Construct a right angle equal to the given one. How to construct an angle equal to a given one

The ability to divide any angle with a bisector is needed not only to get an “A” in mathematics. This knowledge will be very useful for builders, designers, surveyors and dressmakers. In life, you need to be able to divide many things in half. Everyone at school...

Conjugation is a smooth transition from one line to another. To find a mate, you need to determine its points and center, and then draw the corresponding intersection. To solve such a problem, you need to arm yourself with a ruler...

Conjugation is a smooth transition from one line to another. Conjugates are very often used in a variety of drawings when connecting angles, circles and arcs, and straight lines. Constructing a section is a rather difficult task, for which you…

When constructing various geometric shapes, it is sometimes necessary to determine their characteristics: length, width, height, and so on. If we are talking about a circle or circle, then we often have to determine its diameter. The diameter is...

A triangle is called a right triangle if the angle at one of its vertices is 90°. The side opposite this angle is called the hypotenuse, and the sides opposite the two acute angles of the triangle are called the legs. If the length of the hypotenuse is known...

Tasks to construct regular geometric shapes train spatial perception and logic. There are a large number of very simple problems of this kind. Their solution comes down to modifying or combining already...

The bisector of an angle is a ray that begins at the vertex of the angle and divides it into two equal parts. Those. To draw a bisector, you need to find the midpoint of the angle. The easiest way to do this is with a compass. In this case you don't need...

When building or developing home design projects, it is often necessary to build an angle equal to an existing one. Templates and school knowledge of geometry come to the rescue. Instructions 1An angle is formed by two straight lines emanating from one point. This point...

The median of a triangle is a segment connecting any of the vertices of the triangle with the midpoint of the opposite side. Therefore, the problem of constructing a median using a compass and ruler is reduced to the problem of finding the midpoint of a segment. You will need-…

A median is a segment drawn from a certain corner of a polygon to one of its sides in such a way that the point of intersection of the median and the side is the midpoint of that side. You will need - a compass - a ruler - a pencil Instructions 1 Let the given...

This article will tell you how to use a compass to draw a perpendicular to a given segment through a certain point lying on this segment. Steps 1Look at the segment (straight line) given to you and the point (denoted as A) lying on it.2Install the needle...

This article will tell you how to draw a line parallel to a given line and passing through a given point. Steps Method 1 of 3: Along perpendicular lines 1 Label the given line as “m” and the given point as A. 2 Through point A draw...

This article will tell you how to construct a bisector of a given angle (a bisector is a ray dividing the angle in half). Steps 1Look at the angle given to you.2Find the vertex of the angle.3Place the compass needle at the vertex of the angle and draw an arc intersecting the sides of the angle...

Constructing an angle equal to a given one. Given: half-line, angle. Construction. V.A.S. 7. To prove it, it is enough to note that triangles ABC and OB1C1 are congruent as triangles with respectively equal sides. Angles A and O are the corresponding angles of these triangles. It is necessary: ​​to put off from a given half-line into a given half-plane an angle equal to a given angle. C1. IN 1. A. 1. Let's draw an arbitrary circle with its center at vertex A of the given angle. 2. Let B and C be the points of intersection of the circle with the sides of the angle. 3. Using radius AB we draw a circle with the center at point O - the starting point of this half-line. 4. Let us denote the point of intersection of this circle with this half-line as B1. 5. Let us describe a circle with center B1 and radius BC. 6. Point C1 of intersection of the constructed circles in the indicated half-plane lies on the side of the desired angle.

Geometry 7th grade

“Isosceles triangle” - Theorem. A triangle is the simplest closed rectilinear figure. Problem solving. Find angle KBA. Equality of triangles. Guess the rebus. ABC - isosceles. List the congruent elements of triangles. Classification of triangles by sides. In an isosceles triangle AMK AM = AK. Classification of triangles according to the size of their angles. Sides. A triangle with all sides equal. Isosceles triangle.

“Measuring segments and angles” - Comparison of segments. http://www.physicsdepartment.ru/blog/images/0166.jpg. Ф3 = Ф4. MN > CD. 1m =. The middle of the segment. 1km. What is the greatest number of parts that 4 different straight lines can divide a plane into? Other units of measurement. Comparing shapes using overlay. Comparison of angles. The sides of the VM and the EU came together. How many parts can a plane be divided into by 3 different straight lines? http://www.robertagor.it/calibro.jpg.

