Educational - to identify the degree of mastery of knowledge and skills by students on the topic; development of solutions of geometric problems. Developing - to develop the ability to analyze and compare; develop oral and written speech. Educational - to instill interest in geometry, the ability to conduct a cultural discussion.
1. Update the basic knowledge on the topic "Triangle". 2. Check the concept of a triangle. 3. Formulate signs of equality of triangles.4. To consolidate the material through solving problems according to ready-made drawings. 5. Learn to correctly and accurately draw up and solve problems.
Three angles: ABC, DIA, BAC.
Three sides: AC, AB, BC.
Three peaks: A, B, C.
BUT
FROM
AT
A triangle is a figure that consists of three points that do not lie on the same straight line, and three segments connecting these points in pairs. The points are called the vertices of the triangle, and the segments are called its sides.
6
4
1
9
3
8
2
7
5
10
4
1
9
3
8
2
7
5
10
6
4
1
3
8
2
7
5
10
6
9
4
1
3
8
2
5
10
6
9
7
4
1
3
2
5
10
6
9
8
7
III
I
II
On three sides
Three corners
Two corners and a side
On three sides
III
I
II
Two corners and the side between them
On two sides of the angle between them
Three corners
Two corners and a side
On two sides of the angle between them
On three sides
III
I
II
Two corners and the side between them
Three corners
Two corners and a side
Two corners and the side between them
On two sides of the angle between them
On three sides
III
I
II
Three corners
Two corners and a side
4
1
3
2
6
9
8
7
Two corners and the side between them
On two sides of the angle between them
On three sides
4
1
3
2
6
9
8
7
Two corners and the side between them
On two sides of the angle between them
On three sides
4
1
3
2
6
9
8
7
Two corners and the side between them
On two sides of the angle between them
On three sides
4
1
3
2
6
9
8
7
Two corners and the side between them
On two sides of the angle between them
On three sides
4
1
3
2
6
9
8
7
Two corners and the side between them
On two sides of the angle between them
On three sides
Testing
Testing
Answers to testing
Indicate on which of the following figures there are equal triangles, on what basis are they equal?
on two sides and the angle between them
along a side and two adjacent angles
Answers to testing
2. On what basis are triangles equal? a) on two sides and the angle between themb) on a side and two adjacent anglesc) on three sides
1
2
3
4
5
6
7
8
a
in
b
a
a
in
a
b
Three corners
YES
NO
Three corners
DOES NOT EXIST!!!
TRIANGLES ARE NOT EQUAL
"5" 19 - 21 points "4" 16-18 points "3" 10-15 points
On the topic: methodological developments, presentations and notes
A lesson on generalizing and systematizing knowledge on the topic "Signs of equality of triangles" Lesson objectives: Educational: - to consolidate, summarize and systematize the material on the topic "Triangle ...
The presentation contains material for conducting lessons on the topic "Triangle. The first sign of the equality of triangles": proof of the sign itself and a selection of tasks for its application ....
Technological map of the lesson on the topic "Equality of triangles. The first sign of equality of triangles." Geometry grade 7. Type of lesson: lesson of mastering new knowledge. UMK: Geometry 7, authors V.F.Butu...
The figure shows congruent triangles. 1. Determine which of the following entries is correct: a) ABC = PQR; b) ABC = RQP; c) ABC = PRQ. 2. It is known that AC = 5 cm, ے B = 30°. a) What side length of RQP can you specify? b) What angle RQP is known? A C B P Q R 5 cm 30°
Given ΔCDM. Given ΔCDM. a) Name the angles adjacent to side CD. b) Name the angle opposite the side CM. c) Name the angles included between the sides CM and MD, CD and DM. a) Name the angles adjacent to side CD. b) Name the angle opposite the side CM. c) Name the angles included between the sides CM and MD, CD and DM.
Is it possible to complete a triangle if three of its elements are known: two sides and the angle between them? Compare elements of two triangles: EF = MN ED = MS FED = NMS FED = NMS Is it possible to compare triangles without overlapping them? Is it possible to compare triangles without overlapping them?
Given: ABC, A 1 B 1 C 1 AB=A 1 B 1 AC=A 1 C 1 A = A 1 Prove: ABC = A 1 B 1 C 1 sides of equal angles A and A1. The sides of the triangles AB and A1B1, AC and A1C1 are aligned, since AB=A1B1, AC=A1C1. This means that points B and B1, C and C1 will also coincide. Therefore, BC = B1C1 and ABC is fully compatible with A1B1C1. The theorem is proved The theorem is proved.
Consider AOD and BOC It is known that AO = OB (by condition) CO = OD (by condition), ۦ AOD = ۦ BOC (vertical) AOD = BOC by the FIRST (SUS) sign of equality of triangles., Segments AB and CD intersect at a point Oh, which is the middle of each. Prove: AOD = BOC Given: AB CD = O; AO=OB; CO=OD. Prove: AOD = BOC Proof D A B C O Problem 97 O B D A C 2 Consider ABC and CDA. AC - general AD=BC, DAO= BCO - according to the proven. So ABC = CDA on two sides and the angle between them. So AOD = COB on two sides and the angle between them. Therefore, AD=BC, DAO=BCO. Solution: 1 Consider AOD and COB. AO=OC (by condition) BO=OD AOD= BOC as vertical
If you want to learn how to swim,
then boldly enter the water,
If you want to learn how to solve problems,
then solve them.
