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 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° |  |
| deployed | Equals 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 | Equals 360° |  |
The two angles are called related, 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'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 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.
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 the choice of the internal area of the angle. To avoid confusion with pi, the symbol π is generally not used for this purpose. To denote solid angles (see below), the letters ω and Ω are often used.
It is also common to denote an angle with three dot symbols, e.g. ∠ A B C . (\displaystyle \angle ABC.) In such a recording B (\displaystyle B)- top, and A (\displaystyle A) And C (\displaystyle C)- points lying on different sides of the angle. Due to the choice in mathematics of the direction of counting angles counterclockwise, it is customary to list points lying on the sides in the designation of an angle also counterclockwise. This convention allows for unambiguity in distinguishing between two planar angles with common sides but different interior regions. In cases where the choice of the interior region of a plane angle is clear from the context, or otherwise specified, this convention may be violated. Cm. .
Less commonly used are the designations of straight lines forming the sides of an angle. For example, ∠ (b c) (\displaystyle \angle (bc))- here it is assumed that what is meant is internal corner triangle ∠ B A C (\displaystyle \angle BAC), α , which should be designated ∠ (c b) (\displaystyle \angle (cb)).
So, for the figure on the right, notation γ, ∠ A C B (\displaystyle \angle ACB) And ∠ (b a) (\displaystyle \angle (ba)) mean the same angle.
Sometimes lowercase Latin letters ( a, b, c,...) and numbers.
In drawings, corners are marked with small single, double, or triple bows running along the inside area of the corner, centered at the apex of the corner. Equality of angles can be marked by the same multiplicity of the bows or the same number of transverse strokes on the bow. If it is necessary to indicate the direction of the angle, it is marked with an arrow on the bow. Right angles are marked not by arcs, but by two connected equal segments, located 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.
Angular 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 in our division of time and angles.
1 revolution = 2π radians = 360° = 400 degrees.
In nautical terminology, angles are measured in bearings. 1 rhumb is equal to 1 ⁄ 32 from the full circle (360 degrees) of the compass, that is, 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 reference orientation, angles that differ by an integer number of full revolutions are actually equivalent. For example, in such cases, 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 full revolutions are not equivalent.
Some plane angles have special names. In addition to the above-mentioned units of measurement (radian, rhumb, degree, etc.), these include:
- quadrant (right angle, 1 ⁄ 4 circle);
- sextant ( 1 ⁄ 6 circle);
- octant ( 1 ⁄ 8 circles; in addition, in stereometry, an octant is a trihedral angle formed by three mutually perpendicular planes),
Angle counting 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 angle) and intersecting some surface (which is called the surface, contracting given solid angle).
Solid angles are measured in steradians (one of the basic SI units), as well as in non-systemic units - in parts of a complete sphere (that is, a total solid angle of 4π steradians), in square degrees, square minutes and square seconds.
Solid angles are, in particular, the following geometric bodies:
- dihedral angle - part of space limited by two intersecting planes;
- trihedral angle - a part of space limited by three intersecting planes;
- polyhedral angle - a part of space limited 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). rib- the straight line of 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 elements of space x (\displaystyle x) And y. (\displaystyle y.) The scalar product also allows you 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 follows the Cauchy - Bunyakovsky (Cauchy - Schwarz) inequality for the scalar product: | (x, y) | ⩽ | | x | | ⋅ | | y | | , (\displaystyle |(x,y)|\leqslant ||x||\cdot ||y||,) from which it follows that the quantity takes values from −1 to 1, and extreme values are achieved 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 the angle) is zero.
In particular, we can introduce the concept of an angle between continuous lines on a certain 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 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 the angle is defined in a standard way as the ratio of the scalar product of functions to their norms. Functions can also be said to be orthogonal if their dot product (the integral of their product) is zero.
In Riemannian geometry, one can similarly determine 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),) accordingly, the norms of vectors are | | 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 vertex 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.
Measuring angles
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 size. Many tools have been developed to more or less accurately measure angles:
- 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 angle), emanating from one point O (vertex of the angle).
ADJACENT CORNERS- two angles whose sum is 180°. Each of these angles complements the other to the full angle.
Adjacent corners- (Agles adjacets) those that have a common top and a common side. Mostly this name refers to angles of which the remaining 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 sides a1, a2 are additional half-lines.
rice. 3
Figure 3 shows straight line AB, point C is located between points A and B. Point D is a point not lying on straight 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 straight line AB, since points A, B are separated by the starting point C.
Adjacent angle theorem
Theorem: the sum of adjacent angles is 180°
Proof:
Angles a1b and a2b are adjacent (see Fig. 2) Ray b passes between sides a1 and a2 of the unfolded angle. Therefore, the sum of angles a1b and a2b is equal to the developed angle, that is, 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 an 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. And the angle adjacent to an obtuse angle is sharp corner.
