Mbou "Zolotopoleny secondary school".

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Slides captions:

Theme of the lesson: "The sum of the angles of a triangle." "The greatness of a man lies in his ability to think." B. Pascal

The purpose of the lesson: Find out: - What is the sum of the angles of any triangle.

Types of corners 1 2 3 4

Consider drawing a b c 1 2 3 4 d 5

Laboratory work. Instructions for work 1. Construct an arbitrary triangle ABC in a notebook. 2. Measure the degree measures of the angles of the triangle. 3. Write in your notebook:  A =…,  B =…,  C=… 4. Find the sum of the angles of the triangle  A +  B +  C=… 5. Compare the results.

Practical work. Take the paper triangle lying on each desk. Carefully tear off two corners of it. Attach these corners to the third in such a way that they come out of the same vertex.

The sum of the angles of a triangle is Theorem

Consider an arbitrary triangle ABC B A C Given: ∆ABC Document:  A +  B +  C= 180 0

and prove that A B C

Let us draw a straight line through the vertex B parallel to the side AC A C B C

Angles 1 and 4 are cross-lying angles at the intersection of parallel lines and AC and secant AB. A C B 1 4 C

And angles 3 and 5 are cross-lying angles at the intersection of parallel lines and AC and secant BC. A C B C 5 3

Therefore, 4 \u003d 1, 5 \u003d 3 A C 3 B 5 4 1 C

It is obvious that the sum of angles 4, 2 and 5 is equal to the straight angle with vertex B, i.e. A C 2 C B 4 5

From here, given that we get or A 2 C 5 1 3 B 4 4 \u003d 1,

From here, given that we get or A 2 C B 1 3 5 4 5 \u003d 3 4 \u003d 1,

Theorem proven

Approximate proof plan

Historical reference The proof of this fact, stated in modern textbooks, was contained in the commentary to the "Beginnings" of Euclid by the ancient Greek scientist Proclus (5th century AD). Proclus claims that according to Eudemus of Rhodes, this proof was discovered by the Pythagoreans (5th century BC). BC.).

The great scientist Pythagoras was born around 570 BC. on the island of Samos. Pythagoras' father was Mnesarchus, a gem-carver. The name of Pythagoras' mother is unknown. According to many ancient testimonies, the born boy was fabulously handsome, and soon showed his outstanding abilities.

IN A C E 2 1 3 4 5  Try to prove this theorem at home using the drawing of Pythagoras' students.

Triangle Exterior Angle Definition: An exterior angle of a triangle is an angle adjacent to one of the triangle's angles.  4 – outside corner Property. The exterior angle of a triangle is equal to the sum of the two angles of the triangle that are not adjacent to it.  4 =  1 +  2 1 2 3 4

So, really: 1 2 3 4

Oral work: Find the angles of triangles 80 º 70 º? B A C A=30 º

45º? L K M L =45 º

80º? ? N P R N=50º R=50º

At 130º? ? A C B=40 º C=50 º

Is there a triangle with angles: a) 30˚, 60˚, 90˚ b) 46˚, 160˚, 4˚ c) 75˚, 80˚, 25˚ d) 100˚, 20˚, 55˚

Work with the textbook. P.71 No. 223 a) No. 228 a)

Practical application of knowledge. The property of the angles of a right-angled isosceles triangle was known to one of the first creators of geometric science, the ancient Greek scientist Thales. Using it, he measured the height of an Egyptian pyramid by the length of its shadow. According to legend, Thales chose the day and time when the length of his own shadow was equal to his height, since at that moment the height of the pyramid should also be equal to the length of the shadow it casts. Of course, the length of the shadow could be calculated from the midpoint of the square base of the pyramid, but Thales could measure the width of the base directly. Thus, it is possible to measure the height of any tree.

Summary of the lesson. Today in the lesson we have proved by research the theorem on the sum of the angles of a triangle, we have learned to apply the acquired knowledge in practice. Once again we were convinced that geometry is a science that arose from human needs. After all, as Galileo wrote: “Nature speaks the language of mathematics: the letters of this language are circles, triangles and other mathematical figures.”

Homework P.30, No. 223 (b), No. 228 (c). Another way to prove the triangle sum theorem.

