Straight line and segment. presentation for a geometry lesson (grade 7) on the topic
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Slide captions:
Galileo Galilei “Nature speaks the language of mathematics: the letters of this language are circles, triangles and other mathematical figures”
Geometry is one of the most ancient sciences, originating more than 4000 years ago. The word geometry is of Greek origin. Literally it means "land surveying". "geo" - earth in Greek, "metreo" - to measure
This science, like others, arose from human needs: it was necessary to build temples, dwellings, lay roads and irrigation canals, determine boundaries land plots and their sizes. The aesthetic needs of people also played an important role: to paint pictures, decorate clothes and homes. All this contributed to the acquisition and accumulation of geometric information. At the time of the birth of geometry, the rules were derived on the basis of information and facts obtained experimentally, so science was not accurate. Gradually, geometry became a science in which most facts are established through inference, reasoning, and evidence.
The first who began to obtain new geometric facts using reasoning (evidence) was the ancient Greek scientist Thales (VI century BC). Thales (ancient Greek Θαλῆς ὁ Μιλήσιος, 640/624 - 548/545 BC) - ancient Greek philosopher and mathematician from Miletus (Asia Minor). Representative of Ionic natural philosophy and founder of the Milesian (Ionian) school, with which the history of European science begins. Traditionally considered the founder of Greek philosophy (and science)
Greatest influence All subsequent development of geometry was influenced by the works of the Greek scientist Euclid. In the 3rd century. BC. he wrote the essay “Principia”, and for almost 2000 years geometry was studied from this book, and the science was named Euclidean geometry in honor of the scientist. Euclid is the first mathematician of the Alexandrian school. His main job“Principia” contains a presentation of planimetry, stereometry and a number of questions in number theory; in it he summed up the previous development of ancient Greek mathematics and created the foundation for the further development of mathematics.
Geometry planimetry stereometry Part of geometry that deals with figures on a plane (straight line, line segment, ray, angle, polygon) Part of geometry that deals with figures in space (ball, cube, cylinder, pyramid) Geometry is the science that deals with the study of geometric figures
Draw a straight line. How can it be designated? 2. Mark point C, which does not lie on this line, and points D, E, K, lying on the same line. 3. Using symbols of belonging, write down the sentence: “Point K belongs to line AB, point C does not belong to line a.”
Draw two intersecting lines. Mark the lines and the point of intersection. How many common points can two lines have in common? Two lines either have one common point or have no common points.
2. Mark two points A and B. Draw a line passing through these points. 1. Mark point A. Draw three lines a, b and c passing through this point. How many lines can be drawn through a given point A? Draw another line passing through these points. How many lines can be drawn through two points? Can you draw a straight line through any two points? Through any two points you can draw a straight line, and only one. Through a given point A you can draw many straight lines.
The part of the line bounded by two points is called Segment A and B - the ends of the segment AB
1. Draw a straight line, mark it with the letter a. Mark points A, B, C, D lying on this line. Write down all the resulting segments 2. Draw lines m and n intersecting at point K. On line m, mark point M, different from point K. a) Are lines KM and m different lines? b) Are the lines KM and n different lines? c) Can straight line n pass through point M?
1. What is the meaning of the technique “Hanging a straight line”? 2. Where is this technique used in practice? 3. Is it possible to use this technique in educational activities?
1st level of difficulty: 1. No. 2, 5, 6 (textbook) 2nd level of difficulty: 1. How many points of intersection can three straight lines have? Consider all possible cases and make appropriate drawings. 2. Three points are given on a plane. How many lines can be drawn through these points so that at least two of these points lie on each line? ? Consider all possible cases and make appropriate drawings.
1. What is the name of the science that deals with the study of geometric figures 2. What is the name of the part of geometry in which figures on a plane are considered 3. What is the name of the part of geometry in which figures in space are considered 4. How many lines can be drawn through two points? 5. How many points of intersection can two straight lines have?
