Describe each of the triangles.
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
(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 with you wonderful property 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+ FROM |
|||
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 corners 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?
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"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 studied
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.
In the first book of the Elements, Euclid sets out another proof of the triangle sum theorem, which is easy to understand with the help of a drawing.
365 -300 BC
Tasks on finished drawings .
Task #1
Calculate:
Task #2
Calculate:
Task #3
Calculate:
Task #4
Calculate:
Task number 5
Calculate:
Task number 6
Calculate:
Task #7
Calculate:
Task number 8
AK - bisector
Calculate:
Homework .
- P. 3 1 , 223(b),228(b)
- № 229 (optional)
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 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 parallelism of 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 proportionalities of the angles; 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 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 a teacher to demonstrate on the board finding the sum of the angles of a triangle, cards for independent work
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?
Good. Now we again need to show that the sum of the angles in a triangle is what?
(slide 8)
Excellent! 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).
History reference(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).
Letters on the slide 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 use it to solve problem 18, 19.
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.
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.
No.
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 write independent work 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:
Objectives: 1. Introduce the concepts of acute, right and obtuse triangles. 2. With the help of the experiment, bring the children to the formulation of the theorem on the sum of the angles of a triangle, prove it and teach how to apply the knowledge gained in solving problems. 3. Development cognitive activity, thinking, attention. 4. Education industriousness
OBJECTIVES: 1. Consolidate knowledge on the topics: triangle, parallel lines, types of angles; 2. To consolidate the skills of using a protractor; 3. Develop the ability to use a textbook; 4. Develop students' mathematical speech; 5. To form the ability to analyze the material and draw conclusions; 6. To nurture: interest in the subject, the ability to bring things to the end, confidence in their abilities in school.
Lesson plan: 1. Organizational moment. 2. Repetition. 3. Oral work. 4. Statement of the problem, determination of ways to solve it. 5. Putting forward a hypothesis. 6. Confirmation of the hypothesis. 7. Proof of the theorem. 8. Solving assignments to consolidate the studied theorem. 9. Summing up the lesson (reflection), homework.
Lesson progress: 1. Organizational moment Today our class will turn into a "scientific Research institute", and you will become "his employees." And we will not only get acquainted with the work of the “research institute”, but we will also make discoveries ourselves! And so: "research institute" has subdivisions: 1. Laboratory of experiments. 2.Laboratory of scientific evidence. 3. Laboratory testing.
2.Repetition On previous lessons we studied the signs of parallel lines and the properties of angles with parallel lines. And today in the lesson, the knowledge gained on this topic will help to make a discovery. Define parallel lines (Two lines in a plane are called parallel if they do not intersect)
Formulate the signs of parallelism of lines (If at the intersection of two lines of a secant, the lying angles are equal, then the lines are parallel; If at the intersection of two lines of a secant, the corresponding angles are equal, then the lines are parallel; If at the intersection of two lines of a secant, the sum of one-sided angles is 180 °, then the lines are parallel ;)
Formulate the property of angles at parallel lines (If two parallel lines are intersected by a secant, then the crosswise lying angles are equal; If two parallel lines are intersected by a secant, then the corresponding angles are equal; If two parallel lines are intersected by a secant, 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 one straight line, and segments connecting these points in pairs.) 2) Name the elements of a triangle. (Vertices, sides, angles.) 3) What triangles are distinguished? (On the sides: versatile, equilateral, isosceles; cards - triangles) 4) Triangles are also distinguished by the corners.
Let's make a story with you on the topic: ANGLE. To do this, use the plan written on the screen. An angle is a figure, ... (An angle is a figure formed by two rays coming out of one point. The rays are called the sides of the angle, and the point is called the vertex.). 2. If ..., then the angle is called ... (If the angle is 90 °, then the angle is called right. If - 180 °, then deployed. If more than 0 °, but less than 90 °, then it is called acute. If more than 90 °, but less than 180 ° is called obtuse.)
That. angles are obtuse, acute, straight and deployed. Inner corner triangle is ... The inner angle of a triangle is the angle formed by its sides, the vertex of the triangle is the vertex of its angle. So, in a triangle, the angles can be different: obtuse, acute and straight.
Laboratory of experiments Draw a corner: (3 students work at the blackboard, and the rest in place) 1 - row - obtuse; 2 - row - straight; 3 - row is sharp. Complete the drawing to a triangle. What do I need to do? (Take a point on the sides of the corner and connect them with segments.) The resulting triangles can be called: obtuse, right-angled and acute-angled. ((cards - triangles) Please note that an acute triangle has all acute angles.
Are there triangles with right and obtuse angles? With two obtuse corners? With two right angles? How to substantiate this? Make a drawing: Beams BA and SD, CT and OH. KE and PL do not intersect, which means that the triangle will not work. The sum of one-sided angles in the I case is greater than 180°, in the II case it is also greater than 180°, and in the III case 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 have done some practical work, made a justification for the fact that a triangle does not always exist. Its existence depends on the magnitude 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 sharp corner rectangular isosceles triangle? (45°)