Each angle, depending on its size, has its own name:
| Angle view | Size in degrees | Example |
|---|---|---|
| Spicy | Less than 90° | |
| Straight | Equal to 90°. In the drawing, a right angle is usually denoted by a symbol drawn from one side of the angle to the other. |
 |
| Blunt | Greater than 90° but less than 180° |  |
| deployed | Equals 180° A straight angle is equal to the sum of two right angles, and a right angle is half the straight angle. |
 |
| Convex | More than 180° but less than 360° |  |
| Full | Equals 360° |  |
The two corners are called related, if they have one side in common, and the other two sides form a straight line:
corners 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 if the 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 corners are called vertical, if the sides of one angle complement to straight lines the sides of another angle:
Angles 1 and 3, as well as angles 2 and 4, are vertical.
Vertical angles are equal.
Let's prove that 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 sums are equal:
∠1 + ∠2 = ∠3 + ∠2.
In this equality, on the left and on the right there is the same term - ∠2. Equality is not violated if this term on the left and on the right is omitted. Then we get.
In mathematical expressions, angles are often denoted by lowercase Greek letters: α, β, γ, θ, φ, etc. As a rule, these designations are also applied to the drawing to eliminate ambiguity in choosing the internal area of \u200b\u200bthe corner. To avoid confusion with pi, the symbol π is generally not used for this purpose. The letters ω and Ω are often used to denote solid angles (see below).
It is also common to represent an angle with three dot symbols, for example ∠ A B C . (\displaystyle \angle ABC.) In such a record B (\displaystyle B)- top, and A (\displaystyle A) And C (\displaystyle C) are points on different sides of the angle. In connection with the choice in mathematics of the direction of counting angles counterclockwise, it is customary to enumerate the points lying on the sides in the designation of the angle also counterclockwise. This convention allows for unambiguity in distinguishing between two flat corners with common sides but different interior regions. In cases where the choice of the interior area of a flat corner is clear from the context, or indicated in some other way, this convention may be violated. Cm. .
The notation of straight lines forming the sides of an angle is less commonly used. For example, ∠ (b c) (\displaystyle \angle (bc))- here it is assumed that it means inner corner triangle ∠ B A C (\displaystyle \angle BAC), α , which should be denoted ∠ (c b) (\displaystyle \angle (cb)).
So, for the figure on the right, the entries γ , ∠ A C B (\displaystyle \angle ACB) And ∠ (b a) (\displaystyle \angle (ba)) mean the same angle.
Sometimes lowercase Latin letters are used to denote angles ( a, b, c,...) and numbers.
In the drawings, corners are marked with small single, double or triple shackles running along the inside of the corner centered at the apex of the corner. The equality of angles can be marked by the same multiplicity of the arches or by the same number of transverse strokes on the arch. If it is necessary to indicate the direction of the angle reading, it is marked with an arrow on the bow. Right angles are marked not by arches, but by two connected equal segments arranged in such a way that together with the sides they form a small square, one of the vertices of which coincides with the vertex of the angle.
Angle measure
The measurement of angles in degrees goes back to Ancient Babylon, where the sexagesimal number system was used, traces of which have been preserved with us in the division of time and angles.
1 turn = 2π radians = 360° = 400 degrees.
In nautical terminology, angles are measured in points. 1 rhumb is equal to 1 ⁄ 32 from the full circle (360 degrees) of the compass, i.e. 11.25 degrees, or 11°15′.
In some contexts, such as identifying a point in polar coordinates or describing the orientation of an object in two dimensions relative to its base orientation, angles that differ by an integer number of full revolutions are effectively equivalent. For example, in such cases, the angles 15° and 360015° (= 15° + 360°×1000) can be considered equivalent. In other contexts, such as identifying a point on a spiral curve, or describing the cumulative rotation of an object in two dimensions about its initial orientation, angles that differ by a non-zero integer number of complete revolutions are not equivalent.
Some flat corners have special names. In addition to the above units of measurement (radian, rhumb, degree, etc.), these include:
- quadrant (right angle, 1 ⁄ 4 circles);
- sextant ( 1 ⁄ 6 circles);
- octant ( 1 ⁄ 8 circles; in addition, in stereometry, an octant is a trihedral angle formed by three mutually perpendicular planes),
Angles direction
The arrow shows the direction of counting the angles
Solid angle
A generalization of a plane angle to stereometry is a solid angle - a part of space that is the union of all rays emanating from a given point ( peaks corner) and intersecting some surface (which is called a surface, tightening given solid angle).
