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Lesson topic: “Sum of angles of a triangle.” “The greatness of a man lies in his ability to think.” B.Pascal

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

Types of angles 1 2 3 4

Consider figure a b c 1 2 3 4 d 5

Laboratory work. Directions for work 1. Construct an arbitrary triangle ABC in your 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 everyone's desk. Carefully tear off two corners of it. Attach these corners to the third one so that they come out from one vertex.

The sum of the angles of a triangle is equal to Theorem

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

and prove that A B C

and prove that A B C

and prove that A B C

and prove that A B C

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

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

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

Therefore 4 = 1, 5 = 3 A C 3 B 5 4 1 C

Obviously, the sum of angles 4, 2 and 5 is equal to the unfolded angle with vertex B, i.e. A C 2 C B 4 5

Hence, taking into account that we get either A 2 C 5 1 3 B 4 4 = 1,

Hence, taking into account that we get either A 2 C B 1 3 5 4 5 = 3 4 = 1,

The theorem is proven

Rough outline of the proof

Historical background The proof of this fact, set out in modern textbooks, was contained in the commentary to Euclid’s Elements 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 AD). BC.).

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

B A C E 2 1 3 4 5  Try to prove this theorem at home using a drawing from Pythagoras’ students.

External angle of a triangle Definition: An external angle of a triangle is an angle adjacent to one of the angles of the triangle.  4 – external corner Property. An exterior angle of a triangle is equal to the sum of 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? V A C A=30 º

45º? L K M L =45 º

80º? ? N P R N =50 º R =50 º

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˚

Working with the textbook. Page 71 No. 223 a) No. 228 a)

Practical application of knowledge. The property of the angles of a right 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 a day and time when the length of his own shadow was equal to his height, since at that moment the height of the pyramid must 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. This way you can measure the height of any tree.

Lesson summary. Today in class we proved through research the theorem about the sum of the angles of a triangle, and learned to apply the acquired knowledge in practical activities. We are once again 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 angle sum theorem.

Thank you for your attention!

Objectives: 1. Introduce the concepts of acute, right and obtuse triangles. 2. Using an experiment, lead children to the formulation of the theorem on the sum of the angles of a triangle, prove it and teach them to apply the acquired knowledge in solving problems. 3. Development of cognitive activity, thinking, attention. 4. Fostering hard work

OBJECTIVES: 1. Consolidate knowledge on topics: triangle, parallel lines, types of angles; 2. Strengthen the skills of using a protractor; 3. Develop the ability to use the textbook; 4. Develop students’ mathematical speech; 5. Develop the ability to analyze material and draw conclusions; 6. Cultivate: interest in the subject, the ability to complete a task, confidence in one’s abilities in learning.

Lesson plan: 1. Organizational moment. 2. Repetition. 3. Oral work. 4. Statement of the problem, determination of ways to solve it. 5. Proposing a hypothesis. 6. Confirmation of the hypothesis. 7. Proof of the theorem. 8. Solving tasks to consolidate the learned theorem. 9. Summing up the lesson (reflection), homework assignment.

Lesson progress: 1.Organizational moment Today our class will turn into a “research institute”, and you will become “its employees”. And we will not only get acquainted with the work of the “research institute”, but we will also make discoveries ourselves! And so: the “research institute” has divisions: 1. Laboratory of experiments. 2. Laboratory of scientific evidence. 3. Testing laboratory.

2.Repetition In previous lessons, we studied the signs of parallel lines and the properties of angles for parallel lines. And today in the lesson, the knowledge gained on this topic will help make a discovery. Give the definition of parallel lines (Two lines in a plane are called parallel if they do not intersect)

Formulate the signs of parallelism of lines (If, when two lines are intersected by a transversal, the lying angles are equal, then the lines are parallel; If, when two lines are intersected by a transversal, the corresponding angles are equal, then the lines are parallel; If, when two lines are intersected by a transversal, the sum of one-sided angles is equal to 180°, then the lines are parallel ;)

Formulate the property of angles for parallel lines (If two parallel lines are intersected by a transversal, then the angles lying crosswise are equal; If two parallel lines are intersected by a transversal, then the corresponding angles are equal; If two parallel lines are intersected by a transversal, then the sum of one-sided angles is 180°)

1) Formulate the definition of a triangle. (A TRIANGLE is a figure formed by three points that do not lie on the same line, and segments connecting these points in pairs.) 2) Name the elements of a triangle. (Vertexes, sides, angles.) 3) What triangles are distinguished? (On the sides: scalene, equilateral, isosceles; cards - triangles) 4) Triangles are also distinguished by angles.

Let's make up a story on the topic: ANGLE. To do this, we use the plan recorded on the screen. An angle is a figure, ... (An angle is a figure formed by two rays emanating from one point. The rays are called the sides of the angle, and the point is the vertex.). 2. If ..., then the angle is called ... (If the angle is 90°, then the angle is called right. If it is 180°, then it is unfolded. If it is more than 0°, but less than 90°, then it is called acute. If it is more than 90°, but less than 180 °, then they call it stupid.)

