What is a right obtuse and acute angle. Straight, obtuse, acute and straight angles. How to mark an acute angle

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A median is a segment drawn from the vertex of a triangle to the middle opposite side, that is, divides it in half by the intersection point. The point at which the median intersects the side opposite the vertex from which it emerges is called the base. Each median of the triangle passes through one point, called the intersection point. The formula for its length can be expressed in several ways.

Formulas for expressing the length of the median

• Often in geometry problems, students have to deal with a segment such as the median of a triangle. The formula for its length is expressed in terms of sides:

where a, b and c are the sides. Moreover, c is the side on which the median falls. This is how the simplest formula looks like. Medians of a triangle are sometimes required for auxiliary calculations. There are other formulas.

• If during the calculation two sides of a triangle and a certain angle α located between them are known, then the length of the median of the triangle, lowered to the third side, will be expressed as follows.

Basic properties

• All medians have one common point the intersections of O and it are divided in the ratio of two to one, if counted from the vertex. This point is called the center of gravity of the triangle.
• The median divides the triangle into two others whose areas are equal. Such triangles are called equal-area.
• If you draw all the medians, the triangle will be divided into 6 equal figures, which will also be triangles.
• If all three sides of a triangle are equal, then each of the medians will also be an altitude and a bisector, that is, perpendicular to the side to which it is drawn, and bisects the angle from which it emerges.
• IN isosceles triangle the median dropped from a vertex that is opposite a side that is not equal to any other will also be an altitude and a bisector. The medians dropped from other vertices are equal. This is also a necessary and sufficient condition for isosceles.
• If a triangle is the base of a regular pyramid, then the height dropped to this base is projected to the point of intersection of all medians.

• In a right triangle, the median drawn to the longest side is equal to half its length.
• Let O be the intersection point of the triangle's medians. The formula below will be true for any point M.

• The median of a triangle has another property. The formula for the square of its length through the squares of the sides is presented below.

Properties of the sides to which the median is drawn

• If we connect any two points of intersection of medians with the sides on which they are dropped, then the resulting segment will be midline triangle and make up one half of the side of the triangle with which it does not have common points.
• The bases of the altitudes and medians in a triangle, as well as the midpoints of the segments connecting the vertices of the triangle with the point of intersection of the altitudes, lie on the same circle.

In conclusion, it is logical to say that one of the most important segments is the median of the triangle. Its formula can be used to find the lengths of its other sides.

Look at the picture. (Fig. 1)

Rice. 1. Illustration for example

What geometric shapes are you familiar with?

Of course, you saw that the picture consists of triangles and rectangles. What word is hidden in the names of both of these figures? This word is angle (Fig. 2).

Rice. 2. Angle determination

Today we will learn to draw a right angle.

The name of this angle already contains the word “straight”. To correctly depict a right angle, we need a square. (Fig. 3)

Rice. 3. Square

The square itself already has a right angle. (Fig. 4)

Rice. 4. Right angle

He will help us depict this geometric figure.

To correctly depict the figure, we must attach the square to the plane (1), outline its sides (2), name the vertex of the angle (3) and the rays (4).

1.

2.

3.

4.

Let's determine whether among the available angles there are straight lines (Fig. 5). A square will help us with this.

Rice. 5. Illustration for example

Let's find the right angle of the square and apply it to the existing angles (Fig. 6).

Rice. 6. Illustration for example

We see that the right angle coincides with the PTO angle. This means that the PTO angle is straight. Let's do the same operation again. (Fig. 7)

Rice. 7. Illustration for example

We see that the right angle of our square does not coincide with the angle COD. This means that the angle COD is not right. Once again we apply the right angle of the triangle to the angle AOT. (Fig. 8)

Rice. 8. Illustration for example

We see that angle AOT is much larger than a right angle. This means that angle AOT is not right.

In this lesson we learned how to construct a right angle using a square.

The word “angle” gives its name to many things, as well as geometric shapes: rectangle, triangle, square, with which you can draw a right angle.

A triangle is a geometric figure that consists of three sides and three angles. A triangle that has a right angle is called a right triangle.

Class: 2

Presentation for the lesson

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Lesson type: explanation of new material.

Place of the lesson in the structure of the topic: this topic is studied in the section “Tabular addition of single-digit numbers with passing through ten.”

Purpose of the lesson: To introduce students to the concept of “right angle” and teach them to apply the acquired knowledge in practice.

Lesson objectives:

1. Educational:

• Introduce students to the concept of “right angle”;
• Develop practical skills in determining right angle with and without a triangle;
• Continue to improve your skill mental counting within 100;

2. Developmental:

• Development logical thinking, attention, memory, spatial imagination;
• Development of creative skills on the topic for successful implementation assignments;
• Development of the culture of speech and emotions of students.