Solid angles are measured in steradians (one of the basic SI units), as well as in off-system units - in parts of a full sphere (that is, a full solid angle of 4π steradians), in square degrees, square minutes and square seconds.
Solid angles are, in particular, the following geometric bodies:
- dihedral angle - a part of space bounded by two intersecting planes;
- trihedral angle - a part of space bounded by three intersecting planes;
- polyhedral angle - a part of space bounded by several planes intersecting at one point.
A dihedral angle can be characterized by both a linear angle (the angle between the planes forming it) and a solid angle (any point on it can be chosen as a vertex). edge- the direct intersection of its faces). If the linear angle of a dihedral angle (in radians) is φ, then its solid angle (in steradians) is 2φ.
Angle between curves
Both in planimetry and solid geometry, as well as in a number of other geometries, it is possible to determine the angle between smooth curves at the intersection point: by definition, its value is equal to the angle between the tangents to the curves at the intersection point.
Angle and dot product
The concept of an angle can be defined for linear spaces of arbitrary nature (and arbitrary, including infinite dimension), on which a positive definite scalar product is axiomatically introduced (x , y) (\displaystyle (x,y)) between two space elements x (\displaystyle x) And y . (\displaystyle y.) The scalar product also allows us to define the so-called norm (length) of an element as the square root of the product of the element and itself | | x | | = (x , x) . (\displaystyle ||x||=(\sqrt ((x,x))).) From the axioms of the scalar product, the Cauchy-Bunyakovsky (Cauchy-Schwartz) inequality follows for the scalar product: | (x, y) | ⩽ | | x | | ⋅ | | y | | , (\displaystyle |(x,y)|\leqslant ||x||\cdot ||y||,) whence it follows that the value takes values from −1 to 1, and the extreme values are reached if and only if the elements are proportional (collinear) to each other (geometrically speaking, their directions coincide or are opposite). This allows us to interpret the relationship (x, y) | | x | | ⋅ | | y | | (\displaystyle (\frac ((x,y))(||x||\cdot ||y||))) as the cosine of the angle between the elements x (\displaystyle x) And y . (\displaystyle y.) In particular, elements are said to be orthogonal if the dot product (or cosine of an angle) is zero.
In particular, one can introduce the concept of the angle between continuous on some interval [ a , b ] (\displaystyle ) functions if we introduce the standard scalar product (f , g) = ∫ a b f (x) g (x) d x , (\displaystyle (f,g)=\int _(a)^(b)f(x)g(x)dx,) then the norms of the functions are defined as | | f | | 2 = ∫ a b f 2 (x) d x . (\displaystyle ||f||^(2)=\int _(a)^(b)f^(2)(x)dx.) Then the cosine of an angle is defined in a standard way as the ratio of the scalar product of functions to their norms. Functions can also be called orthogonal if their dot product (the integral of their product) is zero.
In Riemannian geometry, one can similarly define the angle between tangent vectors using the metric tensor g i j . (\displaystyle g_(ij).) Dot product of tangent vectors u (\displaystyle u) And v (\displaystyle v) in tensor notation will look like: (u , v) = g i j u i v j , (\displaystyle (u,v)=g_(ij)u^(i)v^(j),) respectively, the norms of the vectors - | | u | | = | g i j u i u j | (\displaystyle ||u||=(\sqrt (|g_(ij)u^(i)u^(j)|))) And | | v | | = | g i j v i v j | . (\displaystyle ||v||=(\sqrt (|g_(ij)v^(i)v^(j)|)).) Therefore, the cosine of the angle will be determined by the standard formula for the ratio of the specified scalar product to the norms of vectors: cos θ = (u, v) | | u | | ⋅ | | v | | = g i j u i v j | g i j u i u j | ⋅ | g i j v i v j | . (\displaystyle \cos \theta =(\frac ((u,v))(||u||\cdot ||v||))=(\frac (g_(ij)u^(i)v^( j))(\sqrt (|g_(ij)u^(i)u^(j)|\cdot |g_(ij)v^(i)v^(j)|))).)
Angle in metric space
There are also a number of works in which the concept of an angle between elements of a metric space is introduced.