D.Poya
Lesson Objectives:
- generalize, expand and deepen knowledge about the triangle;
- Introduce the concept of a theorem and proof of a theorem;
- Prove the first criterion for the equality of triangles;
- Learn to solve problems on the application of the first sign of the equality of triangles.
Template for creating presentations for mathematics lessons. Savchenko E.M.
In geometry lessons, it is very important to be able to look and see, notice and
note different features
geometric shapes.
Template for creating presentations for mathematics lessons. Savchenko E.M.
BUT
FROM
O
AT
What shape is called an angle?
Determining the bisector of an angle.
What are the angles?
Definition of adjacent corners and their properties.
- What is the PDE angle called?
- What is its degree measure?
- How many angles does PDE consist of? name
these corners.
0
Definition of vertical angles and their properties.
Given: 0
Find:
Definition of a triangle, its elements; determining the perimeter of a triangle.
P
FROM
BUT
Name:
- Sides of a triangle
2) Angles of a triangle
3) the angle lying between the sides DN and DL
4) the angle lying between the sides DL and LN
5) the angle lying between the sides LN and ND
From three points it consists of century to century
Because that's how the man came up with it.
The points do not lie on a straight line,
Even though they want to go home to each other.
Three segments unite them all their lives
And they are always connected to each other.
And those points are called vertices,
And stretches of those sides don't forget.
The surface consists
from triangles.
Plato
Template for creating presentations for mathematics lessons. Savchenko E.M.
- In ancient art, images of an equilateral triangle were widespread.
- Chiefs of the North American tribes
Indians wore a symbol of power on their chest: an equilateral triangle with a dot in the center.
- In Africa, women adorned themselves with large plates of equilateral triangles.
Template for creating presentations for mathematics lessons. Savchenko E.M.
Triangles in bridge construction.
http://mirrorsoul.narod.ru/pictures/P1010096_2.htm
High voltage power lines.
Triangles make designs reliable.
http://orsk.ru/index.php?option=com_content&task=view&id=4359&Itemid=110
Starting a game of billiards, you need to arrange the balls in the form of a triangle. To do this, use a special triangular frame.
http://www.bogato.info/index/?node_id=2822
http://www.labirint-shop.ru/screenshot/189362/1/
The arrangement of pins in the bowling game in the form of an equilateral triangle.
http://www.akatuy.ru/bowling.asp?page=./6939/6952/7040/7062
http://rnd.onegintime.ru/game.html?game=3&count=90&limit=10&page_num=8
Triangle- constellation of the northern hemisphere of the sky, contains 25 stars visible to the naked eye.
Bermuda Triangle- an area in the Atlantic Ocean where supposedly mysterious disappearances of ships and aircraft occur. The area is bounded by lines from Florida to Bermuda, then to Puerto Rico and back to Florida via the Bahamas.
bermuda
islands
Florida
http://ru.wikipedia.org/wiki/%D0%91%D0%B5%D1%80%D0%BC%D1%83%D0%B4%D1%81%D0%BA%D0%B8%D0 %B9_%D1%82%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA
Puerto Rico
PHYSICAL MINUTE
In geometry, every statement whose validity is established by reasoning is called theorem , and the reasoning itself is called theorem proof .
The arguments given earlier about the property of adjacent and about the equality of vertical angles were proofs of theorems, although we have not yet called them that.
If a two sides and the angle between them
triangles are respectively equal
two sides and the angle between them
another triangles, then
triangles are equal.
Theorem:
FROM
C1
1
2
AT
BUT
IN 1
A1
Theorem:
(condition) ∆AB C , ∆A₁B₁C ₁, AB = A₁B₁,
AC \u003d A₁C₁, ∠A = ∠A₁.
Given:
(conclusion) ∆AB C \u003d ∆A₁B₁C ₁,
Prove:
FROM
From ₁
1
2
AT
BUT
A₁
B₁
Proof.
Since ∠A = ∠A₁, then ∆AB C can be superimposed on ∆А₁В₁С ₁ so that the vertex A coincides with the vertex A ₁.
Since AB \u003d A₁B₁, AC \u003d A₁C₁, then side AB will be combined with side A₁B₁, and side AC with side A₁C₁.
FROM
Therefore, points B and B₁ will coincide,
C and C₁, therefore compatible
side BC with side B₁C₁.
FROM ₁
AT
BUT
The two triangles are called equal, if, when applied, they are combined.
AT ₁
BUT ₁
So ∆AB C = ∆А₁В₁С ₁, which was to be proved.
Problem solving
Segments AE and DC intersect at point B, which is
the middle of each. a) Prove that ∆AB C = ∆EV D ;
b) find angles A and C in ∆AB C , if in ∆EV D ∠ D = 47°, ∠ E = 42°.
C
E
?
4 2 °
B
4 7 °
?