That. Angles can be obtuse, acute, right or straight. An interior angle of a triangle is... An interior angle of a triangle is the angle formed by its sides, the vertex of a triangle is the vertex of its angle. This means that angles in a triangle can be different: obtuse, acute and right.

Laboratory of experiments Draw an angle: (3 students work at the board, and the rest are on the spot) 1 – row – obtuse; 2 – row – straight; 3 – row sharp. Complete the drawing to a triangle. What do I need to do? (Take a point on the sides of the angle and connect them with segments.) The resulting triangles can be called: obtuse, rectangular and acute. ((cards - triangles) Please note that an acute triangle has all acute angles.

Are there right and obtuse triangles? With two obtuse angles? With two right angles? How to justify this? Make a drawing: Rays VA and SD, CT and OH. KE and PL do not intersect, which means the triangle will not work. The sum of one-sided angles in case I is greater than 180°, in case II it is also greater than 180°, and in case III it is equal to 180°. In case III the lines are parallel, and in the first two cases the lines diverge. They conclude that a triangle cannot have two obtuse or two right angles. Also, a triangle cannot have one obtuse and one right angle at the same time.

We did some practical work, made a substantiation of the fact that a triangle does not always exist. Its existence depends on the size of the angles. How can you find out what the sum of the angles of a triangle is? Practically by measurement, theoretically by reasoning.

Test laboratory (practical application) 1. What is the third angle in a triangle if one of the angles is 40°, the second is 60°? (80°) 2. What is the angle of an equilateral triangle? (60°) 3. What is the sum of the acute angles of a right triangle? (90°) 4. What is the acute angle of a right isosceles triangle? (45°)

Lesson objectives: 1. To consolidate and test students’ knowledge on the topic: “Property of angles formed by the intersection of two parallel lines with a third and signs of parallel lines.” 2. Discover and prove the property of the angles of a triangle. 3. Apply the property when solving simple problems. 4. Use historical material to develop students’ cognitive activity. 5. Instill the skill of accuracy when constructing drawings.

PLAN: 1. Independent work. 2. Practical work. (Preparation for learning new material). 3. Proof of the theorem on the sum of the angles of a triangle. (several ways). 4. Solving problems. (When solving, a theorem is used). Literature: Newspapers “Mathematics”. "A Journey into the History of Mathematics, or How People Learned to Count." Auto. Alexander Svechnikov “Pedagogy” -press. “Physics and Astronomy” - physics textbook 7th grade, author. Pinsky. Soviet encyclopedic dictionary M. 1989 “History of mathematics in school” IV-VI grades M. “Enlightenment” 1981 auto G.I. Glaser.

5) Find angles ABC, Find

Historical reference. 1. Definition of parallel lines - Euclid (III century BC), in the works of “Elements” “Parallel lines are lines that, being in the same plane and being extended in both directions indefinitely on either side, do not meet." 2. Posidonius (1st century BC) “Two straight lines lying in the same plane, equidistant from each other” 3. The ancient Greek scientist Pappus (second half of the 3rd century BC) introduced the symbol for parallelism of lines =. Subsequently, the English economist Ricardo () used this symbol as an equal sign. It was only in the 18th century that the symbol || began to be used.

Discovering the properties of triangle angles. The ancient Greeks, based on observations and practical experience, drew conclusions, expressed their assumptions - hypotheses (Hypotesis - basis, assumption) and then at meetings of scientists - symposiums (symposium - literally a feast, meeting on any scientific issue) they tried to substantiate these hypotheses 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 types of triangles). Determine what angle you will get if you make it from the angles of a triangle. What is its degree measure? (Use models of all types of triangles).

Class 7

Lesson topic: "Sum of the angles of a triangle."

Time : double lesson (pair).

Lesson objectives:

Educational: familiarize yourself with various methods 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 to apply the theorem to find the angles of a triangle in the process of solving problems.

Educational: continue to develop the skills of aesthetically designing notes in a notebook and making drawings, continue to form a positive attitude towards a new academic subject, teach the ability to communicate and listen to others, and cultivate conscious discipline.

Developmental: develop the skill of using the signs of parallelism of lines and the properties of angles for parallel lines to solve problems and prove theorems; develop the skill of finding the angles of triangles at two given angles, with given proportionality of the angles; develop the skill of using the theorem on the sum of the angles of a triangle and its corollary to solve problems; develop the skill of finding the angles of triangles given two given angles, given the proportionality of the angles, given various elements of triangles (equal sides, angles), the ability to find the angles of a triangle if the angle is given bisector, and find the 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, handouts (whole paper triangles and cut cardboard ones), a large paper triangle for the teacher to demonstrate on the board how to find the sum of angles triangle, cards for independent work

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

During the classes:

Stage

lesson

Teacher activities

Student activities

Org.

moment

Homemadeexercise

Learning new material

(Practical work)

Learning new material

Exercise and entertainment. moment

Consolidation of the studied material

Summarizing

Open your diaries and write down your homework: learn notes 22, (p. 33) Numbers for homework 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 a completely different matter -

Fast, accurate and skillful

There are sides to it - there are three of them,

And there are three corners in all of them,

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

Connected by one number.