Let (X , ρ) (\displaystyle (X,\rho))- metric space. Let further x , y , z (\displaystyle x,y,z)- elements of this space.
K. Menger introduced the concept angle between vertices y (\displaystyle y) And z (\displaystyle z) with top at point x (\displaystyle x) as a non-negative number y x z ^ (\displaystyle (\widehat(yxz))), which satisfies three axioms:
In 1932, Wilson considered the following expression as an angle:
Y x z ^ w = arccos ρ 2 (x , y) + ρ 2 (x , z) − ρ 2 (y , z) 2 ρ (x , y) ρ (x , z) (\displaystyle (\widehat ( yxz))_(w)=\arccos (\frac (\rho ^(2)(x,y)+\rho ^(2)(x,z)-\rho ^(2)(y,z)) (2\rho (x,y)\rho (x,z))))
It is easy to see that the introduced expression always makes sense and satisfies Menger's three axioms.
In addition, the Wilson angle has the property that in Euclidean space it is equivalent to the angle between the elements y − x (\displaystyle y-x) And z−x (\displaystyle z-x) in the sense of Euclidean space.
Angle measurement
One of the most common tools for constructing and measuring angles is a protractor (as well as a ruler - see below); as a rule, it is used to construct an angle of a certain magnitude. Many tools have been developed to measure angles more or less accurately:
- goniometer - a device for laboratory measurement of angles;
What is an adjacent angle
Corner- This geometric figure(Fig. 1), formed by two rays OA and OB (sides of the corner), emanating from one point O (vertex of the corner).
ADJACENT CORNERS are two angles whose sum is 180°. Each of these angles complements the other to a full angle.
Adjacent corners- (Agles adjacets) those that have a common top and a common side. Predominantly, this name refers to such angles, of which the other two sides lie in opposite directions of one straight line drawn through.
Two angles are called adjacent if they have one side in common and the other sides of these angles are complementary half-lines.
rice. 2
In Figure 2, angles a1b and a2b are adjacent. They have a common side b, and the sides a1, a2 are additional half-lines.
rice. 3
Figure 3 shows line AB, point C is located between points A and B. Point D is a point not lying on line AB. It turns out that angles BCD and ACD are adjacent. They have a common side CD, and sides CA and CB are additional half-lines of line AB, since points A, B are separated by the initial point C.
Adjacent angle theorem
Theorem: sum of adjacent angles is 180°
Proof:
Angles a1b and a2b are adjacent (see Fig. 2) Beam b passes between sides a1 and a2 of a straightened angle. Therefore, the sum of the angles a1b and a2b is equal to the straight angle, i.e. 180°. The theorem has been proven.
An angle equal to 90° is called a right angle. From the theorem on the sum of adjacent angles it follows that the angle adjacent to a right angle is also a right angle. An angle less than 90° is called acute, and an angle greater than 90° is called obtuse. Since the sum of adjacent angles is 180°, then the angle adjacent to an acute angle is an obtuse angle. An angle adjacent to an obtuse angle sharp corner.
Adjacent corners- two angles with a common vertex, one of the sides of which is common, and the remaining sides lie on the same straight line (not coinciding). The sum of adjacent angles is 180°.
Definition 1. An angle is a part of a plane bounded by two rays with a common origin.
Definition 1.1. An angle is a figure consisting of a point - the vertex of the angle - and two different half-lines emanating from this point - the sides of the angle.
For example, the BOS angle in Fig. 1 Consider first two intersecting lines. When they intersect, lines form angles. There are special cases:
Definition 2. If the sides of an angle are complementary half-lines of one straight line, then the angle is called a straight angle.
Definition 3. A right angle is an angle of 90 degrees.
Definition 4. An angle less than 90 degrees is called an acute angle.
Definition 5. An angle greater than 90 degrees and less than 180 degrees is called an obtuse angle.
intersecting lines.
Definition 6. Two angles, one side of which is common, and the other sides lie on the same straight line, are called adjacent.
Definition 7. Angles whose sides extend each other are called vertical angles.
Figure 1:
adjacent: 1 and 2; 2 and 3; 3 and 4; 4 and 1
vertical: 1 and 3; 2 and 4
Theorem 1. The sum of adjacent angles is 180 degrees.