Thank you for your attention!

Lesson objectives: 1. To consolidate and test students' knowledge on the topic: "The property of the angles formed at the intersection of two parallel lines by a third and signs of parallel lines." 2. Open and prove the property of the angles of a triangle. 3. Apply the property when solving the simplest problems. 4. Use historical material to develop the cognitive activity of students. 5. To instill the skill of accuracy in the construction of drawings.

Lesson plan: 1. Independent work. 2. Practical work. (Preparation for learning new material). 3. Proof of the triangle sum theorem. (several ways). 4. Solving problems. (The theorem is used in solving). Literature: Newspapers "Mathematics". "Journey through the history of mathematics, or how people learned to count". Auth. Alexander Svechnikov "Pedagogy" - press. "Physics and Astronomy" - physics textbook grade 7 auth. Pinsky. Soviet encyclopedic dictionary M. 1989 "History of mathematics at school" IV-VI grades M. "Enlightenment" 1981 ed. G.I. Glaser.

5) Find the angles ABC, Find

Historical reference. 1. Definition of parallel lines - Euclid (3rd century BC), in the works of the "Beginning" meet." 2. Posidonius (I century BC) “Two straight lines lying in the same plane, equally spaced from each other” 3. The ancient Greek scientist Pappus (second half of the 3rd century BC) introduced the symbol of parallel lines =. Subsequently, the English economist Ricardo () used this symbol as an equal sign. It wasn't until the 18th century that the symbol || began to be used.

Opening properties of triangle corners. The ancient Greeks, on the basis of observations and practical experience, drew conclusions, expressed their assumptions - hypotheses (Hypotesis - foundation, assumption) and then at meetings of scientists - symposiums (symposium - literally a feast, a meeting on any scientific issue), these hypotheses tried to substantiate and prove. At that time, there was a statement: “Truth is born in a dispute”

Conjecture about the sum of the angles of a triangle. Practical work. Using a protractor, determine the sum of the angles of a triangle. (Use models of all kinds of triangles). Determine what angle you get if you make it from the angles of the triangle. What is its degree measure? (Use models of all kinds of triangles).

Class 7

Lesson topic: "The sum of the angles of a triangle."

Time : double lesson (pair).

Lesson Objectives:

Educational: familiarize with various ways of proving the theorem on the sum of the angles of a triangle, introduce the concept of an external angle of a triangle, consider its property, learn how to apply the theorem to find the angles of a triangle in the process of solving problems.

Educational: to continue the formation of the skills of aesthetic design of notes in a notebook and the implementation of drawings, to continue to form a positive attitude towards a new academic subject, to teach the ability to communicate and listen to others, to cultivate conscious discipline.

Developing: develop the skill of using signs of parallel lines and the properties of angles with parallel lines for solving problems and proving theorems; develop the ability to find the angles of triangles at two given angles, with given angle proportionalities; develop the skill of using the theorem on the sum of the angles of a triangle and its consequence for solving problems; develop the skill of finding the angles of triangles at two given angles, at given proportional angles, at given different elements of triangles (equal sides, angles), the ability to find the angles of a triangle if an angle is given at bisector, and find angles at the bisector and the base of the triangle, if the angles of the triangle are given; developconscious perception of educational material, visual memory and competent mathematical speech.

Equipment: textbook Pogorelova A.V., Geometry grades 7-9, (pp. 46, 52–53), interactive whiteboard, presentation, handout (whole paper triangles and cut cardboard ones), a large paper triangle for demonstration on the board by the teacher of finding the sum of angles triangles, cards for self-study

Lesson type: a lesson in learning new material and consolidating it (combined lesson).

During the classes:

Stage

lesson

Teacher activity

Student activities

Org.

moment

homemadeexercise

Learning new material

(Practical work)

Learning new material

Fizminutka and entertain. moment

Consolidation of the studied material

Summarizing

Open your diaries and write down your homework: learn abstract 22, (item 33) Homework numbers 19 (2), 22 (2), 24. (slide 2)

Let's start the lesson with you with a poem:

Even a preschooler knows

What is a triangle

And how could you not know.

But it's quite another thing -

Fast, precise and skillful

It has sides - there are three of them,

And there are three corners in all,

And of course there are three peaks.

