Give the concept of the angle between straight lines. Angle between straight lines
Let two nonzero vectors and be given on a plane or in three-dimensional space. Set aside from an arbitrary point O vectors and. Then the following definition is valid.
Definition.
Angle between vectors and called the angle between the rays OA and OB.
The angle between the vectors and will be denoted as.
The angle between vectors can take values \u200b\u200bfrom 0 to or, which is the same, from to.
When vectors are and are co-directed, when vectors are and are oppositely directed.
Definition.
Vectors and are called perpendicularif the angle between them is (radian).
If at least one of the vectors is zero, then the angle is undefined.
Finding the angle between vectors, examples and solutions.
The cosine of the angle between the vectors and, and hence the angle itself, in the general case can be found either using the dot product of vectors, or using the cosine theorem for a triangle built on vectors and.
Let us examine these cases.
By definition, the scalar product of vectors is. If the vectors and are nonzero, then we can divide both sides of the last equality by the product of the lengths of the vectors and, and we get formula for finding the cosine of the angle between nonzero vectors:. This formula can be used if the vector lengths and their dot product are known.
Example.
Calculate the cosine of the angle between the vectors and, and also find the angle itself if the lengths of the vectors and are equal 3 and 6 respectively, and their dot product is -9 .
Decision.
In the problem statement, all the quantities necessary for the application of the formula are given. We calculate the cosine of the angle between vectors and:.
Now we find the angle between the vectors:.
Answer:
There are problems where vectors are given by coordinates in a rectangular coordinate system on a plane or in space. In these cases, to find the cosine of the angle between the vectors, you can use the same formula, but in coordinate form. Let's get it.
The length of a vector is the square root of the sum of the squares of its coordinates, the scalar product of vectors is equal to the sum of the products of the corresponding coordinates. Hence, formula for calculating the cosine of the angle between vectors on the plane has the form, and for vectors in three-dimensional space -.
Example.
Find the angle between the vectors in a rectangular coordinate system.
Decision.
You can immediately use the formula:
Or you can use the formula to find the cosine of the angle between vectors , having previously calculated the lengths of the vectors and the dot product by coordinates:
Answer:
The problem is reduced to the previous case when the coordinates of three points are given (for example AND, IN and FROM) in a rectangular coordinate system and you want to find some angle (for example,).
Indeed, the angle is equal to the angle between vectors and. The coordinates of these vectors are calculated as difference of the corresponding coordinates of the points of the end and the beginning of the vector.
Example.
On a plane in a Cartesian coordinate system, coordinates of three points are given. Find the cosine of the angle between vectors and.
Decision.
Let us determine the coordinates of the vectors and by the coordinates of the given points:
Now let's use the formula to find the cosine of the angle between vectors on the plane in coordinates:
Answer:
The angle between vectors and can also be calculated by cosine theorem... If you postpone from the point O vectors and, then by the cosine theorem in a triangle OAV we can write down, which is equivalent to equality, whence we find the cosine of the angle between the vectors. To apply the resulting formula, we only need the lengths of the vectors and, which are easily found by the coordinates of the vectors and. However, this method is practically not used, since the cosine of the angle between vectors is easier to find by the formula.
Calculation of an orthogonal projection (own projection):
The projection of the vector onto the l-axis is equal to the product of the modulus of the vector by the cosine of the angle φ between the vector and the axis, i.e. pr cosφ.
Doc: If φ \u003d< , то пр l =+ = *cos φ.
If φ\u003e (φ≤), then pr l \u003d - \u003d - * cos (-φ) \u003d cosφ (see figure 10)
If φ \u003d, then pr l \u003d 0 \u003d cos φ.
Consequence: The projection of the vector onto the axis is positive (negative) if the vector forms an acute (obtuse) angle with the axis, and is equal to zero if this angle is a right angle.
Consequence: Projections of equal vectors onto the same axis are equal to each other.
Calculation of the orthogonal projection of the sum of vectors (self-projection):
The projection of the sum of several vectors on the same axis is equal to the sum of their projections on this axis.
Doc: Let, for example, \u003d + +. We have pr l \u003d + \u003d + + -, i.e. pr l (+ +) \u003d pr l + pr l + pr l (see figure 11)
FIG. eleven
Calculating the product of a vector by a number:
When a vector is multiplied by the number λ, its projection onto the axis is also multiplied by this number, i.e. pr l (λ *) \u003d λ * pr l.
Proof: For λ\u003e 0 we have pr l (λ *) \u003d * cos φ \u003d λ * φ \u003d λ * pr l
When λl (λ *) \u003d * cos (-φ) \u003d - * (-cosφ) \u003d * cosφ \u003d λ * pr l.