For proof, consider Fig. 4 adjacent corners AOB and BOS. Their sum is the developed angle AOC. Therefore, the sum of these adjacent angles is 180 degrees.
rice. 4
Relationship between mathematics and music
"Thinking about art and science, about their mutual connections and contradictions, I came to the conclusion that mathematics and music are at the extreme poles of the human spirit, that these two antipodes limit and determine all the creative spiritual activity of a person, and that everything is placed between them, what mankind has created in the field of science and art."
G. Neuhaus
It would seem that art is a very abstract area from mathematics. However, the connection between mathematics and music is conditioned both historically and internally, despite the fact that mathematics is the most abstract of the sciences, and music is the most abstract art form.
Consonance determines the sound of a string that is pleasing to the ear.
This musical system was based on two laws, which bear the names of two great scientists - Pythagoras and Archytas. These are the laws:
1. Two sounding strings determine consonance if their lengths are related as integers forming a triangular number 10=1+2+3+4, i.e. like 1:2, 2:3, 3:4. Moreover, the smaller the number n in relation to n:(n+1) (n=1,2,3), the more consonant the resulting interval.
2. The vibration frequency w of a sounding string is inversely proportional to its length l.
w = a:l,
where a is a coefficient characterizing physical properties strings.
I will also offer your attention a funny parody about a dispute between two mathematicians =)
Geometry around us
Geometry plays an important role in our life. Due to the fact that when you look around, it will not be difficult to notice that we are surrounded by various geometric shapes. We encounter them everywhere: on the street, in the classroom, at home, in the park, in the gym, in the school cafeteria, in principle, wherever we are. But the topic of today's lesson is adjacent coals. So let's look around and try to find corners in this environment. If you look carefully out the window, you can see that some branches of the tree form adjacent corners, and you can see many vertical corners in the partitions on the gate. Give your examples of adjacent angles that you see in the environment.
Exercise 1.
1. There is a book on the table on a book stand. What angle does it form?
2. But the student is working on a laptop. What angle do you see here?
3. What is the angle of the photo frame on the stand?
4. Do you think it is possible for two adjacent angles to be equal?
Task 2.
In front of you is a geometric figure. What is this figure, name it? Now name all the adjacent angles that you can see on this geometric figure.
Task 3.
Here is an image of a drawing and a painting. Look at them carefully and say what types of catch you see in the picture, and what angles in the picture.
Problem solving
1) Two angles are given, related to each other as 1: 2, and adjacent to them - as 7: 5. You need to find these angles.2) It is known that one of the adjacent angles is 4 times larger than the other. What are adjacent angles?
3) It is necessary to find adjacent angles, provided that one of them is 10 degrees greater than the second.
Mathematical dictation for the repetition of previously learned material
1) Draw a picture: lines a I b intersect at point A. Mark the smallest of the formed corners with the number 1, and the remaining angles - sequentially with the numbers 2,3,4; the complementary rays of the line a - through a1 and a2, and the line b - through b1 and b2.2) Using the completed drawing, enter the necessary values and explanations in the gaps in the text:
a) angle 1 and angle .... related because...
b) angle 1 and angle .... vertical because...
c) if angle 1 = 60°, then angle 2 = ..., because ...
d) if angle 1 = 60°, then angle 3 = ..., because ...
Solve problems:
1. Can the sum of 3 angles formed at the intersection of 2 lines equal 100°? 370°?
2. In the figure, find all pairs of adjacent corners. And now the vertical corners. Name these angles.
3. You need to find an angle when it is three times larger than the one adjacent to it.
4. Two lines intersect each other. As a result of this intersection, four corners were formed. Determine the value of any of them, provided that:
a) the sum of 2 angles out of four 84 °;
b) the difference of 2 angles of them is 45°;
c) one angle is 4 times less than the second;
d) the sum of three of these angles is 290°.
Lesson summary
1. name the angles that are formed at the intersection of 2 lines?
2. Name all possible pairs of angles in the figure and determine their type.
Homework:
1. Find the ratio of the degree measures of adjacent angles when one of them is 54 ° more than the second.
2. Find the angles that are formed when 2 lines intersect, provided that one of the angles is equal to the sum of 2 other angles adjacent to it.
3. It is necessary to find adjacent angles when the bisector of one of them forms an angle with the side of the second, which is 60 ° greater than the second angle.
4. The difference of 2 adjacent angles is equal to a third of the sum of these two angles. Determine the values of 2 adjacent angles.