3. Educational:

• In order to solve the problems of moral education, promote the cultivation of humanity and collectivism, observation and curiosity, the development of cognitive activity, and the formation of skills independent work;
• In order to solve problems aesthetic education promote the development of a sense of beauty in students.

DURING THE CLASSES

I. Organizational moment.

Well, check it out, my friend,
Are you ready to start the lesson?
Is everything in place?
Is everything alright?
Pen, book and notebook?
Is everyone sitting correctly?
Is everyone watching carefully?
Everyone wants to receive
Only a “5” rating.

Guys, today we will again go on a journey through the kingdom of Geometry.

3. Oral counting.

– At the gate we are met by King Dot and his daughter, Princess Straight. Before the king and princess introduce us to the inhabitants of their kingdom, they want to test you.

II. Verbal counting.

1) Game “Confused Caterpillar”.

The caterpillar has lost the numbers, look at the remaining ones, guess what rule can be used to continue the series of numbers. (Children say the rule: these are even numbers; each subsequent number is 2 more than the previous one).

What numbers did the caterpillar lose? (2,4,6,8,10,12,14,16)

2) Game “Mathematical Basketball”.

Basketball- a team sports game, the goal of which is to throw a ball into a suspended basket with your hands.

Any of you will score a goal if you solve the example correctly. (Children solve examples in a chain). 30 + 7 25 + 5 32 – 12 66 + 4 80 – 7 28 – 10 45 – 45 53 + 7 59 – 9 90 + 9

Slide 5

Logic task

How many spots do 15 piglets have? (15)

When a goose stands on two legs, it weighs 4 kg. How much will a goose weigh when it stands on one leg?

– You passed all the tests. The king and princess are very pleased with you and are ready to introduce you to the inhabitants of the kingdom of “Geometry”!

(When clicked, the gate leaves open.)

Guys, before you are the inhabitants of the kingdom “Geometry”.

Look at the shapes in each frame. Which one is the odd one out? Why?

(Students name the extra figures and justify their choice).

Divide all remaining figures into two groups. How can I do that? (The remaining shapes can be divided into two groups: lines and polygons.)

Name the types of lines and polygons that you know. (Lines: straight, broken, curved. Polygons: square, trapezoid, rectangle, quadrangle, pentagon, hexagon, polygon).

IV. Working on new material.

(Slide 8)

1) - The crossword puzzle will tell you the topic of the lesson. Crossword “Geometric”.

1) Part of a line that has a beginning but no end. (Ray).

2) Geometric figure, having no corners. (Circle).

4) A geometric figure in the shape of an elongated circle. (Oval).

The topic of our lesson is hidden vertically. Find her. (Corner). (click, geometric shapes fly out).

Please formulate the topic of our lesson.

Guys, why are we going to study angles?

Do you think this knowledge will be useful to you?

(Children's answers)

Angles surround us in everyday life. Give your own examples of where you can find angles around us.

Guys, maybe someone knows what an angle is? (children's opinions are listened to)

We will check the correctness of our formulation a little later.

People of what professions are most likely to encounter angles? (constructor, engineer, designer, builder, architect, sailor, astronomer, architect, tailor, etc.)

Look at the pictures: a connecting corner for pipes and a stationery corner for papers; carpenter's square and drafting square; corner table and corner sofa.

Guys, now the King and Princess offer to play a little.

Slide 10.

Game “The corner gave them a name.”

The angle is an important figure. He helped give names to many figures. Name the figures.

What do the names of the figures have in common? (that they have a square - a common part)

Why is the first part of the words different everywhere? (because there are different numbers of angles)

Fizminutka 11-16 slides

Guys, now step back one cell from the red fields and place point O. Draw two rays from this point.

Draw point O (4-5) on the board in advance. Call 4-5 children to draw rays on the board.

What kind of figures did we get? (corner)

Look how different these angles are.

Guys, now put together a rule from words.

Work in pairs.

(Conclusion: an angle is a geometric figure formed by two different rays

with a common beginning).

Guys, now look at the figure that I drew.

Is it an angle or not.

(The children say no, we return to the rule again, after which we conclude that this is also an angle - a reversed one)

Slide 19. (output by angle)

Poster on blackboard

Point O is the vertex of the angle. An angle can be called by one letter written near its vertex. Angle O. But there can be several angles that have the same vertex. What to do then? (On the sheet there is a drawing of such angles)

Children's answers.