The property is also valid for
Thus, linear operations on vectors lead to corresponding linear operations on the projections of these vectors.
Consisting of two different rays emanating from the same point. The rays are called. sides Y., and their common origin is the top Y. Let [ VA),[ Sun) -
sides of the corner, IN - its top is a plane defined by the sides of Y. The figure divides the plane into two figures. 
i \u003d\u003d l, 2, also called U. or flat angle, called. the inner area of \u200b\u200ba flat W.
Two corners are called. equal (or congruent) if they can be aligned so that their respective sides and vertices match. From any ray on the plane in a given direction from it, you can postpone the only Y. equal to the given Y. Comparison of Y. is carried out in two ways. If U. is considered as a pair of rays with a common origin, then to clarify the question of which of the two U. is greater, it is necessary to combine in one plane the tops of U. and one pair of their sides (see Fig. 1). If the second side of one W. is located inside the other W., then they say that the first W. is less than the second. The second way of comparing Y. is based on comparing each Y. with a certain number. Equal Y. will correspond to the same degrees or (see below), a larger Y. - a larger number, a smaller one - a smaller one.
Two U. called. adjacent if they have a common vertex and one side, and the other two sides form a straight line (see Fig. 2). In general, U. having a common vertex and one common side are called. adjacent. U. called. vertical, if the sides of one are extensions beyond the top of the sides of the other Y. Vertical Y. are equal to each other. U., at which the sides form a straight line, called. deployed. Half of the deployed U. is called. direct Y. Direct Y. can be equivalently defined differently: Y., equal to its adjacent, called. direct. The interior of a flat U. that does not exceed a developed one is a convex domain on the plane. The unit of measurement of W. is taken to be the 90th share of direct W., called. degree.
The measure Y is also used. The numerical value of the radian measure Y is equal to the length of the arc cut by the sides of Y from the unit circle. One radian is attributed to the U. corresponding to the arc, which is equal to its radius. Unfolded U. is equal to radians.
At the intersection of two straight lines lying in one plane, the third straight line is formed by U. (see Fig. 3): 1 and 5, 2 and 6, 4 and 8, Z and 7 - called. corresponding; 2 and 5, 3 and 8 - internal one-sided; 1 and 6, 4 and 7 - external one-sided; 3 and 5, 2 and 8 - internal lying crosswise; 1 and 7, 4 and 6 - lying crosswise.
In practical. problems, it is advisable to consider U. as a measure of the rotation of a fixed ray around its beginning to a given position. Depending on the direction of rotation of Y. in this case, one can consider both positive and negative. Thus, U. in this sense can have any value. U. as the rotation of the ray is considered in the theory of trigonometric. functions: for any values \u200b\u200bof the argument (Y.), you can determine the values \u200b\u200bof trigonometric. functions. The concept of U. in geometric. system, which is based on point-vector axiomatics, is fundamentally different from the definitions of Y. as a figure - in this axiomatics, Y. is understood as a certain metric. a quantity associated with two vectors using scalar vector multiplication. Namely, each pair of vectors a and b defines a certain angle - a number associated with vectors by the formula
where ( a, b) -
dot product of vectors.
The concept of U. as a flat figure and as a certain numerical value is used in various geometrical. problems in which the value is determined in a special way. So, under Y. between intersecting curves that have certain tangents at the point of intersection, understand Y. formed by these tangents.
The angle between the straight line and the plane is taken as the angle formed by the straight line and its rectangular projection onto the plane; it is measured in the range from 0
Encyclopedia of Mathematics. - M .: Soviet encyclopedia... I. M. Vinogradov. 1977-1985.