5. The difference and the sum of 2 adjacent angles are related as 1: 5, respectively. Find adjacent corners.
6. The difference between two adjacent ones is 25% of their sum. How are the values of 2 adjacent angles related? Determine the values of 2 adjacent angles.
Questions:
- What is an angle?
- What are the types of corners?
- What is the feature of adjacent corners?
- (lat. solutio triangulorum) a historical term meaning the solution of the main trigonometric problem: using known data about a triangle (sides, angles, etc.), find the rest of its characteristics. The triangle can be located on ... ... Wikipedia
- (mat.). If we draw straight lines OA and 0B from the point O on this plane, then we get the angle AOB (Fig. 1). Crap. 1. Point 0 ref. the apex of the angle, and the straight lines OA and 0B are the sides of the angle. Suppose that two angles ΒΟΑ and Β 1 Ο 1 Α 1 are given. Let us superimpose them so that ... ...
- (mat.). If we draw straight lines OA and 0B from the point O on this plane, then we get the angle AOB (Fig. 1). Crap. 1. Point 0 ref. the apex of the angle, and the straight lines OA and 0B are the sides of the angle. Let us assume that two angles ΒΟΑ and Β1Ο1Α1 are given. We superimpose them so that the vertices O ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron
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Two straight lines BA and BC (Fig. 13), intersecting at the same point B, form an angle at point B.
Angle detection. An angle is an indefinite part of a plane bounded by two intersecting straight lines. Angle is a quantity that determines the inclination of one straight line to another.
The sides of the corner. The intersecting lines are called the sides of the angle.
corner top. The point where two lines intersect is called the vertex of the angle.. The magnitude of an angle does not depend on the length of the sides, so the sides of the angle can be extended indefinitely.
Angle name. a) Angles are called the letter at the top; so damn angle. 13 is called angle B. b) If there are several corners at the top, then the corners are called three letters, standing at the top and two of its sides. In this case, the letter at the top is pronounced and written in the middle.
Damn it. 13 angle B is called angle ABC. Lines BA and BC are two sides, and point B is the vertex of the corner.
So angle ABC is angle B or
angle ABC = angle B.
Angle sign. The word angle is sometimes replaced by the sign∠ .
Thus, the previous equality is depicted in writing:
In the case when several lines come out of a point, there are several angles at point B.
Damn it. 14 straight lines BA, BC, BD come out of point B and at vertex B there are angles ABC, CBD, ABD.
Adjacent Angles. Two angles are called adjacent when they have a common vertex on one common side, and the other two lie on both sides of a common side.
Angles ABC and CBD (Fig. 14) are adjacent angles. They have a common vertex B, a common side BC, and the other two sides BA and BD lie one above and one below the common side BC.
Angles change their value if the inclination of one side to the other changes. Of two angles that have a common vertex, the angle inside which the other angle is placed is called the greater angle. On the drawing 14
corner ABD > angle ABC and ang. CBD< уг. ABD.
To have an idea of the mutual magnitude of two angles having different vertices, one angle is superimposed on the other. When superimposed, their vertices are combined on one side, then the direction of the other side will make it possible to compare their size. To compare the two angles ABC and DEF (Fig. 15), impose the angle DEF on the angle ABC so that the side EF goes along the side BC, the point E is aligned with the point B; then side ED can take three positions: it can coincide with side BA, fall inside and outside corner ABC.
a) If line ED coincides with line BA, the angles are said to be equal
corner ABC = angle DEF.
b) If line ED falls inside angle ABC and takes position BG, angle ABC will be greater than angle DEF
corner ABC > angle DEF.
c) If line ED falls outside angle ABC in direction BH, angle ABC is less than angle DEF
corner ABC< уг. DEF.
Addition, subtraction, multiplication and division of angles. Two adjacent angles ABC and CBD (Fig. 14) form one angle ABC. Angle ABD is called the sum of angles ABC and CBD. This is expressed in writing as:
∠ABD = ∠ABC + ∠CBD (a)
Equation (a) implies the equality:
∠ABC = ∠ABD - ∠CBD
∠CBD = ∠ABD - ∠ABC,
i.e. angle ABC is the difference between angles ABD and CBD, and angle CBD is the difference between angles ABD and ABC.