Adjacent corners- two angles with a common vertex, one of whose sides 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, angle BOC in Fig.1 Let us first consider two intersecting lines. When straight lines intersect, they form angles. There are special cases:
Definition 2. If the sides of an angle are additional half-lines of one straight line, then the angle is called developed.
Definition 3. A right angle is an angle measuring 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 continue each other are called vertical angles.
In 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 in Fig. 4 adjacent angles AOB and BOC. Their sum is the developed angle AOC. Therefore, the sum of these adjacent angles is 180 degrees.
rice. 4
The connection 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 all creative spiritual activity of man is limited and determined by these two antipodes and that everything lies between them. what humanity has created in the fields 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 determined both historically and internally, despite the fact that mathematics is the most abstract of the sciences, and music is the most abstract form of art.
Consonance determines the pleasant sound of a string
This musical system was based on two laws that 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 the triangular number 10=1+2+3+4, i.e. like 1:2, 2:3, 3:4. Moreover, the smaller the number n in the ratio n:(n+1) (n=1,2,3), the more consonant the resulting interval.
2. The vibration frequency w of the 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 you a funny parody about an argument between two mathematicians =)
Geometry around us
Geometry in our life is of no small importance. 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, basically wherever we are. But the topic of today's lesson is adjacent coals. So let's look around and try to find angles 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 angle does the photo frame form 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 kind of figure is this, 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 painting. Look at them carefully and tell me what types of fish you see in the picture, and what angles you see in the picture.
Problem solving
1) Given two angles 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 the adjacent angles equal to?
3) It is necessary to find adjacent angles, provided that one of them is 10 degrees greater than the second.
Mathematical dictation to review previously learned material
1) Complete the drawing: straight lines a I b intersect at point A. Mark the smaller of the formed angles with the number 1, and the remaining angles - sequentially with the numbers 2,3,4; the complementary rays of line a are through a1 and a2, and line b is through b1 and b2.2) Using the completed drawing, enter the necessary meanings and explanations in the gaps in the text:
a) angle 1 and angle .... adjacent 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 by the intersection of 2 straight lines equal 100°? 370°?
2. In the figure, find all pairs of adjacent angles. And now the vertical angles. Name these angles.
3. You need to find an angle when it is three times larger than its adjacent one.
4. Two straight lines intersected 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 is 84°;
b) the difference between 2 angles is 45°;
c) one angle is 4 times smaller than the second;
d) the sum of three of these angles is 290°.
Lesson summary
1. name the angles that are formed when 2 straight lines intersect?
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 between 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 sum of 2 adjacent angles are in the ratio 1:5 respectively. Find adjacent corners.
6. The difference between two adjacent ones is 25% of their sum. How do the values of 2 adjacent angles relate? Determine the values of 2 adjacent angles.
Questions:
- What is an angle?
- What types of angles are there?
- What is the property of adjacent angles?
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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. An angle is a quantity that determines the inclination of one straight line to another.
Sides of the corner. The intersecting lines are called the sides of the angle.
Top corner. The point of intersection of two lines is called the vertex of the angle. The size of the 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 by the letter at the vertex; so damn angle. 13 is called angle B. b) If there are several angles at the vertex, then the angles are called three letters standing at the vertex and its two sides. In this case, the letter at the top is pronounced and written in the middle.
Fuck it. 13 angle B is called angle ABC. Lines BA and BC are two sides, and point B is the vertex of the angle.
Thus 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 represented in writing:
In the case where several lines come out of a point, there are several angles at point B.
Fuck 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, one common side, and the other two lie on both sides of the common side.
Angles ABC and CBD (Fig. 14) are adjacent angles. They have a common vertex B, a common side BC, and two other sides BA and BD lie one above and the other below the common side BC.
Angles change their magnitude if the inclination of one side to the other changes. Of two angles that have a common vertex, the angle within which the other angle fits is called the major angle. On the drawing 14
ug. ABD > ang. ABC and ug. CBD< уг. ABD.
To have an idea of the mutual magnitude of two angles that have 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 values. To compare two angles ABC and DEF (Fig. 15), angle DEF is superimposed on angle ABC so that side EF goes along side BC, point E coincides with point B; then the side ED can take three positions: it can coincide with the side BA, it can fall inside and outside the angle ABC.
a) If line ED coincides with line BA, the angles are said to be equal
ug. ABC = ang. DEF.
b) If line ED falls inside angle ABC and takes position BG, angle ABC will be greater than angle DEF
ug. ABC > ang. DEF.
c) If line ED falls outside angle ABC in direction BH, angle ABC is less than angle DEF
ug. 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 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 angle AOC, one third angle AOD, one quarter angle AOE.
AOB = ½ AOC = 1/3 AOD = ¼ AOE.
From this we deduce that angles as quantities can not only be added and subtracted, but also multiplied and divided by an abstract number.