In such cases, if you call different angles with the same letter, it will not be clear which angle you are talking about. If this does not happen, you can mark one point on each side of the angle, put a letter near it and designate the angle with three letters, while always writing in the middle the letter indicating the vertex of the angle. Angle AOB. Rays AO and OB are the sides of the angle.

Poster on blackboard

Guys, you have different types of corners on your tables. Please find the same types of angles.

How will you search? (Children's answers)

One person on my models is looking for the same angles.

Guys, look, numbers 6 and 7 matched completely, but 1 and 5 did not. No. 5 is bigger.

What can be concluded? After the children answer, a slide appears.

CONCLUSION: slide 21

• Equal angles coincide when superimposed
• If one angle is superimposed on another and they coincide, then these angles are equal

Making a right angle model.

It is not always convenient to determine a right angle by eye. To do this, use a ruler-square.

What color is used to highlight an angle greater than a right angle? (Blue).

Less direct? (Green).

Which of the three proposed angles is a straight line?

Why did you decide so? (The vertex and sides of the angle coincide with the right angle on the square ruler).

How to determine the type of angle?

• To determine the type of angle, you need to combine its vertex and side, respectively, with the vertex and side of the right angle on the square.

Each of the corners has its own name. An acute angle is an angle that is less than a right angle. An obtuse angle is an angle that is larger than a right angle.

(Tables with the names of the angles appear on the board)

My mother took the piece of paper
And folded the corner
This is the angle for adults
It's called DIRECT.
If the corner is already SHARP,
If wider, then - DUMB.

Guys, is it always possible to overlap the angles?

No. (If drawn in a notebook...)

For this purpose, there is a protractor with which angles are measured. Angles are measured in degrees. Look at the types of protractors.

Very often we can observe angles on the clock. The angles are formed by the hour hands.

Work according to the textbook.

Exercise: Using the right angle model, find right angles and write down their numbers. (Children complete the task independently, then one student names his answer, everyone checks the work).

With the help of a square it is convenient not only to determine right angles, but most importantly - to build them. Let's build a right angle, everyone will name it with one or three letters.

Slide 27-29 (The teacher is on the board, and the children are building a right angle in their notebooks. Mutual testing is carried out in pairs).

I'm SHARP - I want to draw,
Now I’ll take it and draw it.
I lead two straight lines from a point,
It's like two rays
And we see an ACUTE ANGLE,
like the edge of a sword.

And for an obtuse ANGLE
We repeat everything again:
From a point we draw two straight lines,
But let's spread them wider.
Look at my drawing,
He's like scissors inside
If there are two rings
We'll push it all the way.

Practical work to consolidate what has been learned.

There is wire on your desks. Make a right angle out of it and test it with a square, then make it sharp and obtuse.

7. Lesson summary.

Tell me, using a diagram, what did you learn from today's math lesson?

8. Homework.

STRAIGHT, oh, oh; straight, straight, straight, straight and straight. Dictionary Ozhegova. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 … Ozhegov's Explanatory Dictionary

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An angle equal to its adjacent one. * * * RIGHT ANGLE RIGHT ANGLE, an angle equal to its adjacent... encyclopedic Dictionary

An angle equal to its adjacent one; in degree measurement is equal to 90°... Natural science. encyclopedic Dictionary

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1) an angle equal to its adjacent one. 2) Non-system unit. flat angle. Designation L. 1 L = 90° = PI/2 rad 1.570 796 rad (see Radian) ... Big Encyclopedic Polytechnic Dictionary

Straight, direct; straight, straight, straight. 1. Exactly elongated in some way. direction, not crooked, without bends. Straight line. “The straight road ended and was already going downhill.” Chekhov. Straight nose. Straight figure. 2. Direct (railway and unloading). Direct route... ... Ushakov's Explanatory Dictionary

STRAIGHT, oh, oh; straight, straight, straight, straight and straight. 1. Walking smoothly in which no. direction, without bending. Straight line (a line, the image of which can be an endless, tightly stretched thread). Draw a straight line (i.e., a straight line; noun). The road goes... ... Ozhegov's Explanatory Dictionary

angle of the main coil profile- (αb) The angle between the main profile of the involute worm coil and the straight line that makes a right crossing angle with the worm axis. Note The angle of the rectilinear main profile of the involute worm coil αb is equal to the main helix angle... ... Technical Translator's Guide

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• Mathematics. 2nd grade. Textbook. In 2 parts. Part 2, Moro M.I.. The textbook "Mathematics" is included in educational system"School of Russia". The material in the textbook allows you to implement a system-activity approach, organize differentiated learning And…