If at the point O (Fig. 16) there are several equal adjacent angles, i.e. if
∠AOB = ∠BOC = ∠COD = ∠DOE,
then angle AOC equal to the sum of angles AOB and BOC is equal to two angles AOB,
∠AOC = ∠AOB + ∠BOC, next. ∠AOC = 2AOB.
Angle AOD is equal to three angles AOB
Conversely, angle AOB is half of angle AOC, one third of angle AOD, one quarter of angle AOE.
AOB = ½ AOC = 1/3 AOD = ¼ AOE.
Hence we deduce that angles as quantities can not only be added and subtracted, but also multiplied and divided by an abstract number.
If from two adjacent angles ACD and DCB (Fig. 17) two sides CA and CB lie on the same straight line, they are called adjacent.
. Adjacent angles are those that have one side in common and the other two lie on the same straight line.
If the line CD, turning around the point C, takes the position CE, then the decreasing angle ACD will turn into the angle ACE, and the increasing angle BCD will turn into the angle BCE. The line CD, continuing to turn, can assume such a position that two adjacent angles become equal. When two adjacent angles ACD and DCB are equal (Fig. 18), they are called right angles.
In this case line CD is said to be perpendicular to line AB or simply perpendicular to line AB.
In drawing 19, one right angle is drawn without another adjacent to it.
A right angle is one of the congruent adjacent angles.
A perpendicular is a straight line that forms a right angle with another line.
In drawing 18, the angles ACD and DCB, while remaining adjacent and equal, are called right angles. Line DC will be perpendicular to line AB. Such a mutual relation of two lines is sometimes expressed in writing: CD ⊥ AB.
Since the line AB will also be perpendicular to the line CD, then the lines AB and CD will be mutually perpendicular, i.e. if CD ⊥ AB, then AB ⊥ CD.
Perpendicular sole. The point where two perpendicular lines meet is called the foot of the perpendicular.
Point C (drawing 18) is the foot of the perpendicular CD.
At each point on line AB, a perpendicular to line AB can be drawn.
To draw a perpendicular to the line (AB) from a point lying on the line means to set up a perpendicular. To draw a perpendicular (DC) to the line (AB) from a point (D) lying outside the line means to lower the perpendicular(Fig. 18).
oblique line . Any line not perpendicular to another is called a line inclined to it.
In drawing 20 line CE will be inclined to line AB and line CD will be perpendicular to line AB.
Angle ECB is less than a right angle and angle ACE is greater than a right angle. Angle ECB is called acute and angle ACE is called obtuse.
Sharp corner every angle is less than a right angle, A obtuse angle there is a greater than right angle.
Like and different angles. Two acute or two obtuse angles are called homonymous, and two angles, of which one is acute and the other obtuse, are called opposite.
The oblique line CE forms (Fig. 20) with the straight line AB two adjacent angles, one of which is smaller and the other is larger than the right angle, that is, one is acute and the other obtuse.
Theorem 3. From a point taken on a straight line, only one perpendicular can be drawn to it.
Dana line AB and point C on it (Fig. 20).
Required to prove that only one perpendicular can be drawn to it.
Proof. Let us suppose that it is possible to set up two perpendiculars from point C to line AB (Fig. 20) CD and CE. According to the property of the perpendicular
corner DCB = angle ACD(a)
corner BCE = angle A.C.E.
If we apply the angle ECD to the first part of the last inequality, we get the inequality
corner BCE + ang. ECD > ang. ACE, or ang. BCD > ang. A.C.E.
Replacing in this inequality y. BCD equal to its angle ACD (a), we get
corner DCA > angle Ace,
the inequality is obviously absurd, for a part cannot be greater than its whole, hence the assumption that two perpendiculars can be erected leads to absurdity, and therefore it is false. The falsity of the assumption is based on the consideration that from the correct position it is impossible to deduce an incorrect conclusion, therefore, our theorem is true.
The method of proving the validity of a given theorem by pointing out the impossibility and absurdity of any other assumption is called the method of proof by contradiction or the method of reduction to absurdity.
Theorem 4. All right angles are equal.
Suppose we have two pairs of right angles: one pair is made up of angles ACD and DCB, and the other is angles EGH and HGF, hence CD ⊥ AB and HG ⊥ EF (Fig. 21).
It is required to prove that the right angles are equal.
Proof. Let's superimpose the line EF on the line AB with the point G on the point C, then the line GH will go along the line CD, because only one perpendicular can be drawn from the point C, therefore, the right angle DCB = right angle HGF.