If the lengths of all sides

We will find by addition

Then we'll come to the perimeter.

Well, the sum of all angles

In any triangle

Tied to one number.

And today in the lesson we will find out with what number the sum of the angles in any triangle is associated.

Open the notes, write down: note No. 22. The sum of the angles of a triangle (slide 3).

Draw an arbitrary triangle in your notebooks (slide 4). Not very small, about a third of the page. What does random mean?

Right. We draw a triangle. We take a protractor.

And we begin to take turns measuring the angles of the drawn triangle (slide 5). We will measure angles with you.

We take a protractor, apply it to the first measured angle so that the open point on the protractor coincides with the vertex of the angle, and the side of the triangle and the inner straight part of the protractor coincide, forming one straight line.

We measure the angle, and from 0, and not from 180. - note that we have 2 scales, inside and outside the protractor arc. We write down: the angle, for example, B is equal to ... degrees. I got 80 0 . What angles did you get?

And do the same with the rest of the corners.

Did you find all the corners?

Now, let's see, what is our topic?

So what are we going to do with our triangle corners?

Right. We add up your received angles, raise our hands and call how much it turned out.

Well done! Now take, please, paper triangles on your desktops (slide 6). And I'll take a triangle (attached to the board with a magnet). Look at it and thinkhow by bending the angles of this triangle find the sum of its angles.

Not everyone, probably, immediately guessed - we need to add up all the corners. How to do it?

Right! I show again on a large triangle on the board.

Tell me, what is the sum of all the angles, looking at our bent triangle?

Triangles have already been measured twice and still get 180?

(If not, I give an additional triangle). Check if the triangle is formed from these parts?

Did everyone get it right?

Fine. Now we again need to show that the sum of the angles in the triangle is what?

(slide 8)

Great! What are we going to do with the corners?

What happened to us?

Well done boys. Now write it down in your notes. Theorem "On the sum of the angles of a triangle." What do you think she is telling us?

Right! We write down (slide 9).

Historical background (slide 10).

Now we will prove this theorem. This proof you need to write down, parse if something is not clear. If it is difficult, come to additional classes - today 6-7 lessons.

We write down: proof (slide 11)

What have we been given and what do we need to prove?

We write down the given and draw a small arbitrary triangle in a notebook.

Let'sprove this theorem , using the properties of angles known to us with parallel lines and a secant. To do this, we construct through the vertex B a straight lineA parallel to the base - side AC.

And let's denote the resulting angles: those given in the triangle, and two more angles.

We write down:

Let's builda || AC, BÎ a.

How many secant lines are obtained with parallel lines? Name them.

Let's look at one secant first.

What can be said about the angles with our parallel lines and secant AB.

Let's write it down.

Now consider another secant BC. What can be said here about angles with parallel linesa || ACand secant sun?

Right. We write down.

Now let's look at the straight angle B. What is this angle.

Right. What else is he equal to? The sum of what angles?

That's right, you can see it very well in the picture.

Now, looking at the written sum and at the previously proven equalities of the angles, what can be said about the angle B?

Those. what did you get?

Did you prove the theorem?

Fizminutka (slide 12).

On the slide, the letters are written in different colors, which helps to relax the muscles of the eye.

№ 20 (slide 14) - we decide orally. Notebooks with notes are not closed.

Can two angles of a triangle be right angles?

Are the two angles obtuse?

One straight and the other blunt?

What conclusion can be drawn then? What angles can be in a triangle?

Those. acute angles in any triangle should be at least .... ?

Write it down in your notes - this is a consequence of the theorem on the sum of angles of a triangle (slide 15)

Corollary from the theorem:

Every triangle has at least two acute angles.

Oral work with tasks (slides 16-18)

Guys. We go out to the board and solve the numbers indicated on the slide (slide 19):№ 18, № 19 (1), № 22 (1,3),№ 21, №25.

A triangle is drawn on the board - we solve problem 18, 19 using it.

21 orally.

22 - on the board a drawing with a r / w triangle, we solve the problem using it.

№ 25 off the board with the same blueprint.

(20 slide)
(21 slides)

Guys, remember what we learned today.

What is the sum of the angles of any triangle?

How many sharp corners should there be at least in any triangle?

