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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 boy who was born 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!

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).

Material for a geometry lesson in 7th grade

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“Lesson topic: SUM OF ANGLES OF A TRIANGLE”

MBOU "ZOLOTOPOLENSKAYA COMPREHENSIVE SCHOOL"

KIROV DISTRICT OF THE REPUBLIC OF CRIMEA

Lesson in 7th grade on the topic

"Sum of the angles of a triangle"

Teacher: Antipova Galina Ivanovna

Lesson topic: Sum of angles of a triangle.

Lesson type : A lesson in learning new material.

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

Developmental goal: improve the ability to think logically and express your thoughts out loud, develop logical thinking, will, emotions;

Educational purpose: to cultivate in students the desire to improve their knowledge; cultivate interest in the subject.

During the classes

Organizing time

(The teacher holds a triangle in his hands ) 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 line, and 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.

Topic of our lesson: Sum of angles of a triangle. (Slide 1)

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

Goals: ( Slide 2)

Updating basic knowledge.(Slides 3-9)

3.Learning new material

Practical work(entering into the topic of the lesson, preparing for the perception of new material)

Teacher. Answer the question: What tool can you use 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 of practical work on your tables. Take them, use a protractor to measure the angles of the triangles and write the results in tables.

№ p/p				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 arbitrarily, the angles in the triangles were different, but the results were the same for everyone.

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

Teacher. What conclusion can we draw after 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 mutual checking

1. Is there a triangle with angles:

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°, the other is 100°

3.Find the angles of an equilateral triangle.

4. (Slide 25)

Lesson summary. Reflection. (Slide 26,27)

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

Have we achieved it?

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"SUM OF ANGLES OF A TRIANGLE"

C sum of angles of a triangle

Mathematic teacher

Municipal educational institution "Zolotopolenskaya secondary school"

Kirovsky district, Crimea

Antipova Galina Ivanovna

Goals:

• formulate and prove a theorem on the sum of the angles of a triangle;
• consider the tasks of applying proven

Let's repeat studied

Adjacent angles

60 

 AOC+  BOC=

Vertical angles are equal

Amount of one-sided

angles equal to 180 0

Relevant

angles are equal

Crossed angles are equal

a ll b

Calculate all angles.

Practical work

Study

 .

• By “tearing off” the angles of a triangle, you can show 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. p. straight line a || A.C.

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.

In the first book of the Elements, Euclid gives another proof of the theorem about the sum of the angles of a triangle, which can be easily understood with the help of a drawing.

365 –300 BC

Tasks on finished drawings .

Task No. 1

Calculate:

Problem No. 2

Calculate:

Task No. 3

Calculate:

Problem No. 4

Calculate:

Problem No. 5

Calculate:

Problem No. 6

Calculate:

Problem No. 7

Calculate:

Problem No. 8

AK - bisector

Calculate:

Homework .

• P. 3 1 , 223(b),228(b)
• № 229 (optional)

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 learning abilities.

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°)