Conclusion. A right angle is a constant value.
measure of angles. When measuring angles, a right angle, as a constant value, is taken as a unit of comparison. Its value is denoted by the letter d.
In this case
any acute angle< d,
every obtuse angle > d.
All angles are expressed using a straight line. So, for example, they say: a given angle is equal to ½ d, 2/3 d, etc.
Theorem 5. The sum of two adjacent angles is equal to two right angles.
Adjacent angles ACD and DCB are given (Fig. 22).
It is required to prove that ACD + DCB = 2d.
Proof. From the point C we restore the perpendicular CE, then
ACD = ACE + ECD = d + ECD
DCB=ECB-ECD=d-ECD
Adding these equalities, we have:
ACD + DCB = ACE + ECB = 2d (which was to be proved).
Two adjacent angles complete one another to two right angles and are therefore called complementary angles.
Theorem 5 implies consequence. One pair of adjacent angles is equal to the other pair of adjacent angles.
Theorem 6(converse to Theorem 5). If the sum of two adjacent angles is equal to two right angles, then the other two sides lie on the same straight line.
Let the sum of two adjacent angles ACD and DCB be equal to two right angles (Fig. 23).
It is required to prove that ACB is a straight line.
Proof. Let us assume that ACB is a broken line and that the continuation of line AC is line CE, then
Two quantities equal to the same third are equal (axiom 3), therefore
ACD+DCB=ACD+DCE
where does it come from when reduced
the conclusion is absurd (the part is equal to the whole, see ax. 1), hence the line ACB is a straight line (which was to be proved).
Theorem 7. The sum of the angles having a vertex at one point and located on one side of a straight line is equal to two straight lines.
Given angles ACD, DCE, ECF, FCG, GCB, having a common vertex at point C and located on one side of line AB (Fig. 24).
It is required to prove that
ACD + DCE + ECF + FCG + GCB = 2d.
Proof. WE know that the sum of two adjacent angles ACF and FCB is equal to two right angles (point 5).
Since ACF = ACD + DCE + ECF and FCB = FCG + GCB, replacing the angles ACF and FCB with their values, we find:
ACD + DCE + ECF + FCG + GCB = 2d (which was to be proved).
Theorem 8. The sum of all angles located around one point is equal to four right angles.
Given angles AOB, BOC, COD, DOE, EOA, having a common vertex O and located around the point O (Fig. 25).
It is required to prove that
AOB + BOC + COD + DOE + EOA = 4d.
Proof. Let's continue the side EO in the direction of OG (Ch. 25), then
Similar
GOB + BOC + COD + DOE = 2d.
Adding these equalities, we have:
EOA + AOG + GOB + BOC + COD + DOE = 4d.
Since AOG + GOB = AOB, then
EOA + AOB + BOC + COD + DOE = 4d (phd).
Angle ACB with angle DCE and angle BCD with angle ACE are called vertical (Fig. 26).
Vertical angles. Vertical angles are those in which the sides of one are made up of the continuation of the sides of another angle.
Theorem 9. The vertical angles are equal to each other.
Given vertical angles (Fig. 26) ACB and DCE, just like BCD and ACE.
It is required to prove that ACB = DCE and BCD = ACE.
Proof. Based on Theorem 5, the following equalities hold:
ACB + BCD = 2d (as the sum of two adjacent angles)
BCD + DCE = 2d
hence,
ACB+BCD=BCD+DCE
whence, subtracting by equal angle BCD, we find
In the same way, they prove that
∠BCD = ∠ACE.
Equisecant (bisector ) there is a line that bisects the angle.
In drawing 27 BD, there is a bisector if ∠ABD = ∠DBC.
Theorem 10.
Adjacent angles ACB and BCD are given (Fig. 28). Their bisectors lines CF and CE bisect adjacent angles BCD and BCA, hence BCF = FCD, ACE = ECB.
It is required to prove that EC ⊥ CF.
Proof. By condition
ECB = ½ ACB, BCF = ½ BCD
Adding these equalities, we have:
ECB + BCF = ½ ACB + ½ BCD = ½ (ACB + BCD).
Since ACB + BCD = 2d, then
ECB + BCF = ½ 2d = d.
Since ECB + BCF = ECF, then
The angle ECF is right, i.e. the lines CE and CF are mutually perpendicular (PTD).