“Right triangle, its properties” - One of the corners of a right triangle. Solution. Which triangle is called a right triangle? Right triangle. Properties of a right triangle. Warm up. Development of logical thinking. Bisector. Leg of a right triangle. Let's create an equation. Let's look at the drawing carefully. Property of a right triangle. Residents of three houses. Triangle.

“Definition of an angle” - Concepts of angles. Draw the rays. Preparatory stage of the lesson. Corner. Explanation of new material. An angle divides a plane. Concepts of internal and external areas of an angle. Get interested in the subject. The ray in the figure divides the angle. Definition of a straight angle. Development of logical thinking. Obtuse angle. Sharp corner. Opening words. Paint the inside area of ​​the corner. Angles. Ray BM divides angle ABC into two angles.

“The second and third signs of equality of triangles” - Sides. Median in an isosceles triangle. The second and third signs of equality of triangles. Solution. Three sides of one triangle. Base. Prove. Properties of an isosceles triangle. Signs of equality of triangles. Problem solving. Mathematical dictation. Angles. Task. Perimeter of an isosceles triangle.

“Cartesian coordinate system on a plane” - The plane on which the Cartesian coordinate system is specified. Coordinates in people's lives. Geographic coordinate system. Cartesian coordinate system on a plane. Algebra project. Scientists who are the authors of the coordinates. Ancient Greek astronomer Claudius. A cell on the playing field. The point of intersection of the axes. Introduction of simpler notation to algebra. A place in a cinema. The meaning of the Cartesian coordinate system.

Constructing an angle equal to a given one. Given: angle A. A Constructed angle O. B C O D E Prove: A = O Proof: consider triangles ABC and ODE. 1.AC = OE, like the radii of one circle. 2.AB=OD, as the radii of one circle. 3.ВС=DE, as the radii of one circle. ABC = ODE (3rd prize) A = O

Let us prove that the ray AB is a bisector A P L A N 1.Additional construction. 2. Let us prove the equality of triangles ACB and ADB. 3. Conclusions A B C D 1.AC = AD, as the radii of one circle. 2.CB=DB, as the radii of one circle. 3.AB – common side. ACB = ADB, according to the III criterion of equality of triangles Ray AB - bisector Construction of the bisector of an angle.

A N B A C 1 = 2 12 In the r/b triangle AMB, the segment MC is a bisector, and therefore a height. Then, and MN. M Let's prove that a MN Let's look at the location of the compasses. AM=AN=MB=BN, as equal radii. MN-common side. MВN= MAN, on three sides Construction of perpendicular lines. M a

Q P BA ARQ = BPQ, on three sides = 2 Triangle ARV r/b. The segment PO is a bisector, and therefore a median. Then, point O is the middle of AB. О Let us prove that O is the midpoint of the segment AB. Constructing the midpoint of a segment

D C Constructing a triangle using two sides and the angle between them. Angle hk h 1. Let's construct ray a. 2. Set aside a segment AB equal to P 1 Q 1. 3. Construct an angle equal to this one. 4. Let us set aside the segment AC equal to P 2 Q 2. VA Triangle ABC is the desired one. Justify using the first sign. Given: Segments P 1 Q 1 and P 2 Q 2 Q1Q1 P1P1 P2P2 Q2Q2 a k

D C Constructing a triangle using a side and two adjacent angles. Angle h 1 k 1 h2h2 1. Construct ray a. 2. Set aside a segment AB equal to P 1 Q 1. 3. Construct an angle equal to the given h 1 k 1. 4. Construct an angle equal to h 2 k 2. BA A Triangle ABC is the desired one. Justify using the second sign. Given: Segment P 1 Q 1 Q1Q1 P1P1 a k2k2 h1h1 k1k1 N

C 1. Let's build a ray a. 2. Set aside a segment AB equal to P 1 Q 1. 3. Construct an arc with a center at point A and radius P 2 Q 2. 4. Construct an arc with a center at point B and radius P 3 Q 3. BA A Triangle ABC sought after Justify using the third sign. Given: segments P 1 Q 1, P 2 Q 2, P 3 Q 3. Q1Q1 P1P1 P3P3 Q2Q2 a P2P2 Q3Q3 Construction of a triangle using three sides.

math geometry skill lesson

Lesson summary “Constructing an angle equal to a given one. Construction of the angle bisector"

educational: introduce students to construction problems, in solving which only compasses and a ruler are used; teach how to construct an angle equal to a given one, how to construct the bisector of an angle;

developmental: development of spatial thinking, attention;

educational: fostering hard work and accuracy.

