The ability to divide any angle with a bisector is necessary not only in order to get an "A" in mathematics. This knowledge will be very useful to the builder, designer, surveyor and dressmaker. There are many things in life that need to be divided. Everyone at school...
Pairing is a smooth transition from one line to another. To search for a conjugation, it is necessary to determine its points and center, and then draw the corresponding intersection. To solve this problem, you need to arm yourself with a ruler, ...
Pairing is smooth transition one line to another. Conjugation is very often used in a variety of drawings when connecting angles, circles and arcs, straight lines. Building a section is a rather difficult task, for which it is up to 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 a circle, then it is often necessary to determine their diameter. The diameter is…
A right triangle is a triangle whose 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 legs. If you know the length of the hypotenuse...
Tasks for the implementation of the construction of regular geometric shapes train spatial perception and logic. Exists a large number of very simple tasks of this kind. Their solution comes down to modifying or combining already ...
The bisector of an angle is a ray that starts 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 the one already available. Templates and school knowledge of geometry come to the rescue. Instruction 1 Angle is formed by two straight lines emanating from one point. This point...
The median of a triangle is a segment that connects 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 a ruler is reduced to the problem of finding the middle 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 intersection point of the median and the side is the midpoint of this side. You will need a compass-ruler-pencilInstruction 1Let it be given ...
This article will tell you how to draw a perpendicular to a given segment using a compass through a certain point lying on this segment. Steps 1Look at the line segment (line) given to you and the point (denoted as A) lying on it. 2Install the needle ...
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This article will tell you how to construct a bisector of a given angle (a bisector is a ray that bisects an angle). Steps 1Look at the angle you've been given. 2Find the apex of the angle. 3Set the compass needle at the apex of the angle and draw an arc across the sides of the angle...
Often it is necessary to draw (“build”) an angle that would be equal to a given angle, and the construction must be done without the help of a protractor, but using only a compass and a ruler. Knowing how to build a triangle on three sides, we can solve this problem. Let on a straight line MN(dev. 60 and 61) is required to be built at the point K angle equal to angle B. This means that it is necessary from the point K draw a straight line constituting MN angle equal to B.
To do this, mark a point on each side of a given angle, for example BUT and FROM, and connect BUT and FROM straight line. Get a triangle ABC. Let's build now on a straight line MN this triangle so that its apex AT was at the point To: then this point will have an angle equal to the angle AT. Build a triangle on three sides Sun, VA and AC we can: postpone (dev. 62) from the point To line segment kl, equal Sun; get a point L; around K, as near the center, we describe a circle with a radius VA, and around L- radius SA. point R connect the intersections of the circles with To and Z, - we get a triangle KPL, triangular ABC; it has a corner To= ang. AT.
This construction is faster and more convenient if from the top AT set aside equal segments (with one dissolution of the compass) and, without moving its legs, describe with the same radius a circle around the point TO, like near the center.
How to cut a corner in half
Let it be required to divide the angle BUT(Fig. 63) into two equal parts using a compass and ruler, without using a protractor. We'll show you how to do it.
From the top BUT draw equal segments on the sides of the angle AB and AC(Fig. 64; this is done with one dissolution of the compass). Then we put the tip of the compass at the points AT and FROM and describe with equal radii the arcs intersecting at the point D. straight line connecting BUT and D divides the angle BUT in half.
Let's explain why. If the point D connect with AT and C (Fig. 65), then you get two triangles ADC and adb, u which have a common side AD; side AB equal to side AC, a BD is equal to CD. Triangles are equal on three sides, so the angles are equal. bad and DAC, lying opposite equal sides BD and CD. Therefore, a straight line AD divides the angle YOU in half.
Applications
12. Construct an angle of 45° without a protractor. At 22°30'. At 67°30'.
Solution. Dividing the right angle in half, we get an angle of 45 °. Dividing the angle of 45° in half, we get an angle of 22°30'. By constructing the sum of the angles 45° + 22°30', we get an angle of 67°30'.
How to draw a triangle given two sides and an angle between them
Let it be required on the ground to find out the distance between two milestones BUT and AT(device 66), separated by an impenetrable swamp.
How to do it?
We can do this: aside from the swamp, we choose such a point FROM, from where both milestones are visible and it is possible to measure distances AC and Sun. Corner FROM we measure with the help of a special goniometric device (called an astrolabe). According to these data, i.e., according to the measured sides AC and Sun and corner FROM between them, build a triangle ABC somewhere in a convenient location as follows. Having measured one known side in a straight line (Fig. 67), for example AC, build with it at the point FROM corner FROM; on the other side of this angle, a known side is measured Sun. Ends of known sides, i.e. points BUT and AT connected by a straight line. It turns out a triangle in which two sides and the angle between them have pre-specified dimensions.
It is clear from the method of construction that only one triangle can be constructed given two sides and the angle between them. therefore, if two sides of one triangle are equal to two sides of another and the angles between these sides are the same, then such triangles can be superimposed on each other by all points, i.e., their third sides and other angles must also be equal. This means that the equality of the two sides of the triangles and the angle between them can serve as a sign of the complete equality of these triangles. Shortly speaking:
Triangles are equal under two sides and angles between them.
When building or developing home design projects, it is often necessary to build an angle equal to the one already available. Templates and school knowledge of geometry come to the rescue.
Instruction
- An angle is formed by two straight lines emanating from the same point. This point will be called the vertex of the corner, and the lines will be the sides of the corner.
- Use three letters to designate corners: one at the top, two at the sides. They call the corner, starting with the letter that stands at one side, then they call the letter at the top, and then the letter at the other side. Use other ways to mark corners if you prefer otherwise. Sometimes only one letter is called, which is at the top. Can you mark angles? Greek letters, for example, α, β, γ.
- There are situations when it is necessary to draw an angle so that it is equal to an already given angle. If it is not possible to use a protractor when constructing a drawing, you can only get by with a ruler and a compass. Suppose, on a straight line, indicated in the drawing by the letters MN, you need to build an angle at the point K, so that it is equal to the angle B. That is, from point K it is necessary to draw a straight line forming an angle with line MN, which will be equal to angle B.
- First, mark a point on each side of this corner, for example, points A and C, then connect points C and A with a straight line. Get triangle ABC.
- Now construct the same triangle on line MN so that its vertex B is on the line at point K. Use the rule for constructing a triangle on three sides. Set aside the segment KL from point K. It must be equal to the segment BC. Get point L.
- From point K, draw a circle with a radius equal to the segment BA. From L draw a circle with radius CA. Connect the resulting point (P) of the intersection of two circles with K. Get the triangle KPL, which will be equal to the triangle ABC. So you get angle K. It will be equal to angle B. To make this construction more convenient and faster, set aside equal segments from vertex B, using one compass solution, without moving the legs, describe the circle with the same radius from point K.
Lesson Objectives:
- Formation of skills to analyze the studied material and skills to apply it to solve problems;
- Show the significance of the concepts being studied;
- Development of cognitive activity and independence in obtaining knowledge;
- Raising interest in the subject, a sense of beauty.
Lesson objectives:
- To form skills in constructing an angle equal to a given one using a scale ruler, compass, protractor and drawing triangle.
- Check students' ability to solve problems.
Lesson plan:
- Repetition.
- Constructing an angle equal to a given one.
- Analysis.
- Construction of the first example.
- Construction of the second example.
Repetition.
Corner.
flat corner- unlimited geometric figure, formed by two rays (sides of the corner) coming out of one point (the vertex of the corner).
An angle is also called a figure formed by all points of the plane enclosed between these rays (Generally speaking, two such rays correspond to two angles, since they divide the plane into two parts. One of these angles is conditionally called internal, and the other external.
Sometimes, for brevity, an angle is called an angular measure.
To designate an angle, there is a generally accepted symbol: , proposed in 1634 by the French mathematician Pierre Erigon.
Corner- this is a geometric figure (Fig. 1), formed by two rays OA and OB (corner sides), emanating from one point O (corner apex).
An angle is denoted by a symbol and three letters indicating the ends of the rays and the vertex of the angle: AOB (moreover, the letter of the vertex is the middle one). The angles are measured by the amount of rotation of the ray OA around the vertex O until the ray OA passes into position OB. There are two commonly used units for measuring angles: radians and degrees. For radian measurement of angles, see below under "Arc length" and also in the chapter "Trigonometry".
Degree system for measuring angles.
Here, the unit of measure is the degree (its designation is °) - this is the rotation of the beam by 1/360 of a full turn. Thus, a full rotation of the beam is 360 o. One degree is divided into 60 minutes (notation ‘); one minute - respectively for 60 seconds (designation “). An angle of 90 ° (Fig. 2) is called right; an angle less than 90° (Fig. 3) is called acute; an angle greater than 90 ° (Fig. 4) is called obtuse.
Straight lines forming a right angle are called mutually perpendicular. If the lines AB and MK are perpendicular, then this is denoted: AB MK.
Constructing an angle equal to a given one.
Before starting construction or solving any problem, regardless of the subject, it is necessary to carry out analysis. Understand what the task is about, read it thoughtfully and slowly. If after the first time there are doubts or something was not clear or clear but not completely, it is recommended to read it again. If you are doing an assignment in class, you can ask the teacher. Otherwise, your task, which you misunderstood, may not be solved correctly, or you may find something that is not what was required of you and it will be considered incorrect and you will have to redo it. As for me - it is better to spend a little more time studying the task than to redo the task again.
Analysis.
Let a be a given ray with vertex A, and let (ab) be the desired angle. We choose points B and C on the rays a and b, respectively. Connecting points B and C, we get triangle ABC. In equal triangles, the corresponding angles are equal, and hence the method of construction follows. If points C and B are chosen in some convenient way on the sides of a given angle, a triangle AB 1 C 1 equal to ABC is constructed from the given ray to the given half-plane (and this can be done if all sides of the triangle are known), then the problem will be solved.
When carrying out any constructions Be extremely careful and try to carry out all the constructions carefully. Since any inconsistencies can result in some kind of errors, deviations, which can lead to an incorrect answer. And if a task of this type is performed for the first time, then the error will be very difficult to find and fix.
Construction of the first example.
Draw a circle centered at the vertex of the given angle. Let B and C be the points of intersection of the circle with the sides of the angle. Draw a circle with radius AB centered at point A 1 - the starting point of this ray. The point of intersection of this circle with the given ray will be denoted by B 1 . Let's describe a circle with center B 1 and radius BC. The intersection point C 1 of the constructed circles in the specified half-plane lies on the side of the required angle.
Triangles ABC and A 1 B 1 C 1 are equal on three sides. Angles A and A 1 are the corresponding angles of these triangles. Therefore, ∠CAB = ∠C 1 A 1 B 1
For greater clarity, we can consider the same constructions in more detail.
Construction of the second example.
The task also remains to postpone from the given half-line to the given half-plane an angle equal to the given angle.
Construction.
Step 1. Let's draw a circle with an arbitrary radius and centers at the vertex A of the given angle. Let B and C be the intersection points of the circle with the sides of the angle. And draw the segment BC.
Step 2 Draw a circle with radius AB centered at point O, the starting point of this half-line. Denote the point of intersection of the circle with the ray B 1 .
Step 3 Now let's describe a circle with center B 1 and radius BC. Let the point C 1 be the intersection of the constructed circles in the specified half-plane. 
Step 4 Let's draw a ray from point O through point C 1 . Angle C 1 OB 1 will be the desired one.
Proof.
Triangles ABC and OB 1 C 1 are congruent as triangles with corresponding sides. And therefore the angles CAB and C 1 OB 1 are equal.
Interesting fact:
In numbers.
