Look at the picture. (Fig. 1)
Rice. 1. Illustration for example
What geometric shapes are familiar to you?
Of course, you saw that the picture consists of triangles and rectangles. What word is hidden in the name of both these figures? This word is an angle (Fig. 2).
Rice. 2. Determining the angle
Today we will learn how to draw a right angle.
The name of this angle already has the word "straight". To correctly depict a right angle, we need a square. (Fig. 3)
Rice. 3. Square
The square itself already has a right angle. (Fig. 4)
Rice. 4. Right angle
He will help us to depict this geometric figure.
To correctly depict the figure, we must attach the square to the plane (1), circle its sides (2), name the vertex of the angle (3) and the rays (4).
1. 
2. 
3. 
4. 
Let's determine if there are straight lines among the available angles (Fig. 5). A square will help us with this.
Rice. 5. Illustration for example
Let's find the right angle of the square and apply it to the existing angles (Fig. 6).
Rice. 6. Illustration for example
We see that the right angle coincided with the PTO angle. This means that the PTO angle is right. Let's do the same operation again. (Fig. 7)
Rice. 7. Illustration for example
We see that the right angle of our square did not coincide with the COD angle. This means that the angle COD is not a right angle. Once again we apply the right angle of the square to the angle AOT. (Fig. 8)
Rice. 8. Illustration for example
We see that the AOT angle is much larger than the right angle. This means that the AOT angle is not a right angle.
In this lesson, we learned how to build a right angle using a square.
The word "angle" gave the name to many things, as well as geometric shapes: a rectangle, a triangle, a square, with which you can draw a right angle.
A triangle is a geometric figure that consists of three sides and three angles. A triangle that has a right angle is called a right triangle.
DIRECT, oh, oh; straight, straight, straight, straight and straight. Explanatory dictionary of Ozhegov. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 ... Explanatory dictionary of Ozhegov
right angle- — Topics oil and gas industry EN right angle …
right angle is an angle equal to its adjacent angle. * * * RIGHT ANGLE RIGHT ANGLE, an angle equal to its adjacent ... encyclopedic Dictionary
RIGHT ANGLE- an angle equal to its adjacent one; in degrees is 90° ... Natural science. encyclopedic Dictionary
Right angle- see Angle... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron
RIGHT ANGLE- 1) an angle equal to its adjacent one. 2) Off-system unit. flat corner. Designation L. 1 L \u003d 90 ° \u003d PI / 2 rad 1.570 796 rad (see Radian) ... Big encyclopedic polytechnic dictionary
STRAIGHT- direct, direct; straight, straight, straight. 1. Exactly elongated in some kind. direction, no curve, no bends. Straight line. “The straight road broke off and was already going down.” Chekhov. Straight nose. Straight figure. 2. Direct (railroad and open). Direct route ... ... Explanatory Dictionary of Ushakov
STRAIGHT- DIRECT, oh, oh; straight, straight, straight, straight, straight. 1. Exactly walking in what n. direction, no bends. Straight line (a line, an endless tightly stretched thread can serve as a way to swarm). Draw a straight line (i.e., a straight line; n.). The road goes... ... Explanatory dictionary of Ozhegov
angle of the main profile of the coil- (αb) Angle between the main coil profile of an involute worm and a straight line making a right crossing angle with the worm axis. Note The angle of the rectilinear main profile of the involute worm coil αb is equal to the main angle of elevation ... ... Technical Translator's Handbook
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When decorating and building, a clear geometry is sometimes needed: perpendicular walls and other structures that require a right angle of 90 degrees. An ordinary square cannot allow you to check or mark corners with sides of several meters. The described method is excellent for marking or checking any corners - the length of the sides is not limited. The main measuring tool is a tape measure.
We will look at the exact marking of a right angle, as well as a method for checking already marked angles on walls and other objects.
Pythagorean theorem
The theorem is based on the assertion that in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. This is written as a formula:
a²+b²=c²
Sides a and b are legs, between which the angle is exactly 90 degrees. Therefore, side c is the hypotenuse. Substituting two known quantities into this formula, we can calculate the third, unknown. And therefore we can mark right angles, as well as check them.
The Pythagorean theorem is also known as the "Egyptian triangle". This is a triangle with sides 3, 4 and 5, and it does not matter at all in what units the length is. There are exactly ninety degrees between sides 3 and 4. Let's check this statement with the above formula: a²+b²=c² = (3×3)+(4×4) = 9+16 = (5×5) = 25 - everything converges!
Now let's put the theorem into practice.
Right Angle Check
Let's start with the simplest - checking the right angle using the Pythagorean theorem. The most common example in decoration and construction is checking perpendicularity walls. Perpendicular walls are walls located at a right angle of 90° to each other.
So, we take any checked internal corner. On the walls (at the same height) or on the floor, we mark segments of arbitrary lengths on both walls. The length of these segments is arbitrary, if possible, you need to mark as much as possible, but so that it is convenient to measure the diagonal between the marks on the walls. For example, we marked 2.5 meters (or 250 cm) on one wall and 3 meters (or 300 cm) on the other. Now we square the length of the segment of each wall (multiply by itself) and add the resulting products. It looks like this: (2.5 × 2.5) + (3 × 3) \u003d 15.25 - this is the diagonal squared. Now we need to extract from this number the square root √15.25≈3.90 - 3.9 meters should be the diagonal between our marks. If the measurement with a tape measure shows a different diagonal length, the angle being checked is unfolded and has a deviation from 90 °.
Right Angle Diagonal Calculator
Attention! For the calculator to work, support must be enabled JavaScript in your browser!
Length a
Length b
Diagonal c
Extracting the square root has never attracted me - an ordinary person cannot do without a calculator, moreover, not all calculators on mobile devices can extract it. Therefore, a simplified method can be used. You just need to remember: at a right angle with sides exactly 100 centimeters, the diagonal is 141.4 cm. Thus, at a right angle with sides of 2 m, the diagonal is 282.8 cm. That is, for each meter of the plane there are 141.4 cm. This method has one drawback: from the measured angle, you need to set aside the same distances on both walls and these segments must be a multiple of a metre. I will not say, but in my humble practice - it is much more convenient. Although you should not forget about the original method completely - in some cases it is very relevant.
The question immediately arises: what deviation from the calculated length of the diagonal is considered the norm (error), and what is not? If the checked angle with marked sides of 1 m is 89 °, then the diagonal will decrease to 140 cm. From understanding this dependence, we can make an objective conclusion that a diagonal error of 141.4 cm of a few millimeters will not give a deviation of one whole degree.
How to check the outer corner? Checking the outer corner is essentially the same, you just need to extend the lines of each wall on the floor (or ground, with a cord) and measure the resulting inner corner in the usual way.
How to mark a right angle with a tape measure
The markup can be based both on the general Pythagorean theorem and on the principle of the "Egyptian triangle". However, it is only in theory that the lines are simply drawn on paper, while “catching” all the selected sizes with stretched cords or lines on the floor is a more difficult task.
Therefore, I propose a simplified method based on a diagonal of 141.4 cm from a triangle with sides of 100 cm. The entire marking sequence is shown in the pictures below. It is important not to forget: the diagonal of 141.4 cm must be multiplied by the number of meters in the segment A-B. Segments A-B and A-C must be equal and correspond to a whole number in meters. Pictures enlarge with a click!
How to mark an acute angle
Much less often there is a need to create sharp corners, in particular 45 °. For the formation of such figures, the formulas are more complex, but this is not the most problematic. It is much more difficult to reduce all the lines drawn or stretched with cords - this is not an easy task. Therefore, I suggest using a simplified method. First, a right angle of 90 ° is marked, and then the diagonal 141.4 is divided into the required number of equal parts. For example, to get 45 °, the diagonal must be divided in half and from point A draw a line through the division. This will give us two 45 degree angles. If you divide the diagonal into 3 parts, you get three angles of 30 degrees. I think the algorithm is clear to you.
Actually, I told everything that I could tell, I hope I explained everything in an understandable language and you will no longer have questions about how to mark and check right angles. It is worth adding that any finisher or builder should be able to do this, because relying on a small construction square is unprofessional.
Look at the picture. (Fig. 1)
Rice. 1. Illustration for example
What geometric shapes are familiar to you?
Of course, you saw that the picture consists of triangles and rectangles. What word is hidden in the name of both these figures? This word is an angle (Fig. 2).
Rice. 2. Determining the angle
Today we will learn how to draw a right angle.
The name of this angle already has the word "straight". To correctly depict a right angle, we need a square. (Fig. 3)
Rice. 3. Square
The square itself already has a right angle. (Fig. 4)
Rice. 4. Right angle
He will help us to depict this geometric figure.
To correctly depict the figure, we must attach the square to the plane (1), circle its sides (2), name the vertex of the angle (3) and the rays (4).
1.
2.
3.
4.
Let's determine if there are straight lines among the available angles (Fig. 5). A square will help us with this.
Rice. 5. Illustration for example
Let's find the right angle of the square and apply it to the existing angles (Fig. 6).
Rice. 6. Illustration for example
We see that the right angle coincided with the PTO angle. This means that the PTO angle is right. Let's do the same operation again. (Fig. 7)
Rice. 7. Illustration for example
We see that the right angle of our square did not coincide with the COD angle. This means that the angle COD is not a right angle. Once again we apply the right angle of the square to the angle AOT. (Fig. 8)
Rice. 8. Illustration for example
We see that the AOT angle is much larger than the right angle. This means that the AOT angle is not a right angle.