Textbook: paragraphs 1, 2; questions 1-3 (p. 25) Textbook: No. 1, 3, 4, 7. Additional task: How many different lines can be drawn through four points? Consider all cases and make appropriate drawings.
On the topic: methodological developments, presentations and notes
Introductory geometry lesson in 7th grade "A brief history of the origin and development of geometry. Basic geometric information"
Introductory geometry lesson in 7th grade using multimedia" Short story the emergence and development of geometry. Basic geometric information"Type: combined, with...
Municipal budgetary educational institution "Nizhneshitsinskaya secondary school Sabinsky district of the Republic of Tatarstan"
Methodological development open lesson geometry in 7th grade Topic: Basic geometric information. Dots. Direct. Segments
Mathematics teacher Gulyusa Airatovna Gafiyatova
Saba 2013
Lesson type:lesson - introduction to a new subject.
Methods and techniques for teaching a lesson:
1.Working with the textbook
2. Frontal work with the class
3.Individual work with students.
Lesson objectives:
1.
Educational:
acquaintance with the structure, basic concepts and history of the development of geometry.
2.
Educational:development of spatial imagination, creative thinking, cognitive interest of students, interdisciplinary connections, culture of mathematical speech.
3.
Educational:nurturing respect among students for each other in the process of educational activities, self-control and self-esteem, respect for educational work
Equipment:interactive whiteboard, computer, models of geometric shapes, landscape sheets, colored markers, reference notes. Lesson structure
DURING THE CLASSES
I.Organizing time
Teacher:- Hello guys! Sit down! Today we are starting to study a new subject - geometry. You probably have questions: -What is “geometry”? What is she studying? Teacher: Geometry is integral part big science - mathematics. It would be wrong to say that until now you have not studied geometry at all and know nothing about it. You have often encountered triangles and pyramids, squares and cubes, circles and balls. Maybe not much, but you know something about these bodies and figures, you have a good idea of what they look like, and you understand that they all have to do with geometry. The statement that we are beginning to study geometry means, first of all, that we are beginning a systematic course in geometry. This, in turn, means that we will gradually, step by step, build a geometric theory, consistently proving our statements, deriving them from those already known in accordance with mathematical laws. First of all, what is geometry? Have you ever heard the word “geometry”? You have been studying the subject “geography” since the sixth grade. And you probably know what the word “geo” means. What about “metrics”? (Students' answers) The word “geometry” is Greek, it is composed of two parts “geo” and “metry” and is literally translated into Russian as “land-measurement”.
Teacher:Let's continue our fairy tale. And Dunno has more questions:
- Why did the same teacher come who taught mathematics last year? A very smart teacher probably knows several subjects? And whoever came up with this - geometry, now suffer, learn another subject.
Teacher: - Yes, because geometry is only one of many branches of mathematics. The word "mathematics" comes from the ancient Greek μάθημα (máthēma), which means studying, knowledge, the science.
Mathematics how academic discipline
is divided into several sections: 1.
Arithmetic (this section is studied in primary and 5-6 grades.) 2.
Elementary algebra and elementary geometry. Therefore, at school mathematics, algebra and geometry are taught by one teacher, a mathematics teacher. II.Familiarization with historical material
-And, if we look at the history of geometry, we will see a lot of interesting things. (Student's speech) How did geometry come about? As Eudemus of Rhodes said: “Geometry was discovered by the Egyptians and arose in the measurement of the earth. This dimension was him
necessary due to the flooding of the Nile, which constantly washed away the borders. It is not unusual that this science, like others, arose from human needs.” This means that geometry arose from the practical activities of people.
It was necessary to build dwellings, temples, build roads, irrigation canals, establish the boundaries of land plots and determine their sizes.
Satisfying their aesthetic needs, people decorated their homes and clothes with ornaments. Mastering the world around them, people became familiar with geometric shapes, they began to learn to measure areas, lengths, and volumes.