Solution
A
D
- AB = BE, and CB = AT D, since, by condition, point B is the midpoint of the segments AE and DC . ∠CBA = ∠EB D, since these angles are vertical. According to the first criterion for the equality of triangles ∆ AB C = ∆ E AT D .
2) In equal triangles against respectively equal
sides are equal angles, so ∠ A = ∠ E = 42° ,
FROM = ∠ D = 47°,
Answer: ∠ A = 42° , ∠С =47° .
- item 15- teach (proof of the theorem)
- Solve #93, #95
- Draw your mood triangle
- Draw your mood triangle
- Draw your mood triangle
Teacher of mathematics "Education Center No. 18" Postnikova Elena Alekseevna
slide 2
Lesson Objectives
To systematize and consolidate knowledge, skills and abilities on the topic “Signs of equality of triangles”.
slide 3
Equal Triangles
Triangles are called equal if their corresponding sides and angles are equal.
slide 4
Signs of equality of triangles
The first sign of equality of triangles: If two sides and the angle between them of one triangle are equal, respectively, to two sides and the angle between them of another triangle, then such triangles are congruent
slide 5
The second sign of equality of triangles: If the side and angles adjacent to it of one triangle are equal, respectively, to the side and angles adjacent to it of another triangle, then such triangles are equal
slide 6
The third sign of the equality of triangles: If three sides of one triangle are equal, respectively, to three sides of another triangle, then such triangles are congruent
Slide 7
Properties of Equal Triangles
Equal triangles have all the corresponding elements equal (sides, angles, heights, medians, bisectors) Equal triangles have equal angles opposite equal sides, and equal sides opposite equal angles.
Slide 8
Dictation
1. Indicate the numbers of the figures in which the triangles are equal in: first feature: second feature: third feature:
Slide 9
2. Triangles DFG and PQR are equal. It is known that DFG = PQR; FGD=QRP; DF=7cm, DG=14cm. What are the corresponding sides of triangle PQR? 3. In equal triangles DEA and FEB: D= F. Determine the form ∆AEB. E D A B F F G D R P Q
Slide 10
Answers to the dictation
1. On two sides and the angle between them: 2,8,9,13. Along the side and adjacent angles: 3,6,12,14. On three sides: 1,10,11. 2.PR=14, HQ=7. 3. ∆AEB - isosceles.
To use the preview of presentations, create a Google account (account) and sign in: https://accounts.google.com
Slides captions:
The second sign of the equality of triangles.
Objectives: to study the second sign of the equality of triangles, to develop the skills of using them in solving problems. systematize, expand and deepen students' knowledge of the triangle, consolidate skills and abilities in solving problems using definitions and theorems on this topic. Developing: to develop the mathematical speech of students, their memory, attention, observation, the ability to compare, generalize, reasonably draw conclusions, develop the ability to overcome difficulties in solving problems, as well as the cognitive interest of students. Educational: education of control and self-control skills, education of correct self-esteem, accuracy, attentiveness, positive attitude to learning.
Lesson 1 Course of the lesson 1. Organizational moment 2. Repetition 3. Learning new material 4. Consolidation from the material 5. Homework
"Geometry is the most powerful tool for the refinement of our mental faculties and enables us to think and reason correctly." Galileo Galilei
Task 1: Fill in the gaps so that you get sentences corresponding to this drawing. 1. Degree measure of angles
Task 2: Highlight the condition and conclusion in the listed statements. 1. If the triangles are equal, then the corresponding angles are equal in them. Condition: Conclusion: 2. If the triangles are equal, then their perimeter is equal. Condition: Conclusion: 3. There are two equal sides in an isosceles triangle. Condition: Conclusion: 4. In an isosceles triangle, the angles at the base are equal. Condition: Conclusion: 5. In an isosceles triangle, the medians drawn to the sides are equal to each other. Condition: Conclusion:
Oral: Insert the appropriate words into the sentences to make correct statements. 1. The perimeter of an equilateral triangle is three times the length of its side 2. If triangle ABC and MNK are equal, then in triangle ABC there is an angle equal to angle NMK 3. If AK and BN are medians of triangle ABC, then the third median of this triangle will pass through the intersection point median AK and BN . 4. If two sides and the angle between them of one triangle are respectively equal to two sides and the angle between them of another triangle, then such triangles.
If a side and adjacent angles of one triangle are respectively equal to a side and adjacent angles of another triangle, then such triangles are congruent. Given: ∆ ABC , ∆ MNK AB = MN ,
Consolidation of the studied material. Problem No. 1. Segments AB and CD intersect at point O . Prove that triangles ACO and DOB are congruent if it is known that angle ACO is equal to angle DBO and BO = CO .
Solution: Consider ∆ ACO and ∆ DBO: BO = CO (by convention)
Problem 2. Segments AC and BD intersect at point O . Prove that triangles BAO and DCO are equal if angle BAO is known to be equal to angle DCO , AO = CO . .
Solution: Consider ∆ BAO and ∆ DCO . AO = CO (by condition)
In class No. 121, No. 123 Homework: item 19, question 14 p. 50, No. 122, No. 124
