will be convenient to consider this angle, bringing together all its main properties, which is what we did in this list:

1. The sides of the rotated angle are antiparallel and form a straight line.
2. The rotated angle is always 180˚.
3. Two adjacent angles together always form a straight angle.
4. A full angle, which is 360˚, consists of two unfolded ones and is equal to their sum.
5. Half of a straight angle is a right angle.

So, knowing all these characteristics of this type of angles, we can use them to solve a number of geometric problems.

Problems with rotated angles

To see if you have grasped the concept of a straight angle, try answering the following few questions.

1. What is the magnitude of a straight angle if its sides form a vertical line?
2. Will two angles be adjacent if the first is 72˚ and the other is 118˚?
3. If a complete angle consists of two reverse angles, then how many right angles does it have?
4. A straight angle is divided by a ray into two angles such that their degree measures are in the ratio 1:4. Calculate the resulting angles.

Solutions and answers:

1. No matter how the rotated angle is located, it is always, by definition, equal to 180˚.
2. Adjacent angles have one side in common. Therefore, to calculate the size of the angle they make together, you just need to add the value of their degree measures. This means 72 +118 = 190. But by definition, a reversed angle is 180˚, which means that two given angles cannot be adjacent.
3. A straight angle contains two right angles. And since the complete one has two unfolded ones, it means there will be 4 straight lines.
4. If we call the desired angles a and b, then let x be the coefficient of proportionality for them, which means that a=x, and accordingly b=4x. The rotated angle in degrees is 180˚. And according to its properties that the degree measure of an angle is always equal to the sum of the degree measures of those angles into which it is divided by any arbitrary ray that passes between its sides, we can conclude that x + 4x = 180˚, which means 5x = 180˚ . From here we find: x = a = 36˚ and b = 4x = 144˚. Answer: 36˚ and 144˚.

If you were able to answer all these questions without prompts and without peeking at the answers, then you are ready to move on to the next geometry lesson.

An angle is a geometric figure that consists of two different rays emanating from one point. In this case, these rays are called sides of the angle. The point that is the beginning of the rays is called the vertex of the angle. In the picture you can see the angle with the vertex at the point ABOUT, and the parties k And m.

Points A and C are marked on the sides of the angle. This angle can be designated as angle AOC. In the middle there must be the name of the point at which the vertex of the angle is located. There are also other designations, angle O or angle km. In geometry, instead of the word angle, a special symbol is often written.

Developed and non-expanded angle

If both sides of an angle lie on the same straight line, then such an angle is called expanded angle. That is, one side of the angle is a continuation of the other side of the angle. The figure below shows the expanded angle O.

It should be noted that any angle divides the plane into two parts. If the angle is not unfolded, then one of the parts is called the internal region of the angle, and the other is called the external region of this angle. The figure below shows an undeveloped angle and marks the outer and inner regions of this angle.

In the case of a developed angle, either of the two parts into which it divides the plane can be considered the outer region of the angle. We can talk about the position of a point relative to an angle. A point can lie outside the corner (in the outer region), can be located on one of its sides, or can lie inside the corner (in the inner region).

In the figure below, point A lies outside angle O, point B lies on one side of the angle, and point C lies inside the angle.

Measuring angles

To measure angles there is a device called a protractor. The unit of angle is degree. It should be noted that each angle has a certain degree measure, which is greater than zero.

Depending on the degree measure, angles are divided into several groups.