If of 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 in which one side is common and the other two lie on the same straight line.
If line CD, turning around point C, takes position CE, then angle ACD, decreasing, will turn into angle ACE, and angle BCD, increasing, will turn into angle BCE. Line CD, continuing to rotate, can take 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 called 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 equal adjacent angles.
A perpendicular is a straight line that forms a right angle with another line.
In drawing 18, angles ACD and DCB, while remaining adjacent and equal, are called right angles. Line DC will be perpendicular to line AB. This mutual relationship of two lines is sometimes expressed in writing: CD ⊥ AB.
Since line AB will also be perpendicular to line CD, then line AB and CD will be mutually perpendicular, i.e. if CD ⊥ AB, then AB ⊥ CD.
Perpendicular sole. The point of mutual meeting of two perpendicular lines is called the foot of the perpendicular.
Point C (Fig. 18) is the bottom of the perpendicular CD.
At each point on line AB you can draw a perpendicular to line AB.
To draw a perpendicular to a line (AB) from a point lying on the line means to construct a perpendicular. To draw a perpendicular (DC) to a line (AB) from a point (D) lying outside the line means to lower the perpendicular(Figure 18).
Sloping 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 right, and angle ACE is more than right. Angle ECB is called acute and angle ACE is called obtuse.
Sharp corner there is every angle less than a right angle, A obtuse angle there is an angle greater than a right angle.
Same and unlike angles. Two acute or two obtuse angles are called identical, and two angles, one acute and the other obtuse, are called opposite.
The inclined line CE forms (Fig. 20) with the straight line AB two adjacent angles, of which one is less and the other is greater than the right angle, that is, one is acute and the other is obtuse.
Theorem 3. From a point taken on a straight line, only one perpendicular can be constructed to it.
Dana straight line AB and point C on it (Fig. 20).
Required to prove, that only one perpendicular can be restored to it.
Proof. Let us assume that it is possible to construct two perpendiculars (Fig. 20) CD and CE from point C to line AB. According to the property of perpendicular
ug. DCB = ang. ACD(a)
ug. BCE = ang. ACE.
If we apply the angle ECD to the first part of the last inequality, we obtain the inequality
ug. BCE + ang. ECD > ang. ACE, or ug. BCD > ang. ACE.
Replacing ug in this inequality. BCD by angle ACD (a) equal to it, we get
ug. DCA > ang. ACE,
the inequality is obviously absurd, because a part cannot be greater than its whole, therefore the assumption that two perpendiculars can be constructed leads to absurdity, therefore it is false. The falsity of the assumption is based on the consideration that an incorrect conclusion cannot be drawn from a correct position, 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 made up of angles EGH and HGF, therefore, CD ⊥ AB and HG ⊥ EF (Fig. 21).
You need to prove that right angles are equal.
Proof. Let's superimpose line EF on line AB with point G on point C, then line GH will go along line CD, because from point C only one perpendicular can be constructed, therefore, 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 complement each other to two right angles and are therefore called supplementary angles.
It follows from Theorem 5 consequence. One pair of adjacent angles is equal to the other pair of adjacent angles.
Theorem 6(the converse of 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), therefore the line ACB is a straight line (which was what needed to be proven).
Theorem 7. The sum of angles that have a vertex at the same point and are located on the same side of a straight line is equal to two right angles.
Angles ACD, DCE, ECF, FCG, GCB are given, having a common vertex at point C and located on one side of the straight 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, then replacing the angles ACF and FCB with their values, we find:
ACD + DCE + ECF + FCG + GCB = 2d (which is what needed to be proven).
Theorem 8. The sum of all angles around one point is equal to four right angles.
Angles AOB, BOC, COD, DOE, EOA are given, having a common vertex O and located around point O (Fig. 25).
It is required to prove that
AOB + BOC + COD + DOE + EOA = 4d.
Proof. Let's continue the EO side in the OG direction (Fig. 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 (NOT).
Angle ACB with angle DCE and angle BCD with angle ACE are called vertical (Fig. 26).
Vertical angles. Vertical angles are those angles in which the sides of one are made up of the continuation of the sides of the other angle.
Theorem 9. Vertical angles are equal to each other.
The vertical angles (drawing 26) ACB and DCE are given, as well as BCD and ACE.
You need 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
from where, subtracting BCD by an equal angle, we find
In a similar way it is proved that
∠BCD = ∠ACE.
Equisecant (bisector ) there is a line dividing the angle in half.
In drawing 27 BD has a bisector if ∠ABD = ∠DBC.
Theorem 10.
Adjacent angles ACB and BCD are given (Figure 28). Their bisector lines CF and CE bisect adjacent angles BCD and BCA, hence BCF = FCD, ACE = ECB.
We need 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
Angle ECF is right, i.e. lines CE and CF are mutually perpendicular (CPH).