Occupations of people in ancient times:
ü
Construction of temples and houses;
ü
Decorating dishes and dwellings with ornaments;
ü
Marking the ground, measuring distances and areas, volumes of vessels.
Several centuries BC, in Babylon, China, Egypt, and Greece, initial geometric knowledge already existed, which was obtained experimentally and then systematized. The first who began to obtain new geometric facts using reasoning was the ancient Greek mathematician Thales (6th century BC). Gradually geometry becomes a science. WITH V century BC, the attempt of Greek scientists to bring geometric facts into a system begins. The work of the Greek scientist Euclid, Elements, was the main book used to study geometry for almost 2000 years. The geometry set forth in it came to be called Euclidean geometry.
Euclid is a famous ancient Greek mathematician, born in Athens around
Founders of geometry:
Plato founded a school whose motto is “Those who do not know geometry are not admitted!” (2400 years ago), Thales of Miletus (640-
Teacher:If you want to learn more about the history of geometry and get to know the founders of geometry better, you can click on the names of famous mathematicians and find out detailed information.
Watch a video about the importance of geometry.
III.Learning new material. Dive into the problem
-Pay attention to the board. There are geometric shapes there. And we need to divide them into two groups. What two groups will we divide them into?
Yes, right. By what principle are these geometric figures written in two different groups? (1 on the plane, 2 in space). The part of geometry that deals with figures on a plane is called planimetry, and the other part of geometry that deals with figures in space is called stereometry. We will begin our study of geometry with planimetry. Teacher:
The topic of today's lesson: “Dots. Direct. Segments." Write the topic of the lesson in your notebook. The tools needed for construction are a pencil and a ruler.-In geometry lessons we will need: Pencil, ruler, compass, protractor. And therefore, every student should have these tools in geometry lessons. Now you and I will complete tasks.The largest building is made up of small bricks, and complex geometric shapes are made up of the simplest shapes. One of them - dot.
The point is the result of an instant touch, an injection.Teacher:
Points are designated in capital Latin letters. In our case, we marked points A. 2.Draw a straight line. How can it be designated? (Direct a or MR) 1.
Mark point C, which does not lie on the given line, and points D, E. K , lying on the same line.. WITH
Teacher:
In mathematics, there are special symbols that allow you to briefly write down a statement. The symbols € and € are called membership symbols. They mean “belongs” and “does not belong” respectively.1.
Using membership symbols, write down the sentence “Point P belongs to line AB, but points K and C do not belong to line a.”2.
(P €AB, K, C € a)
3.
Using the drawing and membership symbols, write down which points belong to the line c and which do not?- How many lines can be drawn through a given point A? (Through a given point A, many straight lines can be drawn.)- How many lines can be drawn through two points? (one straight line)- Can you draw a straight line through any two points? (Yes)- What conclusion can we draw?So, through any two points you can draw a straight line and, moreover, only one.
6.Draw lines AB and MT intersecting at point O.In order to briefly write down that lines AB and MT intersect at point O, use the symbol ∩ and write it like this: AB∩MT=O
7. On straight line a, mark points A, B, X, Y in sequence. Write down all the resulting segments.
Physical education minute
Teacher:
And now it's time to rest. I will tell you geometric figures, if they are viewed on a plane, then you must sit down, and if they are viewed in space, jump on the spot.Straight line, cube, broken line, cylinder, segment, ball, ray, cone, rectangle, pyramid, square, parallelepiped. I.Solving fun problems.
Solve the puzzle
I.Checking the degree of mastery of the material
2.
Crossword solution
Test in the program
Excel
VI. Summing up the lesson
- What does geometry study?- What can we say about two lines passing through the same two points?- How many points in common can two straight lines have?Homework assignment
Teacher:
Open your diaries and write down your homework: point 1, solve No. 1, 4, do all drawings only with drawing tools.Choose a face that suits your mood after the lesson and draw it in your notebook. The lesson is over. All the best, goodbye.
on the topic: “Initial concepts of planimetry. Straight line and segment. Beam and Angle."