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

Open your notes, write down: note No. 22. Sum of angles of a triangle (slide 3).

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

Right. Draw a triangle. We pick up a protractor.

And we begin to measure the angles of the drawn triangle one by one (slide 5). We will measure the angles together with you.

We take a protractor, apply it to the first angle to be measured 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: angle, for example, B is equal to ... degrees. I got 80 0 . What angles did you get?

And I do the same with the other corners.

Did you find all the corners?

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

So what do we do with our triangle angles?

Right. Add up your resulting angles, raise your hands and say how many you got.

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

Not everyone probably guessed right away - we need to add all the corners. How to do it?

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

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

Have you already measured the triangles twice and still get 180?

(If not, I give an additional triangle). Check to see if a triangle can be made from these parts?

Did everyone succeed?

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

(slide 8)

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

What did we get?

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! Let's write it down (slide 9).

Historical background (slide 10).

Now we will prove this theorem. You need to write down this evidence and review it if something is not clear. If it’s difficult, come to additional classes - today 6-7 lessons.

We write down: proof (slide 11)

What has been given to us and what needs to be proven?

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

Let'slet's prove this theorem , using the properties of angles known to you and me for parallel lines and transversals. To do this, construct a straight line through vertex BA parallel to the base - side AC.

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

We write down:

Let's builda || AC,BÎ a.

How many secants are there for parallel lines? Name them.

Let's look at one secant first.

What can we say about the angles at our parallel lines and secant AB.

Let's write this down.

Now consider another secant of the sun. What can we say here about angles at parallel lines?a || A.C.and secant sun?

Right. Let's write it down.

Now let's look at the developed angle B. What is this angle equal to?

Right. What else is it equal to? The sum of which angles?

That's right, this is very clearly visible in the figure.

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

Those. what did you get?

Have you proven the theorem?

Physical exercise (slide 12).

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

№ 20 (slide 14) – we decide orally. We don’t close notebooks with notes.

Can two angles of a triangle be right?

Are two angles obtuse?

One is straight and the other is stupid?

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

Those. There must be at least... acute angles in any triangle. ?

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

Corollary of the theorem:

Any triangle has at least two acute angles.

Oral work with tasks (slides 16-18)

Guys. We go 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 - use it to solve problem 18, 19.

21 orally.

22 – there is a drawing on the board with a r/b triangle, using it we solve the problem.

№ 25 at the board with the same drawing.

(20 slide)
(21 slides)

Guys, let's remember what we learned today.

What is the sum of the angles of any triangle?

Tell me, how many acute angles should there be at least in any triangle?

Can there be 2 stupid ones?

Well done!

I'll see you at the next lesson after the bell.

Open diaries and write down homework.

They open their notes and write.

Any.

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

Yes.

Sum of angles of a triangle.

Let's add it up. And let's find what the sum is equal to.

They count and say the answers. Everyone should be 180.

They look at the triangles, try to fold them, and come to a solution.

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

The unfolded angle is 180 degrees.

Yes.

Yes.

Yes, it adds up.

Exactly.

180.

Add them together to show their total.

Again, the rotated angle is 180.

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

Write down the theorem.

They listen and ask questions.

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

Write down the given information and draw a picture:

Given:

ABC

Prove:

РА+РВ+РС=180°

They build behind the teacher (the teacher scrolls through the animation on the slide).

Two? AB and BC.

Ð 4= Ð 1 , like crosswise angles with parallel linesa || A.C.and secant AB.

Ð 5= Ð 2, like cross lying angles with parallel linesa || A.C.and secant sun.

180, because it's unfolded.

Ð 4 + Ð 3+ Ð 5 = 180°, becauseÐ B – expanded (Ð B = 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.

They proved it.

Repeat the exercises (physical training) after the teacher.

No.

No.

No.

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

Two!

Recorded from dictation or from a slide.

They solve puzzles.

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

180 degrees.

At least two sharp corners.

No.

Continuation of the topic

Reinforcing the material learned

Self-work

Summarizing

So, how many angles are there in a triangle?

Then since two angles are always acute, 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...

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

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 board and in notebooks (slide 27)

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

Guys, tell me, what did we learn and remember today?

Well done!

Lesson grades are given...

anyone.

Acute angular.

Rectangular.

Obtuse.

Obtuse, because there is an obtuse angle.

Acute angular, because all corners are sharp.

Rectangular, because 180 – 40 -50 = 90.

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

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

By the angle sum theorem D:
RA = (180 0 – РС) : 2 =
= (180 0 – 90 0 ) : 2 = Ð45 0

Solve problems with the help of a teacher.

Write independent work on 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 the ruler and compass.

Task 2 .

Given:

Find:

Ð1 and Ð 2Solution:

Task 3.

Given:

Find:

Ð1 and Ð 2Solution: