Geometric figure angle: definition of angle, measurement of angles, notations and examples. Right angle
Each angle, depending on its size, has its own name:
| Angle type | Size in degrees | Example |
|---|---|---|
| Spicy | Less than 90° | |
| Straight | Equal to 90°. In a drawing, a right angle is usually denoted by a symbol drawn from one side of the angle to the other. |
 |
| Blunt | More than 90° but less than 180° |  |
| Expanded | Equal to 180° A straight angle is equal to the sum of two right angles, and a right angle is half of a straight angle. |
 |
| Convex | More than 180° but less than 360° |  |
| Full | Equal to 360° |  |
The two angles are called adjacent, if they have one side in common, and the other two sides form a straight line:
Angles MOP And PON adjacent, since the beam OP- the common side, and the other two sides - OM And ON make up a straight line.
The common side of adjacent angles is called oblique to straight, on which the other two sides lie, only in the case when adjacent angles are not equal to each other. If adjacent angles are equal, then their common side will be perpendicular.
The sum of adjacent angles is 180°.
The two angles are called vertical, if the sides of one angle complement the sides of the other angle to straight lines:
Angles 1 and 3, as well as angles 2 and 4, are vertical.
Vertical angles are equal.
Let us prove that the vertical angles are equal:
The sum of ∠1 and ∠2 is a straight angle. And the sum of ∠3 and ∠2 is a straight angle. So these two amounts are equal:
∠1 + ∠2 = ∠3 + ∠2.
In this equality, there is an identical term on the left and right - ∠2. Equality will not be violated if this term on the left and right is omitted. Then we get it.
STRAIGHT, oh, oh; straight, straight, straight, straight and straight. Ozhegov's explanatory dictionary. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 … Ozhegov's Explanatory Dictionary
right angle- — Topics oil and gas industry EN right angle …
An angle equal to its adjacent one. * * * RIGHT ANGLE RIGHT ANGLE, an angle equal to its adjacent... encyclopedic Dictionary
An angle equal to its adjacent one; in degree measurement is equal to 90°... Natural science. encyclopedic Dictionary
See Angle... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron
1) an angle equal to its adjacent one. 2) Non-system unit. flat angle. Designation L. 1 L = 90° = PI/2 rad 1.570 796 rad (see Radian) ... Big Encyclopedic Polytechnic Dictionary
Straight, direct; straight, straight, straight. 1. Exactly elongated in some way. direction, not crooked, without bends. Straight line. “The straight road ended and was already going downhill.” Chekhov. Straight nose. Straight figure. 2. Direct (railway and unloading). Direct route... ... Ushakov's Explanatory Dictionary
STRAIGHT, oh, oh; straight, straight, straight, straight and straight. 1. Walking smoothly in which no. direction, without bending. Straight line (a line, the image of which can be an endless, tightly stretched thread). Draw a straight line (i.e., a straight line; noun). The road goes... ... Ozhegov's Explanatory Dictionary
angle of the main coil profile- (αb) The angle between the main profile of the involute worm coil and the straight line that makes 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 helix angle... ... Technical Translator's Guide
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- Mathematics. 2nd grade. Textbook. In 2 parts. Part 2, Moro M.I.. The textbook “Mathematics” is included in the educational system “School of Russia”. The textbook material allows you to implement a system-activity approach, organize differentiated training and...
The angle is the main geometric figure, which we will analyze throughout the entire topic. Definitions, methods of setting, notation and measurement of angle. Let's look at the principles of highlighting corners in drawings. The whole theory is illustrated and has a large number of visual drawings.
Definition 1Corner– 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 straight line and a plane. For a thorough study, you need to delve deeper into 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 straight line a on the plane. Let us denote a certain point O on it. A straight line is divided by a point into two parts, each of which has a name Ray, and point O – beginning of the beam.
In other words, the beam or half-straight – 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, point O.
The beam designation is allowed in two variations: one lowercase or two uppercase letters of the Latin alphabet. When designated 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 determining an angle.
Definition 3
Corner is a figure located in a given plane, formed by two divergent rays that have a common origin. Angle side is a ray vertex– common origin of the sides.
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 expanded.
The picture below shows a rotated corner.
A point on a straight line is the vertex of an angle. Most often it is designated by the point O.
An angle in mathematics is denoted by the sign “∠”. When the sides of an angle are designated by small Latin letters, then to correctly determine the angle, letters are written in a row corresponding to the sides. If two sides are designated k and h, then the angle is designated ∠ k h or ∠ h k.
When the designation is in capital letters, then, respectively, the sides of the angle are named O A and O B. In this case, the angle has a name made up 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 angles do not have names or letter designations. Below is a picture where angles are indicated in different ways.
An angle divides a plane into two parts. If the angle is not turned, then one part of the plane is called 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 developed angle on a plane, any of its parts is considered to be the interior region of the developed angle.
The inner area of the angle is an element that serves for the second definition of the angle.
Definition 5
Angle called a geometric figure consisting of two divergent rays that have a common origin and a corresponding internal angle area.
This definition is more strict 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 emanating from one point. When it is necessary to perform actions with an angle, the definition means the presence of two rays with a common beginning and an internal area.
Definition 6The two angles are called adjacent, if there is a common side, and the other two are additional half-lines or form a straight angle.
The figure shows that adjacent angles complement each other, since they are a continuation of one another.
Definition 7
The two angles are called vertical, if the sides of one are complementary half-lines of the other or are continuations of the sides of the other. The picture below shows an image of vertical angles.
When straight lines intersect, 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. Let's look at which angle is considered larger, 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.
Two angles are given. It is necessary to come to a conclusion whether these angles are equal or not.
It is known that there is an overlap of the vertices of two angles and the sides of the first angle with any other side of the second. That is, if there is a complete coincidence when the angles are superimposed, the sides of the given angles will align completely, the angles equal.
It may be that when superimposed the sides may not align, then the corners unequal, smaller of which consists of another, and more contains a complete different angle. Below are unequal angles that were not aligned when overlaid.
Straight angles are equal.
Measuring angles begins with measuring the side of the angle being measured and its internal area, filling which with unit angles and applying them to each other. It is necessary to count the number of laid angles, they predetermine the measure of the measured angle.
The angle unit can be expressed by any measurable angle. There are generally accepted units of measurement that are used in science and technology. They specialize in other titles.
The concept most often used degree.
Definition 8
One degree called an angle that has one hundred and eightieth part of a straight angle.
The standard designation for a degree is “°”, then one degree is 1°. Therefore, a straight angle consists of 180 such angles of one degree. All available corners are tightly laid to each other and the sides of the previous one are aligned with the next one.
It is known that the number of degrees in an angle is the very measure of the angle. An unfolded angle has 180 stacked angles in its composition. The figure below shows examples where the angle is laid 30 times, that is, one sixth of the unfolded, and 90 times, that is, half.
Minutes and seconds are used to accurately measure angles. They are used when the angle value is not a whole degree designation. These fractions of a degree allow for more accurate calculations.
Definition 9
in a minute called one sixtieth of a degree.
Definition 10
In a second called one sixtieth of a minute.
A degree contains 3600 seconds. Minutes are designated """, and seconds are """. The designation takes place:
1 ° = 60 " = 3600 "" , 1 " = (1 60) ° , 1 " = 60 "" , 1 "" = (1 60) " = (1 3600) ° ,
and the designation for an angle of 17 degrees 3 minutes and 59 seconds is 17 ° 3 "59"".
Definition 11
Let's give an example of the designation of the degree measure of an angle equal to 17 ° 3 "59 "". The entry has another form: 17 + 3 60 + 59 3600 = 17 239 3600.
To accurately measure angles, use a measuring device such as a protractor. When denoting the angle ∠ A O B and its degree measure of 110 degrees, a more convenient notation is used ∠ A O B = 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 rotation angles. The value of angles is always expressed as a real number. Right angle- This is an angle that has 90 degrees. Sharp corner– 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 has a correspondingly larger degree measure than a smaller one. The degree measure of one angle is the sum of all available degree measures of interior angles. Below is a figure showing the angle AOB, consisting of 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, we can conclude that sum everyone adjacent angles are equal to 180 degrees, because they all make up a straight angle.
It follows that any vertical angles are equal. If we consider this as an example, we find that the angles 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 this 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 denoting the angles of polygons. What is a radian called?
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 dot, with two points on the circle connected and transformed into radii O A and O B. By definition, this triangle A O B is equilateral, which means the length of the arc A B is equal to the lengths of the radii O B and O A.
The designation of the angle is taken to be “rad”. That is, writing 5 radians is abbreviated as 5 rad. Sometimes you can find a notation called pi. Radians do not depend on the length of a given circle, since the figures have a certain limitation by the angle and its arc with the center located at the vertex of the 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 arc length of the central angle by the length of its radius.
In practice they use converting degrees to radians and radians to degrees for more convenient problem solving. This article contains information about the connection between the degree measure and the radian, where you can study in detail the conversions from degrees to radians and vice versa.
Drawings are used to visually and conveniently depict arcs and angles. It is not always possible to correctly depict and mark this or that angle, arc or name. Equal angles are designated by the same number of arcs, and unequal angles by a different number. The drawing shows the correct designation of acute, equal and unequal angles.
When more than 3 corners need to be marked, special arc symbols are used, such as wavy or jagged. It's not that important. Below is a picture showing their designation.
Angle symbols should be kept simple so as not to interfere with other meanings. When solving a problem, it is recommended to highlight only the angles necessary for the solution, so as not to clutter the entire drawing. This will not interfere with the solution and proof, and will also give an aesthetic appearance to the drawing.
If you notice an error in the text, please highlight it and press Ctrl+Enter
During finishing work and construction, clear geometry is sometimes needed: perpendicular walls and other structures that require a right angle of 90 degrees. An ordinary square cannot check or mark corners with sides of several meters. The described method is excellent for marking or checking any angles - the length of the sides is not limited. The main tool for measurements is a tape measure.
We will look at accurately marking right angles, as well as a method for checking already marked angles on walls and other objects.
Pythagorean theorem
The theorem is based on the statement 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. By substituting two known quantities into this formula, we can calculate the third, unknown one. Therefore, we can mark right angles and also 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 in what units the lengths are. Between sides 3 and 4 is exactly ninety degrees. 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.
Checking right angle
Let's start with the simplest thing - checking a right angle using the Pythagorean theorem. The most common example in finishing and construction is checking perpendicularity walls Perpendicular walls are walls located at right angles of 90° to each other.
So, we take any tested internal angle. On the walls (at the same height) or on the floor, mark segments of arbitrary lengths on both walls. The length of these segments is arbitrary; if possible, you need to mark as many 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)=15.25 - this is the diagonal squared. Now we need to take the square root of this number √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 rotated and has a deviation from 90°.
Right angle diagonal calculator
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Length a
Length b
Diagonal c
Extracting the square root has never attracted me - an ordinary person cannot do without a calculator, and besides, not all mobile devices have calculators that can extract it. Therefore, you can use a simplified method. You just need to remember: at a right angle with sides exactly 100 centimeters, the diagonal is 141.4 cm. Thus, for a right angle with sides of 2 m, the diagonal is 282.8 cm. That is, for every meter of the plane there are 141.4 cm. This method has one drawback: from the measured angle it is necessary to set off the same distances on both walls and these segments must be multiples of a meter. I won’t claim it, but in my humble experience, it’s much more convenient. Although you should not forget about the original method completely - in some cases it is very relevant.
The question immediately arises: which deviation from the calculated length of the diagonal is considered normal (error), and which is not? If the angle being tested with marked sides of 1 m is 89°, then the diagonal will decrease to 140 cm. From understanding this dependence, we can draw an objective conclusion that an error of a few millimeters in the diagonal of 141.4 cm will not give a deviation of one whole degree.
How to check the outer corner? Checking the external corner is essentially no different, you just need to extend the lines of each wall on the floor (or ground, using a cord) and measure the resulting internal angle in the usual way.
How to mark a right angle with a tape measure
The marking can be based both on the general Pythagorean theorem and on the principle of the “Egyptian triangle”. However, this is only in theory, lines are simply drawn on paper, but “catching” all selected sizes with stretched cords or lines on the floor is a more difficult task.
Therefore, I propose a simplified method based on the diagonal of 141.4 cm for 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 segment A-B. Segments A-B and A-C must be equal and correspond to a whole number in meters. Pictures enlarge by clicking!
How to mark an acute angle
Much less often there is a need to create acute angles, in particular 45°. To form such figures, the formulas are more complex, but this is not the most problematic. It is much more difficult to connect 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°, you need to divide the diagonal in half and draw a line from point A through the division point. This way we get 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 presented everything in 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 you familiar with?
Of course, you saw that the picture consists of triangles and rectangles. What word is hidden in the names of both of these figures? This word is angle (Fig. 2).
Rice. 2. Angle determination
Today we will learn to draw a right angle.
The name of this angle already contains 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 depict this geometric figure.
To correctly depict the figure, we must attach the square to the plane (1), outline its sides (2), name the vertex of the angle (3) and the rays (4).
1. 
2. 
3. 
4. 
Let's determine whether among the available angles there are straight lines (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 coincides with the PTO angle. This means that the PTO angle is straight. Let's do the same operation again. (Fig. 7)
Rice. 7. Illustration for example
We see that the right angle of our square does not coincide with the angle COD. This means that the angle COD is not right. Once again we apply the right angle of the triangle to the angle AOT. (Fig. 8)
Rice. 8. Illustration for example
We see that angle AOT is much larger than a right angle. This means that angle AOT is not right.
In this lesson we learned how to construct a right angle using a square.
The word “angle” gives its name to many things, as well as geometric shapes: rectangle, triangle, 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.