Lesson type - ONZ.
Lesson objectives:
I Educational:
Systematize information about the relative positions of points and lines;
Consider the properties of a straight line;
Learn to designate points and lines in a drawing;
Introduce the concept of a segment;
Remind students what a ray and an angle are; introduce the concepts of internal and external areas of an undeveloped angle, introduce various notations for rays and angles;
Start learning the ability to isolate from the text of a geometric problem what is given and what needs to be found, reflect the situation given in the conditions of the problem and arising in the course of solving it in a drawing, briefly and clearly write down the solution to the problem.
II Developmental:
Development of students' cognitive interest;
Development of students' memory;
Developing students' curiosity.
III Educational:
Mental education (formation of logical, abstract, systematic thinking; mastery of intellectual skills and mental operations - analysis and synthesis, comparison, generalization);
Formation of such personality qualities as organization, discipline, accuracy.
IV Meta-subject: development of cognitive interest in the subject, the ability to find analogies and connections with other sciences.
During the classes
I. Organizing time.
Teacher: “The bell rang, the students are ready for the lesson. Let's start our lesson."
II. Report the topic of the lesson with a note in a notebook. Setting lesson goals for students.
III. Introductory conversation about the emergence and development of geometry.
Conversation plan:
1. The origin of geometry.
2. From practical geometry to the science of geometry.
3. Geometry of Euclid.
4. History of the development of geometry.
5. Geometric shapes.
Slides No. 2-5.
Geometry arose as a result of the practical activities of people: it was necessary to build houses, temples, lay roads, irrigation canals, establish the boundaries of land plots and determine their sizes. Translated from Greek, the word “geometry” means “land surveying” (“geo” means earth in Greek, and “metreo” means to measure). This name is explained by the fact that the origin of geometry was associated with various measuring works.
The aesthetic needs of people also played an important role: the desire to decorate their homes and clothes, to paint pictures of the life around them. All this contributed to the formation and accumulation of geometric information.
Several centuries BC in Babylon, China, Egypt and Greece, basic geometric knowledge already existed, which was obtained mainly experimentally, but it was not yet systematized and was passed on from generation to generation in the form of rules and recipes, for example, rules for finding areas figures, volumes of bodies, construction of right angles, etc.
There was no proof of these rules yet, and their presentation did not constitute a scientific theory. The first who began to obtain geometric facts using reasoning (proofs) was the ancient Greek mathematician Thales(6th century BC), who in his research used bending the drawing, rotating part of the figure, and so on, that is, what in modern geometric language is called movement.
Gradually, geometry becomes a science in which most facts are established through conclusions, reasoning, and evidence.
Attempts by Greek scientists to bring geometric facts into a system began already in the 5th century. BC e. The greatest influence on all subsequent development of geometry was exerted by the works of the Greek scientist Euclid, who lived in Alexandria in the 3rd century. BC e. Euclid's work “Elements” served as the main book for studying geometry for almost 2000 years. In the “Principles” the geometric information known by that time was systematized, and geometry first appeared as a mathematical science.
This book was translated into the languages of many peoples of the world, and the geometry presented in it began to be called Euclidean geometry.
The school geometry course is divided into planimetry And stereometry. The branch of geometry that studies the properties of figures on a plane is called planimetry (from the Latin word “planum” - plane and the Greek “metreo” - I measure). In stereometry, the properties of figures in space, such as a parallelepiped, a sphere, a cylinder, and a pyramid, are studied. We will begin our study of geometry with planimetry.
In geometry, the shapes, sizes, and relative positions of objects are studied, regardless of their other properties: mass, color, etc. Abstracting from these properties and taking into account only the shape and size of objects, we come to the concept of a geometric figure.
Geometry not only gives an idea of shapes, their properties, and relative positions, but also teaches one to reason, pose questions, analyze, draw conclusions, that is, to think logically.
In mathematics lessons you became acquainted with some geometric figures and can imagine what point, straight line, segment, ray, angle, how they can be located relative to each other.
IV. Presentation of new material.
Slide number 7.
Construct two pairs of points and draw lines through the points using a ruler. How many lines can be drawn through two different points?
The first characteristic property of the line is established.
Slide number 8.
The student concludes that there is only one straight line passing through two different points.
The teacher introduces students to the sign of belonging and 
. The main purpose of the slide is to encourage children to identify the second property of a straight line: you can construct any point on it, a straight line has “as many” points as you like. Students naturally accept replacing the phrase “as many points as you like” with the phrase “infinitely many points.”
Slide number 9.
Working with this slide, students realize that the model of the straight line has not yet been obtained: the construction should be continued by moving the ruler to the right or left. The question arises: how far can you “go” with such a construction? The clarity of the operation prompts the answer: as far as you like, infinitely far, both to the right and to the left. This means that the line is infinite, this is its second property. That is why, as the textbook says, “from any point on a straight line you can lay off segments of any length in both directions.” The teacher reads a phrase from the textbook: “A straight line, unlike a segment, has neither beginning nor end.” But a circle has neither beginning nor end. Maybe a straight line “looks” like a circle? Now we should tackle the second question of the slide: will a crocodile and a bee meet, constructing a straight line, one to the left, the other to the right. Usually children answer: “They won’t meet, a straight line is not like a circle, it is not closed” (another answer is also logical, but students may not be aware of it).
If in this visual way we find out the property of non-closedness of a straight line, then students will be able to then understand how a ray is “produced” and see the origin of the concept.
Slide number 10.
This slide is shown to summarize. The ability to refer to this or that property will indicate that the concept of a straight line has been formed in the student’s thinking.
Students performing physical education to improve cerebral circulation:
And physical exercises for the eyes:
Slide number 11.
It is natural to ask students: is it possible to explain how a segment is obtained? We use a slide. In this case, the term “between” is perceived by intuition.
Slides No. 12 and 13.
Students solve problem No. 5 and problem No. 7 (the text of the problems is given on the slides). These problems can be solved along with the teacher's comments (or the answer can be shown so that the student checks his solution).
Slide number 14.
The teacher introduces the concept of a ray. A straight line AB and a point O belonging to it are constructed. Drawing received. The teacher suggests coloring point O and part of the line lying to the right of point O, for example, in pink color. The result is a new figure - a ray. Its production is described on the “beam” slide. The rays are constructed, the notation is introduced, and the children find out why the ray is infinite away from the beginning. The ray is obtained as the union of a point on a line and one of the parts into which this point divides the line.
Slide number 15.
To consolidate the concept, children complete task No. 8 of the textbook (the text of the task is given on the slide).
Slide number 16.
The formation of the concept of an angle is carried out in approximately the same way as the concepts of intersection and union of figures (for example, as the ray was previously introduced). Students build two different beams with a common beginning. Remembering that the ray is infinite, children find out that the constructed two rays with a common origin divide the plane into two regions. One of the areas is proposed to be painted over. The fact that the rays and the selected area are colored the same color means that their union has been constructed. The resulting figure is called an angle. How is the angle constructed? The teacher encourages students to create a description of the concept using this slide. Enter the designation of angles.
Slide number 17.
Slides No. 18 and 19.
Students perform exercises that promote the formation of the concept of an angle and the formation of the concept of intersection of figures. These exercises are especially interesting; they will allow you to find out whether the concept has been formed.
Students performing physical education for the eyes:Close your eyes tightly (count to 3, open them and look into the distance (count to 5). Repeat 4 - 5 times.
V. Consolidation of the material being studied.
Slide number 20.
The teacher asks students to complete the following tasks independently:
Based on Figure 1, answer the questions:
1. Write down all the segments.
2. Write down all the lines.
3. Which points belong to the line AD and which do not? Write your answer using mathematical symbols.
4. Indicate a point that belongs to both the straight line BC and the straight line AC. What else can you call the indicated point?
5. According to Figure 2, write down the points belonging to:
A) the outer area of the corner;
B) the inner area of the corner;
Self-test answers:
1. AB, BD, AD, DC, BC, DM, AM.
Students summarize the lesson and answer the teacher’s questions orally:
1) what new did they learn?
2) what is “geometry”?
3) what branches of geometry exist?
4) what basic concepts were covered in the lesson?
5) what is “straight”? "line segment"? "Ray"? "corner"?
VII. Giving a grade for a lesson with a comment from the teacher.
VIII. Homework(slide number 22):
Literature:
1) Atanasyan L. S., Butuzov V. F. and others. Geometry: textbook. for 7-9 grades. general education institutions. - M.: Education, 2010.
2) Gavrilova N. F. Lesson developments in geometry. 7th grade. M.: "VAKO", 2010.
Geometry is one of the most ancient sciences. The first geometric facts are found in Babylonian cuneiform tables and Egyptian papyri (III millennium BC), as well as in other sources. The name of the science “geometry” is of ancient Greek origin, it is composed of two ancient Greek words: “ge” - “earth” and “metreo” - “I measure” (I measure the earth).
Geometry - is a branch of mathematics that studies geometric figures and their properties.
1 . Draw a straight line. How can it be designated?
2 . Mark point C, which does not lie on this line, and points D , E , K , lying on the same line .
Symbols of belonging
belongs does not belong
3 . Using affiliation symbols, write the sentence “Point D belongs to the line AB, and point C does not belong to the line A ".
4 . Using the drawing and membership symbols, write down which points belong to the line b , and which ones are not.
— How many straight lines can be drawn through a given point? A?
— How many lines can be drawn through two points?
-Can a straight line be drawn through any two points?
5 .Draw straight lines XY And MK , intersecting at a point ABOUT .
In order to briefly write down that straight lines XYAndMK intersect at a point ABOUT, use the symbol ∩ and write it like this: XY∩ MK = Oh.
- How many common points can two straight lines have?
6. On a straight line A mark the points sequentially A, B, C,D . Write down all the resulting segments.
7 . Draw straight lines A And b , intersecting at a point M. On straight A mark the point N , different from the point M.
a) Are straight lines MN And A different straight lines?
b) Can a straight line b pass through a point N ?
Solve problems:
1) How many points of intersection can three straight lines have? Consider all possible cases and make appropriate drawings.
Didactic material
To test theoretical knowledge for a 7th grade geometry course.
1. Mark correct statements with a “+” sign and erroneous statements with a “-” sign.
1. Examples of geometric figures on a plane are a point, a straight line, a square, a cube, a ball.
2. Examples of geometric figures on a plane are a point, a straight line, a ray, a segment, a polygon.
3. Two lines either have only one common point or have no common points.
4. Three straight lines can be drawn through any two points.
5. A segment is a part of a straight line.
6. A ray is a part of a line, consisting of all points of this line that lie on one side of a given point on it.
7. The beginning of the ray AB is point B.
8. An angle is a geometric figure consisting of a point and two rays emanating from this point.
9. Any angle can have several vertices.
10. The point of a segment dividing it in half is called the midpoint of the segment.
11. An undeveloped angle is always larger than a developed one.
12. An undeveloped angle is always smaller than a developed angle.
13. The bisector of an angle is a ray emanating from the vertex of an angle, dividing the angle into two equal angles.
14. The length of a segment is the distance between any of its points.
15. Any point lying on a segment splits it into two parts.
16. If point B belongs to the segment AK, then AK = AB – BK.
17. A straight angle has a degree measure of 90 0.
18. An angle is called right if it is equal to 60 0.
19. An acute angle is always smaller than a right angle.
20. Two angles in which one side is common, and the other two are continuations of one another, are called adjacent.
21. The sum of adjacent angles is 180 0.
22. The sum of vertical angles is always 100 0.
23. If two adjacent angles are equal, then they are right angles.
Basic geometric information.