Or maybe 2 stupid ones?

Well done!

See you at the next lesson after the bell.

Open diaries and write down homework.

Open notes, write.

Any.

For example, 30 0 , 120 0 , 50 0 , 90 0 ….

Yes.

The sum of the angles of a triangle.

Add up. And find out what the sum is.

Count and say the answers. Everyone should have 180.

Consider triangles, try to add, come to a solution.

Just bend the triangle so that all the corners come together.

The expanded angle is 180 degrees.

Yes.

Yes, it does.

Exactly.

180.

Add them together to show their sum.

Again, the expanded angle is 180.

That the sum of all the angles of a triangle is 180.

Write down the theorem.

Listen, ask questions.

Dan, triangle, arbitrary. And you need to prove that the sum of its angles is 180 0 .

Write down the given and draw a picture:

Given:

ABC

Prove:

РА+РВ+РС=180°

They build after the teacher (the teacher flips through the animation on the slide).

Two? AB and VS.

Ð 4= Ð 1 , as crosswise lying angles with parallel linesa || ACand secant AB.

Ð 5= Ð 2, as cross lying angles with parallel linesa || ACand secant sun.

180, because it is expanded.

Ð 4 + Ð 3+ Ð 5 = 180°, becauseÐ B - deployed (Ð H = 180°)

BecauseÐ4=Ð1 and Ð5=Ð2, THEN

Ð 4 + Ð 3+ Ð 5 = Ð 1 + Ð 3+ Ð 2 = 180.

That the sum of the angles of a triangle is 180.

Proved.

Repeat exercises (physical minute) after the teacher.

No.

Two sharp and one blunt, one straight and two sharp, all three sharp.

Two!

Recorded from dictation or from a slide.

Guessing puzzles.

Theorem on the sum of angles in a triangle. And a consequence of it.

180 degrees.

At least two sharp corners.

No.

Continuation of the topic

Consolidation of the studied material

Self.work

Summarizing

So, how many angles are there in a triangle?

Then since two angles are always sharp, then the third one can be ... what?

Then we will determine the type of triangle by the third angle.

Look at the slide (slide 22). Name the angle and determine the type of triangle.

If two angles of a triangle are acute and the third is also acute, then the triangle...

If two angles of a triangle are acute and the third is also right, then the triangle is...

If two angles of a triangle are acute and the third is also obtuse, then the triangle is...

Well done!

Historical moment (slide 23)

Now we solve oral problems.

(slide 24)

Determine the type of triangle if:

one of its angles is 40 0 , and the other is 100 0 ,

one of its angles is 60 0 , and the other - 70 0 ,

one of its angles is 40 0 , and the other - 50 0 .

(Slide 25-26)

Now we solve problems at the blackboard and in notebooks (slide 27)

Now we are writing an independent work on options, three tasks.

Guys, tell me what we learned and remembered today?

Well done!

Lesson grades are...

anyone.

Acute-angled.

Rectangular.

obtuse.

obtuse, because there is an obtuse angle.

Acute-angled, because all corners are sharp.

Rectangular, because 180 - 40 -50 = 90.

By the angle sum theorem D:
RW = 180 0 – (РС + РВ) =
= 180 0 – (90 0 + 50 0 ) = Ð40 0

Because D ABC is isosceles, then РА = РВ, by the property of r/b D.

By the angle sum theorem D:
RA = (180 0 – РС) : 2 =
= (180 0 – 90 0 ) : 2 = R45 0

Solve problems with the help of a teacher.

Write independent work in cards.

- The sum of the angles of any triangle is 180.

Types of triangles - acute, obtuse, rectangular.

We learned that the most ancient tools in geometry were a ruler and a compass.

Task 2 .

Given:

Find:

Р1 and Р2Solution:

Task 3.

Given:

Find:

Р1 and Р2Solution:

Material for a geometry lesson in grade 7

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"The topic of the lesson is the SUM OF THE ANGLES OF A TRIANGLE"

MBOU "ZOLOTOPOLENSKY COMPREHENSIVE SCHOOL"

KIROV DISTRICT OF THE REPUBLIC OF CRIMEA

Lesson in 7th grade on the topic

"The sum of the angles of a triangle"

Teacher: Antipova Galina Ivanovna

Lesson topic: The sum of the angles of a triangle.