In this lesson, we learned how to build a right angle using a square.
The word "angle" gave the name to many things, as well as geometric shapes: a rectangle, a triangle, a square, with which you can draw a right angle.
A triangle is a geometric figure that consists of three sides and three angles. A triangle that has a right angle is called a right triangle.
The angle is the main geometric figure, which we will analyze throughout the topic. Definitions, methods of setting, notation and measurement of the angle. Let's analyze the principles of selecting corners in the drawings. The whole theory is illustrated and has a large number of visual drawings.
Definition 1Injection- a simple important figure in geometry. The angle directly depends on the definition of a ray, which in turn consists of the basic concepts of a point, a line and a plane. For a thorough study, you need to delve into the topics straight line on a plane - necessary information and plane - necessary information.
The concept of an angle begins with the concepts of a point, a plane, and a straight line depicted on this plane.
Definition 2
Given a line a on a plane. Denote some point O on it. The line is divided by a point into two parts, each of which has a name Ray, and the point O is beam start.
In other words, a beam or half-line - it is a part of a line, consisting of points of a given line, located on the same side relative to the starting point, that is, the point O.
The designation of the beam is allowed in two variations: one lowercase or two uppercase letters of the Latin alphabet. When denoted by two letters, the beam has a name consisting of two letters. Let's take a closer look at the drawing.
Let's move on to the concept of defining an angle.
Definition 3
Injection- this is a figure located in a given plane, formed by two mismatched rays that have a common origin. side corner is a beam vertex- the common beginning of the parties.
There is a case when the sides of an angle can act as a straight line.
Definition 4
When both sides of an angle are located on the same straight line or its sides serve as additional half-lines of one straight line, then such an angle is called deployed.
The figure below shows a flattened corner.
A point on a straight line is the vertex of the angle. Most often, it is denoted by the dot O.
An angle in mathematics is denoted by the sign "∠". When the sides of an angle are denoted by small Latin, then for the correct definition of the angle, letters are written in a row, respectively, according to the sides. If two sides are denoted k and h, then the angle is denoted as ∠ k h or ∠ h k .
When there is a designation in capital letters, then, respectively, the sides of the corner have the names O A and O B. In this case, the angle has a name of three letters of the Latin alphabet, written in a row, in the center with a vertex - ∠ A O B and ∠ B O A . There is a designation in the form of numbers when the corners do not have names or letters. Below is a figure where angles are indicated in different ways.
An angle divides the plane into two parts. If the angle is not developed, then one part of the plane has the name inner corner area, the other - outer corner area. Below is an image explaining which parts of the plane are external and which are internal.
When divided by a straight angle on a plane, any of its parts is considered to be the interior of the straight angle.
The inner area of the corner is an element that serves for the second definition of the corner.
Definition 5
corner a geometric figure is called, consisting of two non-coinciding rays having a common origin and a corresponding inner region of the angle.
This definition is more rigorous than the previous one, as it has more conditions. It is not advisable to consider both definitions separately, because an angle is a geometric figure transformed using two rays coming out of one point. When it is necessary to perform actions with an angle, then the definition means the presence of two rays with a common origin and an internal region.
Definition 6The two corners are called related, if there is a common side, and the other two are complementary half-lines or form a straight angle.
The figure shows that adjacent corners complement each other, as they are a continuation of one another.
Definition 7
The two corners are called vertical, if the sides of one are complementary half-lines of the other or are extensions of the sides of the other. The figure below shows an image of the vertical corners.
When crossing lines, 4 pairs of adjacent and 2 pairs of vertical angles are obtained. Below is shown in the picture.
The article shows the definitions of equal and unequal angles. We will analyze which angle is considered large, which is smaller, and other properties of the angle. Two figures are considered equal if, when superimposed, they completely coincide. The same property applies to comparing angles.
Given two angles. It is necessary to come to the conclusion whether these angles are equal or not.
It is known that the vertices of two corners and the side of the first corner overlap with any other side of the second. That is, in case of complete coincidence, when the angles are superimposed, the sides of the given angles will coincide completely, the angles equal.
It may be that when superimposing the sides may not be combined, then the corners unequal, smaller of which consists of another, and more incorporates a complete other angle. Below are unequal angles not aligned when superimposed.
The developed angles are equal.
The measurement of angles begins with the measurement of the side of the measured angle and its inner region, filling which with unit angles, they are applied to each other. It is necessary to count the number of stacked corners, they predetermine the measure of the measured angle.
An angle unit can be expressed in any measurable angle. There are generally accepted units of measurement that are used in science and technology. They specialize in other titles.
The most commonly used concept degree.
Definition 8
one degree is called an angle that has one hundred and eightieth of a straightened angle.
The standard notation for a degree is "°", then one degree is 1°. Therefore, a straight angle consists of 180 such angles, consisting of one degree. All available corners are tightly stacked to each other and the sides of the previous one are aligned with the next.
It is known that the number of degrees in an angle is the same measure of the angle. The developed corner has 180 stacked corners in its composition. The figure below shows examples where the angle is laid 30 times, that is, one sixth of the expanded, and 90 times, that is, half.
Minutes and seconds are used to accurately determine angle measurements. They are used when the angle value is not an integer degree designation. Such parts of a degree allow you to perform more accurate calculations.
Definition 9
minute called one sixtieth of a degree.
Definition 10
second called one sixtieth of a minute.
A degree contains 3600 seconds. Minutes denote """, and seconds """". The designation takes place:
1°=60"=3600"", 1"=(160)°, 1"=60"", 1""=(160)"=(13600)°,
and the notation for the angle 17 degrees 3 minutes and 59 seconds is 17° 3 "59"".
Definition 11
Let's give an example of the notation of the degree measure of an angle equal to 17 ° 3 "59" ". The entry has another form 17 + 3 60 + 59 3600 \u003d 17 239 3600.
To accurately measure angles, a measuring device such as a protractor is used. When designating the angle ∠ A O B and its degree measure of 110 degrees, a more convenient notation is used ∠ A O B \u003d 110 °, which reads "Angle A O B is equal to 110 degrees."
In geometry, an angle measure from the interval (0 , 180 ] is used, and in trigonometry an arbitrary degree measure is called turning angles. The value of the angles is always expressed as a real number. Right angle is an angle that has 90 degrees. Sharp corner is an angle that is less than 90 degrees, and blunt- more.
An acute angle is measured in the interval (0, 90) , and an obtuse angle - (90, 180) . Three types of angles are clearly shown below.
Any degree measure of any angle has the same value. A larger angle, respectively, has a larger degree measure than a smaller one. The degree measure of one angle is the sum of all available degree measures of interior angles. The figure below shows the angle AOB, consisting of the angles AOC, COD and DOB. In detail, it looks like this: ∠ A O B = ∠ A O C + ∠ D O B = 45 ° + 30 ° + 60 ° = 135 °.
Based on this, it can be concluded that sum all adjacent angles is 180 degrees because they all make up an expanded angle.
It follows from this that any vertical angles are equal. If we consider this with an example, we get that the angle A O B and C O D are vertical (in the drawing), then the pairs of angles A O B and B O C, C O D and B O C are considered adjacent. In such a case, the equality ∠ A O B + ∠ B O C = 180 ° together with ∠ C O D + ∠ B O C = 180 ° are considered uniquely true. Hence we have that ∠ A O B = ∠ C O D . Below is an example of the image and designation of vertical catches.
In addition to degrees, minutes and seconds, another unit of measurement is used. It is called radian. Most often it can be found in trigonometry when designating the angles of polygons. What is called a radian.
Definition 12
One radian angle called the central angle, which has a radius of a circle equal to the length of the arc.
In the figure, the radian is depicted as a circle, where there is a center, indicated by a point, with two points on the circle connected and converted into radii O A and O B. By definition, this triangle A O B is equilateral, which means that the length of the arc A B is equal to the lengths of the radii O B and Oh A.
The designation of the angle is taken as "rad". That is, an entry in 5 radians is abbreviated as 5 rad. Sometimes you can find a designation that has the name pi. Radians do not depend on the length of a given circle, since the figures have some kind of restriction with the help of an angle and its arc with a center located at the vertex of a given angle. They are considered similar.
Radians have the same meaning as degrees, only the difference is in their magnitude. To determine this, it is necessary to divide the calculated length of the arc of the central angle by the length of its radius.
In practice, they use convert degrees to radians and radians to degrees for easier problem solving. The specified article has information about the connection between the degree measure and the radian, where you can study in detail the translations from degree to radian and vice versa.
For a visual and convenient depiction of arcs, angles, drawings are used. It is not always possible to correctly depict and mark a particular angle, arc or name. Equal angles have the designation in the form of the same number of arcs, and unequal in the form of different ones. The drawing shows the correct designation of sharp, equal and unequal angles.
When more than 3 corners need to be marked, special arc designations are used, such as wavy or jagged. It doesn't matter that much. The figure below shows their designation.
The designation of the angles should be simple so as not to interfere with other values. When solving a problem, it is recommended to select only the corners necessary for solving so as not to clutter up the entire drawing. This will not interfere with the solution and proof, and will also give an aesthetic appearance to the drawing.
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