2. Mark correct statements with a “+” sign and erroneous statements with a “-” sign.
1. Two straight lines always have a common point.
2. A segment is a part of a line consisting of all points of this line lying between two given points.
3. An angle is a geometric figure consisting of a point and three rays emanating from this point.
4. Geometric figures are called equal if all their sides are pairwise equal.
5. Geometric figures are called equal if they coincide when superimposed.
6. An angle is called developed if both its sides lie on the same straight line.
7. Any ray emanating from the vertex of an angle divides it into two equal angles.
8. The length of a segment is the distance between its ends.
9. The length of a segment is equal to the sum of the lengths of its parts into which it is divided by any of its points.
10. Units for measuring angles are degrees.
11. An obtuse angle is always less than a right angle.
12. Two angles are called vertical. If the sides of one angle are continuations of the sides of another.
13. Adjacent angles are equal.
14. Two lines are called perpendicular if they form two right angles.
15. Two lines perpendicular to the third do not intersect.
16. Equal angles have equal degrees.
17. The straight angle is 180 0.
18. If two adjacent angles are equal, then they are acute.
19.If two lines are perpendicular to a third, then they are parallel.
20. Two adjacent angles can both be obtuse.
Triangles.
1. A triangle is a three-dimensional figure.
2. A triangle is a geometric figure consisting of three points connected in pairs by segments.
3. A triangle is a geometric figure consisting of three points that do not lie on the same straight line and are connected in pairs by segments.
4. If two triangles are equal, then their corresponding elements are always equal.
5. The first sign of equality of triangles is a sign of equality along a side and two angles.
6. When perpendicular lines intersect, four acute angles are obtained.
7. The median of a triangle drawn from a given vertex is a straight line connecting this vertex to the midpoint of the opposite side.
8. The median of a triangle drawn from a given vertex is a segment connecting this vertex with the midpoint of the opposite side.
9. In any triangle you can draw only three bisectors.
10. The bisector of any triangle is a segment.
11. The bisectors of any triangle always intersect at one point.
12. The altitude of a triangle dropped from a given vertex is the perpendicular drawn from the vertex to the opposite side of the triangle.
13. The altitude of a triangle dropped from a given vertex is the perpendicular drawn from the vertex to the line containing the opposite side of the triangle.
14. Equal sides of an isosceles triangle are called lateral.
15. Equal sides of an isosceles triangle are called bases.
16. An isosceles triangle has two sides and one base.
17. The angles at the base of an isosceles triangle are equal.
18. In an isosceles triangle, all angles are equal.
19. If the perimeter of a triangle is 60 cm and the triangle is equilateral, then the length of each side is 20 cm.
20. The third sign of equality of triangles is the sign of equality on two sides and an angle.
21. The third sign of equality of triangles is a sign of equality on three sides.
22. A circle is a figure consisting of points on a plane located at a given distance from a given point.
23. Diameter is the largest chord.
24. The radius is a chord.
Triangles.
1. A triangle is a flat figure.
2. In triangle ABC, the sides adjacent to angle CAB are AC and BC.
3. In triangle AMC, the side opposite to angle AMC is side AC.
4. The perimeter of a triangle MSC with sides 7 cm, 11 cm, 8 cm is equal to 26 cm.
5. The first sign of equality of triangles is the sign of equality on the sides and angles.
6. The first sign of equality of triangles is the sign of equality on the sides and the angle between them.
7. When perpendicular lines intersect, four right angles are obtained.
8. In any triangle, only three medians can be drawn.
9. In any triangle you can only draw one median.
10. The bisector of a triangle drawn from a given vertex is the ray emerging from this vertex, passing between the sides of the angle and dividing the angle in half.
