Angles which are with the pattern. What are the angles? Verbal counting. Math warm-up
“The little son came to his father, and asked the Little One: “What are the corners?”. But father forgot the answer. This is very bad!".
In our article, we propose to recall the lessons of mathematics and find answers to the questions of the Baby.
What is an angle
What is an angle is of course easier to show than to explain. From elementary grades, we know that a flat angle:
- This is a geometric figure.
- It is formed by two sides - rays.
- The rays come out from one vertex - a point.
- Measured in degrees.
That is, if you put a point on any plane, and then draw two rays from this point (a ray is a straight line that has a beginning but no end), then we get an angle, and not one, but two. This is because the rays divided the plane into two parts. We have formed two corners - internal and external.
Angle designation
An angle is denoted in mathematics by such a sign - "˪" and Greek letters: β, δ, φ. You can also designate angles in small or large Latin letters. Lowercase (d, c, b) denote the rays forming an angle, therefore, the name will consist of two letters and the icon - ˪ab. Capital Latin letters denote the three points of an angle: two on the sides and one vertex (˪DEF). Moreover, the letter of the top will always be in the middle of the name, and how to read DEF or FED, this already makes no difference.
Types of corners
Depending on the degrees (measured value), the angles are divided into:
- Acute (> 90 degrees);
- Direct (exactly 90);
- Dull (180);
- Expanded (equal to 180);
- Non-convex (more than 180, but less than 360);
- Full (360);
All angles that are not right or straight are called oblique.
Also, what are the angles?
- Adjacent - they have one side in common, while the others lie, not coinciding, on the same plane. The sum of these angles will always be 180.
- Vertical - angles formed by two intersecting straight lines and they do not have common sides, but their rays come out from one point. That is, the side of one corner is a continuation of the other. These angles are equal.
- Central - An angle whose vertex is the center of the circle.
- Inscribed angle. Its vertex is on a circle, and the rays forming it intersect this circle.
Now you know what a right angle is, and you can also tell which angle is acute. Remembering this is not difficult, and other types of angles also have characteristic names.
When two beam (AO And OB) come from one point, then the figure formed by these rays (together with the part of the plane bounded by them) is called corner .
Rays forming an angle are called parties. The point from which they originate - summit angle.
Side angle should be thought of as infinitely extended from the top.
Corner usually denoted by three letters, of which the middle is placed at peaks, and extreme at some points of the sides. For example, say "angle AOB or angle SAI". But it is possible to designate an angle with one letter placed at the vertex, if there are no other angles at this vertex. We will sometimes denote an angle by a number placed inside the angle at the vertex. The word "angle" in writing is often replaced by the sign / .
When two beams come from one points, then we strictly say that they form not one angle, but two angles.
These two angles are equal to each other only if the rays AO And OB constitute one direct .
Such an angle is called angle.
Two corners count equal angles if, when superimposed, they can be combined.
We take it for granted that inside any angle from its vertex it is possible to draw a ray (and only one) that divides this angle in half. Such a beam is called angle bisector .
two corners ( AOB And BOC) are called related, if they have one side in common, and the other two sides are straight line.
Damn 1. Damn.2
When two adjacent corner are equal (Fig. 2), then their common side OB called perpendicular to a straight line AC on which the other sides lie.
If adjacent angles are unequal (Fig. 1), then the common side OB called oblique To AC.
In both cases, the point O called basis(perpendicular or oblique).
From any point of the straight line, on either side of this straight line, one can restore to it perpendicular and only one .
Each of the equal adjacent angles is called direct. The right angle is permanent a value equal to 90 0 (it is usually denoted by the sign d, i.e. the initial letter of the French word "droit" - straight). As a result, ordinary angles are compared in magnitude with a right angle.
Any deployed angle is 2 d= 180°.
Every corner ( AOC), smaller than the right angle ( AOB) is called sharp.
Every corner ( AOD) the larger direct is called stupid.
Sections: Primary School
Class: 4
Lesson Objectives:
- Acquaintance with the concepts of "detailed angle", "adjacent angles". Clarification of the concept of "acute" and "obtuse" angle.
- Practicing solving problems for percentage content.
- The development of mental operations.
- Formation of a holistic view of the world.
Equipment : clock faces, fans, pencils, angle sets, textbooks "Mathematics", grade 4, Peterson G., explanatory dictionary of the Russian language.
During the classes.
1. Organizational moment. Motivation.
The teacher begins the lesson with a poetic appeal to the children:
Well check it out buddy
Are you ready to start the lesson?
Everything is in place, everything is in order,
Pen, book and notebook?
Is everyone seated correctly?
Is everyone watching closely?
Everyone wants to receive
Only a rating of "5".
Here are the ideas and tasks,
Games, jokes, everything for you!
We wish you good luck -
To work, have a good time!
So let's start the math lesson. And mathematics is gymnastics for the mind. Why do you think this expression came about? Why do you think you should study mathematics?
2. Checking homework.
The teacher addresses the children.