In the objects of the world around you, first of all, you notice their individual properties that distinguish one object from another.
The abundance of particular, individual properties overshadows the general properties inherent in absolutely all objects, and therefore it is always more difficult to discover such properties.
One of the most important common properties of objects is that all objects can be counted and measured. We reflect this common property of objects in the concept of number.
People mastered the process of counting, that is, the concept of number, very slowly, for centuries, in a stubborn struggle for their existence.
In order to count, one must not only have objects that can be counted, but already have the ability to be distracted when considering these objects from all their other properties, except for number, and this ability is the result of a long historical development based on experience.
Every person now learns to count with the help of numbers imperceptibly even in childhood, almost simultaneously with how he begins to speak, but this counting habitual to us has gone a long way of development and has taken different forms.
There was a time when only two numbers were used to count objects: one and two. In the process of further expansion of the number system, parts were involved human body and, first of all, fingers, and if there were not enough such “numbers”, then also sticks, pebbles and other things.
N. N. Miklukho-Maclay in his book "Travels" talks about a funny way of counting used by the natives of New Guinea:
Questions:
- What is the definition of an angle?
- What are the types of corners?
- What is the difference between diameter and radius?
List of sources used:
- Mazur K. I. "Solving the main competitive problems in mathematics of the collection edited by M. I. Scanavi"
- Mathematical ingenuity. B.A. Kordemsky. Moscow.
- L. S. Atanasyan, V. F. Butuzov, S. B. Kadomtsev, E. G. Poznyak, I. I. Yudina "Geometry, 7 - 9: a textbook for educational institutions"
Worked on the lesson:
Levchenko V.S.
Poturnak S.A.
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Subjects > Mathematics > Mathematics Grade 7Constructing an angle equal to a given one. Given: angle A. A Constructed angle O. B C O D E Prove: A \u003d O Proof: consider triangles ABC and ODE. 1.AC=OE, as radii of one circle. 2.AB=OD, as the radii of one circle. 3.BC=DE, as radii of one circle. ABC \u003d ODE (3 prizes) A \u003d O
Let us prove that the ray AB is a bisector A P L A N 1. Additional construction. 2. Let's prove the equality of triangles ACB and ADB. 3. Conclusions A B C D 1.AC=AD, as radii of one circle. 2.CB=DB, as radii of one circle. 3.AB - common side. ASV \u003d ADB, according to the III sign of equality of triangles Beam AB - bisector Construction of the bisector of the angle.
A N B A C 1 = 2 12 In the r/b triangle AMB, the segment MC is a bisector, and hence the 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 is the common side. MBN= MAN, on three sides Construction of perpendicular lines. M a
Q P VA APQ \u003d BPQ, on three sides \u003d 2 Triangle ARV r / b. The segment RO is a bisector, and therefore a median. Then point O is the midpoint of AB. О Let us prove that О is the midpoint of the segment AB. Construction of the middle of the segment
D С Construction of a triangle given two sides and an angle between them. Angle hk h 1. Let's build a beam a. 2. Set aside the segment AB, equal to P 1 Q 1. 3. Construct an angle equal to this one. 4. Set aside the segment AC, equal to P 2 Q 2. B A The triangle ABC is the desired one. Justify using the I sign. Given: Segments P 1 Q 1 and P 2 Q 2 Q1Q1 P1P1 P2P2 Q2Q2 a k
D С Construction of a triangle by a side and two angles adjacent to it. Angle h 1 k 1 h2h2 1. Let's build a beam a. 2. Set aside the 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. B 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 construct a ray. 2. Set aside the segment AB, equal to P 1 Q 1. 3. Construct an arc centered at point A and radius P 2 Q 2. 4. Construct an arc centered at point B and radius P 3 Q 3. B A Triangle ABC desired. Justify using the III sign. Given: segments P 1 Q 1, P 2 Q 2, P 3 Q 3. Q1Q1 P1P1 P3P3 Q2Q2 and P2P2 Q3Q3 Construction of a triangle on three sides.
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