11. The bisector of a triangle drawn from a given vertex is the segment of the bisector of the angle of the triangle connecting this vertex with a point on the opposite side.
12. In any triangle you can draw as many heights as you like.
13. In any triangle you can only draw three altitudes.
14. An isosceles triangle is one whose two sides are equal.
15 . An isosceles triangle is one in which three sides are equal.
16. An equilateral triangle is one in which all sides are equal.
17. In an equilateral triangle, all angles are equal.
18. The second sign of equality of triangles is the sign of equality along a side and two angles.
19. The second sign of equality of triangles is a sign of equality along a side and two adjacent angles.
20. A circle is a figure consisting of all points of the plane located at a given distance from a given point.
21. In a circle, all radii have different lengths.
22. In a circle, all chords are equal.
23. Diameter is a chord passing through the center.
24. The diameter of a circle is twice the radius of the same circle.
25. In a circle, all radii are equal.
Parallel lines
1. Mark the correct statements with a “+” sign and the incorrect ones with a “-” sign.
1. Parallel lines are lines that do not intersect.
2. Only two parallel lines can be drawn.
3. If a certain line intersects one of two parallel lines, then it also intersects the other.
4. If two lines are parallel to a third, then they cannot be parallel.
5. If two lines are perpendicular to the third, then they are parallel.
6. When two straight lines intersect with a third, four undeveloped angles are formed.
3 4 7. Angles 3 and 5, 4 and 6 are called crosswise.
8. Angles 3 and 6, 5 and 4 are called crosswise.
9. Angles 3 and 5, 4 and 6 are called one-sided.
5 6 10. Angles 3 and 7, 2 and 6 are called corresponding.
7 8 11. Angles 4 and 6, 5 and 4 are called one-sided.
12. Through a point not lying on a given line there pass many lines parallel to the given one.
13. If a line intersects one of two parallel lines, then it is perpendicular to the other line.
14. If, when two straight lines are intersected crosswise, the lying angles are equal, then the straight lines are parallel.
15. If, when two lines intersect with a transversal, the sum of the crosswise angles is equal to 180 0, then the lines are parallel.
16. If two parallel lines are intersected by a transversal, then the intersecting angles are equal.
17. If two parallel lines are intersected by a transversal, then the sum of one-sided angles is equal to 180 0.
2. Mark correct statements with a “+” sign and erroneous statements with a “-” sign.
1. Parallel lines are lines that lie on a plane and do not intersect.
2. Only three parallel lines can be drawn.
3. Through any point not lying on a given line, you can draw in the plane a line parallel to it, and only one.
4. If two lines are parallel to a third, then they are parallel to each other.
5. When two straight lines intersect with a third, eight undeveloped angles are formed.
6. When two straight lines intersect with a third, two pairs of cross-lying angles are formed.
7. An axiom is a mathematical statement about the properties of figures.
8. An axiom is a mathematical statement about the properties of geometric figures, accepted without proof.
9. A straight line passes through any two points, and only one.
10. Through a point not lying on a given line there passes only one line parallel to the given one.
11. Through a point not lying on a given line there pass only two lines parallel to the given one.
12. If two lines are parallel to a third, then they are perpendicular to each other.
13. If two lines are parallel to a third, then they are parallel to each other.
14. If, when two lines intersect with a transversal, the corresponding angles are equal, then the lines are parallel.
15. If, when two lines intersect with a transversal, the sum of the corresponding angles is equal to 180 0, then the lines are parallel.
16. If, when two lines intersect with a transversal, the sum of one-sided angles is equal to 180 0, then the lines are parallel.
17. If a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.
18. If two parallel lines are intersected by a transversal, then the corresponding angles are equal.