Lesson type : Lesson learning new material.

Lesson Objectives : Learning goal: prove the triangle sum theorem;
to teach how to apply the proven theorem in solving problems, to introduce the concept of an external angle of a triangle;

Development goal: improve the ability to think logically and express your thoughts aloud, develop logical thinking, will, emotions;

educational goal: to educate students in the desire to improve their knowledge; cultivate interest in the subject.

During the classes

Organizing time

(The teacher is holding a triangle ) The triangle plays a special role in geometry. Without exaggeration, we can say that all or almost all geometry is built on a triangle.

So what is a triangle?(A triangle is a figure formed by three points that do not lie on the same straight line, and line segments connecting these points in pairs.)

Look at the triangle (fig. 1). What is B equal to? (formulation of the problem)

So today in the lesson we will try to formulate and prove the wonderful property of a triangle , which will help us answer this question.

The topic of our lesson: The sum of the angles of a triangle. (slide 1)

Write the date and topic of the lesson in your notebook.

Goals: ( slide 2)

Updating of basic knowledge.(Slides 3-9)

3. Studying new material

Practical work(entrance to the topic of the lesson, preparation for the perception of new material)

Teacher. Answer the question: What instrument can be used to measure the angles of a triangle? Check your readiness for the lesson, does everyone have a protractor, pencil, ruler?

Part 1 (Work in pairs ) (Slide 10)

Teacher. Guys, you have sheets with practical work on your tables. Take them, use a protractor to measure the angles of the triangles and write the results in the tables.

№ p/n				A+ B + WITH

Teacher. Find the sum of the angles of your triangles and write the results in tables. What is it equal to? What did you notice? (All sums are close to 180º.) Look guys! The triangles were taken arbitrary, the angles in the triangles were different, and the results were the same for everyone.

What explains the small difference? Is it because there is no regularity, or because there is a regularity, but with our tools we cannot establish it with sufficient accuracy?

Teacher. What conclusion can we draw from this practical work?

Students conclude: the sum of the angles of a triangle is 180 degrees.

Part 2 (working with models on desks) slide 11)

Statement and proof of the theorem(Slide 12, 13)

Historical information. (Slides 14,15)

Consolidation.(Slides 16-24)

Tasks on finished drawings

2) Independent work with peer review

1. Is there a triangle with corners:

a) 30 o, 60 o, 90 o; b) 46 o, 160 o, 4 o; c) 75 o, 90 o, 25 o?

2. Determine the type of triangle if one angle is 40 o, the other is 100 o

3. Find the angles of an equilateral triangle.

4. (Slide 25)

Lesson results. Reflection. (Slide 26.27)

What was the main purpose of today's lesson? (Prove the theorem on the sum of angles of a triangle. Learn to solve problems on the application of the theorem on the sum of angles of a triangle)

Have we achieved it?

View presentation content
"SUM OF ANGLES OF A TRIANGLE"

C umma of triangle angles

Mathematic teacher

Municipal educational institution "Zolotopolenskaya school"

Kirovsky district, Crimea

Antipova Galina Ivanovna

Goals:

formulate and prove the triangle sum theorem;
consider tasks for the application of the proven

Let's repeat learned

Adjacent corners

60 

 AOC+  BOC=

Vertical angles are equal

Amount of unilateral

angles is 180 0

Respective

angles are equal

Crosswise angles are equal

a ll b

Compute all angles.

Practical work

Study

 .

By "tearing off" the angles of a triangle, it can be shown that the sum of the angles of a triangle is 180  .

Theorem: The sum of the angles of a triangle is 180  .

Given: ∆ABC

Prove:  A+  B +  C =180 

Proof:

1)D. n. straight line a || AC

2)  4 =  1

3) Because  4+  2+  5=180  ,

then  1 +  2+  3 =180 

or  A+  B+  C=180 

... As for mortals, the truth is clear,

That two stupid people cannot fit into a triangle. Dante A.

Pythagoras

The proof of the theorem on the sum of the angles of a triangle "The sum of the interior angles of a triangle is equal to two right angles" is attributed to Pythagoras .

580 - 500 BC e.