- Guys, at home you had to try to solve a logical problem. Which of you completed the task? Tell me, will the mouse overtake the cat? (No. A cat needs to run 70 unit segments to a mink, and a mouse only 20. A cat moves at a speed of 10 unit segments per unit of time, and a mouse - 3 unit segments per unit of time. A cat will need 7 units of time to reach a mink, and a mouse more than 6 , but less than 7. Therefore, the cat will not catch up with the mouse).
- To check task number 14, use the standard card. Who doesn't have a single mistake in this task? Well done!
- What should have been done in task number 8 (Compare angles. Write down the name of the famous ruler of Ancient Egypt, for whom the largest pyramid was built.)
What angles are shown in the picture? (2 sharp, 1 straight, 2 blunt).
For which ruler was the largest pyramid built in Egypt? (Pharaoh Cheops).
- Who will remember the most important discovery of the Ancient Egyptians, which we still use today? (Calendar.)
3. Oral account. Math workout.
– Do you want to know which city was the capital of Ancient Egypt in the third millennium BC?
– Complete task number 8, page 7.
– Work in pairs by doing the calculations of 2 algorithms. You can work on the options individually by performing the calculations of 1 algorithm.
- List the responses you received. Enter the required letters. Got the name of the city
4. Goal setting. Formulation of the problem.
Who can say that about himself?
The top serves me as my head,
What do you think of as legs?
Everyone is called parties.
Enlarge my sides whenever you please
You can quite freely
After all, I'm on the plane.
When straight lines meet
We will always be between them. (Corner)
So, who can guess what the topic of our lesson is? (Corner.)
- What is an angle? Two rays coming out of the same point are vertices.
We are already familiar with the concept of angle.
- Look at the drawing. How many angles do you see? (Students assume there are 4).
– Do you want to find the answer? To do this, you need to discover new knowledge. Who is ready?
- I propose to answer the following questions in the lesson:
- What is a folded corner?
- What angles are called adjacent?
Does anyone know the answer to these questions already?
- What are the objectives of the lesson?(Students formulate tasks for the lesson).
- Answer questions by observing and draw conclusions.
- Learn to find new types of angles.
5. Problem solving.
6. Physical Minute.
We walk, we walk
We raise our hands higher
We don't lower our heads
We breathe evenly, deeply.
Suddenly we see from the bush,
The chick fell out of the nest.
Quietly we take a chick
And put it back in the nest.
Ahead from behind a bush
Looks sly fox.
We will outwit the fox
Let's run on toes.
We go to the meadow
We find many berries there.
Strawberries are so fragrant
That we are not too lazy to bend over.
7. Primary fastening.
We will learn to apply our knowledge.
1st task.
- What angle does the hour and minute hands form on the clock face at 6 o'clock, 14 o'clock, 15 hours 25 minutes, 22 hours 15 minutes. (Textbook assistants after the answers of the students show the dial).
2nd task.
Now work in groups. Together build from sticks or pencils one angle model: acute, obtuse, straight, deployed. Complete the model of each corner so that adjacent corners are obtained. (Students build models of angles).
- Count how many pencils you need for this?
3rd task. Practical work.
- Guys, I suggest you work in pairs. Open the textbook on page 6, read task number 3 (a). Do it together. Then the first option will complete task number 3 (b), and the second option will perform task number 3 (c). Discuss with each other the result obtained and be prepared to answer questions on this assignment.
4th task. Practical work. Individual implementation followed by discussion and frontal verification.
The teacher gives the students the following task.
Take the envelope with task number 4. It contains models of five different angles. Find a pair of corners that are adjacent. Build a new model out of them. Write your answers on a card. Get ready to verbally justify your opinion.
The teacher checks the correctness of the assignment.
What difficulties did you experience while completing the task? Assess the difficulty of tasks using the +, + /–, – icons.
8. Repetition. Solving percentage problems.
The teacher addresses the class:
- Take card number 5. Read the condition of the problem carefully. Choose the right solution. Discuss in groups the correctness of the solution. Justify your answer.
- What was the problem?
9. The result of the lesson.
Guys, this is the end of our lesson. You did a good job today. I am very satisfied with you. What new did you learn? What have you learned? What task did you find the most difficult? What would you like to tell your friends or parents? What else would you like to know about this topic?
10. Homework.
- Guys, at home you can once again test your knowledge on this topic by completing task number 7 on page 7.
- And for the savvy and everyone, I suggest that you additionally complete task No. 15 or No. 16 of your choice on page 8.
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 come out of one point, which is called the apex of the corner. Based on these signs, we can make a definition: an angle is a geometric figure that consists of two rays (sides) emerging from one point (vertex).
They are classified by degrees, by location relative to each other and relative to the circle. Let's start with the types of angles by their size.
There are several varieties of them. Let's take a closer look at each type.
There are only four main types of angles - right, obtuse, acute and developed angle.
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 quadrangles as a square and a rectangle have them.
Blunt
It looks like this:
The degree measure is always greater than 90 degrees, but less than 180 degrees. It can occur in such quadrangles as a rhombus, an arbitrary parallelogram, in polygons.