Equipment: tables with the order of solving construction problems; compass and ruler.

During the classes:

1. Updating of basic theoretical concepts (5 min).

First, you can conduct a frontal survey on the following questions:

• 1. What figure is called a triangle?
• 2. Which triangles are called equal?
• 3. Formulate the criteria for the equality of triangles.
• 4. Which segment is called the bisector of a triangle? How many bisectors does a triangle have?
• 5. Define a circle. What are the center, radius, chord and diameter of a circle?

To repeat the signs of equality of triangles, you can suggest.

Exercise: indicate which of the pictures (Fig. 1) contains equal triangles.

Rice. 1

A repetition of the concept of a circle and its elements can be organized by offering the class the following exercise, with one student performing it on the board: given a line a and a point A lying on the line and a point B not lying on the line. Draw a circle with a center at point A and passing through point B. Mark the points of intersection of the circle with line a. Name the radii of the circle.

2. Studying new material (practical work) (20 min)

Constructing an angle equal to a given one

To review new material, it is useful for the teacher to have a table (Table No. 1 of Appendix 4). Work with the table can be organized in different ways: it can illustrate the teacher’s story or a sample solution record; You can invite students, using the table, to talk about the solution to the problem, and then complete it independently in their notebooks. The table can be used when questioning students and when repeating material.

Task. Subtract an angle from a given ray equal to a given one.

Solution. This angle with vertex A and the ray OM are shown in Figure 2.

Rice. 2

It is required to construct an angle equal to angle A, so that one of the sides coincides with the ray OM. Let us draw a circle of arbitrary radius with its center at vertex A of the given angle. This circle intersects the sides of the angle at points B and C (Fig. 3, a). Then we draw a circle of the same radius with the center at the beginning of this ray OM. It intersects the beam at point D (Fig. 3, b). After this, we will construct a circle with center D, the radius of which is equal to BC. Circles with centers O and D intersect at two points. Let us denote one of these points by the letter E. Let us prove that the angle MOE is the desired one.

Consider triangles ABC and ODE. Segments AB and AC are the radii of a circle with center A, and OD and OE are the radii of a circle with center O. Since, by construction, these circles have equal radii, then AB = OD, AC = OE. Also by construction BC = DE. Therefore, ABC = ODE on three sides. Therefore DOE = YOU, i.e. the constructed angle MOE is equal to the given angle A.

Rice. 3

Constructing the bisector of a given angle

Task. Construct the bisector of the given angle.

Solution. Let us draw a circle of arbitrary radius with its center at vertex A of the given angle. It will intersect the sides of the angle at points B and C. Then we draw two circles of the same radius BC with centers at points B and C (Figure 4 shows only parts of these circles). They will intersect at two points. We will denote the one of these points that lies inside the angle BAC with the letter E. Let us prove that the ray AE is the bisector of this angle.

Consider triangles ACE and ABE. They are equal on three sides. Indeed, AE is the general side; AC and AB are equal, like the radii of the same circle; CE=BE by construction. From the equality of triangles ACE and ABE it follows that CAE = BAE, i.e. ray AE is the bisector of a given angle.

Rice. 4

The teacher can ask students to use this table (Table No. 2 of Appendix 4) to construct the bisector of an angle.

The student at the board performs a construction, justifying each step of the actions performed.

The teacher shows the proof; it is necessary to dwell in detail on the proof of the fact that as a result of the construction, equal angles will actually be obtained.

3. Consolidation (10 min)

It is useful to offer students the following task to reinforce the material covered:

Task. Obtuse angle AOB is given. Construct the ray OX so that the angles HOA and HOB are equal obtuse angles.

Task. Construct angles of 30° and 60° using a compass and ruler.

Task. Construct a triangle using a side, an angle adjacent to its side, and the bisector of the triangle emanating from the vertex of the given angle.

• 4. Summing up (3 min)
• 1. During the lesson we solved two construction problems. Studied:
  • a) construct an angle equal to the given one;
  • b) construct the bisector of the angle.
• 2. In the course of solving these problems:
  • a) remembered the signs of equality of triangles;
  • b) used the construction of circles, segments, rays.
• 5. To home (2 min): No. 150-152 (see Appendix 1).