Spicy
It looks like this:
The degree measure of an acute angle is always less than 90°. It occurs in all quadrilaterals, except for a square and an arbitrary parallelogram.
deployed
The expanded angle looks like this:
It does not occur in polygons, but it is no less important than all the others. A straight angle is a geometric figure, the degree measure of which is always 180º. You can build on it by drawing one or more rays from its vertex in any direction.
There are several other secondary types of angles. They are not studied in schools, but it is necessary to know at least about their existence. There are only five secondary types of angles:
1. Zero
It looks like this:
The very name of the angle already speaks of its magnitude. Its interior area is 0 o, and the sides lie on top of each other as shown in the figure.
2. Oblique
Oblique can be straight, and obtuse, and acute, and developed angle. Its main condition is that it should not be equal to 0 o, 90 o, 180 o, 270 o.
3. Convex
Convex are zero, right, obtuse, acute and developed angles. As you already understood, the degree measure of a convex angle is from 0 o to 180 o.
4. Non-convex
Non-convex are angles with a degree measure from 181 o to 359 o inclusive.
5. Full
An angle with a measure of 360 degrees is a complete angle.
These are all types of angles according to their size. Now consider their types by location on the plane relative to each other.
1. Additional
These are two acute angles that form one straight line, i.e. their sum is 90 o.
2. Related
Adjacent angles are formed if a ray is drawn in any direction through a deployed, more precisely, through its top. Their sum is 180 o.
3. Vertical
Vertical angles are formed when two 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
The central angle is the one with the 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 one whose vertex lies on the circle and whose sides intersect it. Its degree measure is equal to half of the arc on which it rests.
It's all about the corners. Now you know that in addition to the most famous - sharp, obtuse, straight and deployed - in geometry there are many other types of them.
Students are introduced to the concept of angle in elementary school. But as a geometric figure with certain properties, they begin to study it from the 7th grade in geometry. Seems, pretty simple shape what can be said about her. But, acquiring new knowledge, schoolchildren understand more and more that you can learn quite interesting facts about her.
In contact with
When are studied
The school geometry course is divided into two sections: planimetry and solid geometry. Each of them has a lot of attention. given to the corners:
- In planimetry, their basic concept is given, acquaintance with their types in size takes place. The properties of each type of triangles are studied in more detail. New definitions for students appear - these are geometric shapes formed at the intersection of two lines with each other and the intersection of several lines of a secant.
- In stereometry, spatial angles are studied - dihedral and trihedral.
Attention! This article discusses all types and properties of angles in planimetry.
Definition and measurement
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 get a geometric figure - an angle, consisting of the following elements:
- the vertex - the point from which the rays were drawn, is indicated by a capital letter of the Latin alphabet;
- the sides are half-line drawn from the top.
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 intuitively. To begin with, students are introduced to the concept of a developed angle.
Important! An angle is said to be developed if the half-lines emanating from its vertex form a straight line. An unfolded angle is all other cases.
If it is divided into 180 equal parts, then it is customary to consider the measure of one part 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 subdivided according to such criteria as degree measure, the nature of their formation and the categories below.
By size
Given the magnitude, the angles are divided into:
- deployed;
- straight;
- blunt;
- spicy.
What angle is called deployed was presented above. Let's define the concept of a straight line.
It can be obtained by dividing the deployed into two equal parts. In this case, it is easy to answer the question: a right angle, how many degrees is it?
Divide 180 degrees by 2 to get right angle is 90 degrees. This is a wonderful figure, since many facts in geometry are associated with it.
It also has its own characteristics in the designation. To show a right angle in the figure, it is indicated not by an arc, but by a square.
The angles that are obtained by dividing an arbitrary ray of a straight line are called acute. According to the logic of things, 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 greater than a right angle, but less than a straight angle. Its degree measure varies from 90 to 180 degrees.
This element can be divided into different types of figures under consideration, excluding the expanded one.
Regardless of how the non-rotated angle is broken, the basic axiom of planimetry is always used - “the main property of measurement”.
At dividing the 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 level of the 7th grade, the types of angles in their magnitude end there. But to increase erudition, it can be added that there are other varieties that have a degree measure of more than 180 degrees. They are called convex.
Figures at the intersection of lines
The next types of angles that students are introduced to are the elements formed when two lines intersect. Figures that are placed opposite each other are called vertical. Their distinguishing feature is that they are equal.
Elements that are adjacent to the same line are called adjacent. The theorem mapping their property says that Adjacent angles add up to 180 degrees.
Elements in a triangle
If we consider the figure as an element in a triangle, then the angles are divided into internal and external. The triangle is bounded by three segments and consists of three vertices. The angles located inside the triangle at each vertex, 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 180 degrees.
Intersection of two straight lines
Line intersection
When two straight lines intersect, angles are also formed, which are usually distributed in pairs. Each pair of elements has its own name. It looks like this:
- internal cross-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 the secant intersects two lines, all these pairs of angles have certain properties:
- Internal crosswise lying and corresponding figures are equal to each other.
- Internal one-sided 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, studying, it will be necessary to recall all the properties of the angles formed at the intersection of two parallel lines of a secant. When studying the features of triangles, it is necessary to remember what adjacent angles are. Having switched to stereometry, all three-dimensional figures will be studied and built based on planimetric figures.