Synonyms:See what "ANGLE" is in other dictionaries:
coal - angle / ek / ... Morphemic and spelling dictionary
Husband. fracture, fracture, knee, elbow, protrusion or hall (depression) about one face. The angle is linear, every two opposite lines and their interval; plane or plane angle, meeting of two planes or walls; the corner is thick, solid, meeting in one ... Dahl's Explanatory Dictionary
Angle, about the angle, at (in) the corner and (mat.) In the angle, m. 1. Part of the plane between two straight lines emanating from one point (mat.). Top of the corner. The sides of the corner. Angle measurement by degrees. Right angle. (90 °). Sharp corner. (less than 90 °). Obtuse angle.… … Ushakov's Explanatory Dictionary
ANGLE - (1) the attack angle between the direction of the air flow incident on the airplane wing and the chord of the wing section. The value of the lift depends on this angle. The angle at which the lift is maximum is called the critical angle of attack. Do ... ... Big Polytechnic Encyclopedia
- (flat) geometrical figure formed by two rays (sides of an angle) emanating from one point (apex of the angle). Any angle with a vertex in the center of some circle (central angle) defines on the circle an arc AB, bounded by points ... ... Big Encyclopedic Dictionary
Head of the corner, around the corner, bearish corner, no end corner, in all corners .. Dictionary of Russian synonyms and expressions similar in meaning. under. ed. N. Abramova, M .: Russian dictionaries, 1999. top corner, corner point; bearing, shelter, nine, point, ... ... Synonym dictionary
angle - angle, genus. angle; offer about the corner, in (on) the corner and in the speech of mathematicians in the corner; pl. angles, genus. corners. In prepositional and stable combinations: around the corner and it is permissible for the corner (go in, wrap, etc.), from corner to corner (move, position, etc.), corner ... ... Dictionary of pronunciation and stress difficulties in modern Russian
ANGLE, corner, about a corner, on (in) corner, husband. 1. (in the corner.). In geometry: a flat figure formed by two rays (in 3 values) emanating from one point. Top of the corner. Direct y. (90 °). Sharp u. (less than 90 °). Stupid y. (more than 90 °). External and internal ... ... Ozhegov's Explanatory Dictionary
angle - ANGLE, angle, m. A quarter of a bet, when announced, the edge of the card is folded. ◘ Ace and Queen of Spades with an Angle // Killed. A.I. Polezhaev. A day in Moscow, 1832. ◘ After dinner, he scatters gold pieces on the table, shuffles the cards; ponters crack with decks, ... ... 19th century card terminology and jargon
This material is devoted to such a concept as the angle between two intersecting straight lines. In the first paragraph, we will explain what it is and show it in illustrations. Then we will analyze in what ways you can find the sine, cosine of this angle and the angle itself (we will separately consider the cases with a plane and three-dimensional space), we will give the necessary formulas and show with examples how exactly they are applied in practice.
In order to understand what the angle formed when two straight lines intersect, we need to remember the very definition of the angle, perpendicularity and intersection point.
Definition 1
We call two lines intersecting if they have one common point. This point is called the intersection point of the two lines.
Each straight line is divided by an intersection point into rays. In this case, both straight lines form 4 angles, of which two are vertical and two are adjacent. If we know the measure of one of them, then we can determine the other remaining ones.
Suppose we know that one of the angles is equal to α. In this case, the angle that is vertical with respect to it will also be α. To find the remaining angles, we need to calculate the difference 180 ° - α. If α equals 90 degrees, then all angles will be right. Lines intersecting at right angles are called perpendicular (a separate article is devoted to the concept of perpendicularity).
Take a look at the picture:
Let's move on to formulating the main definition.
Definition 2
The angle formed by two intersecting lines is a measure of the smaller of the 4 angles that the two lines form.
An important conclusion must be drawn from the definition: the size of the angle in this case will be expressed by any real number in the interval (0, 90]. If the straight lines are perpendicular, then the angle between them in any case will be equal to 90 degrees.
The ability to find the measure of the angle between two intersecting straight lines is useful for solving many practical problems. The solution method can be selected from several options.
For starters, we can take geometric methods. If we know something about additional angles, then we can associate them with the angle we need using the properties of equal or similar shapes. For example, if we know the sides of a triangle and we need to calculate the angle between the straight lines on which these sides are located, then the cosine theorem is suitable for us. If we have a right-angled triangle in the condition, then knowledge of the sine, cosine and tangent of an angle will also come in handy for calculations.
The coordinate method is also very convenient for solving problems of this type. Let us explain how to use it correctly.
We have a rectangular (Cartesian) coordinate system O x y, in which two lines are given. Let us denote them by the letters a and b. In this case, the straight lines can be described using any equations. The original lines have an intersection point M. How to determine the required angle (let us designate it as α) between these lines?
Let's start by formulating the basic principle of finding an angle under given conditions.
We know that such concepts as direction and normal vector are closely related to the concept of a straight line. If we have an equation of some straight line, we can take the coordinates of these vectors from it. We can do this for two intersecting lines at once.
The angle formed by two intersecting straight lines can be found using:
- the angle between the direction vectors;
- angle between normal vectors;
- the angle between the normal vector of one straight line and the direction vector of the other.
Now we will consider each method separately.
1. Suppose that we have a straight line a with a direction vector a → \u003d (a x, a y) and a straight line b with a direction vector b → (b x, b y). Now we will postpone two vectors a → and b → from the intersection point. After that, we will see that they will each be located on their own line. Then we have four options for their relative position. See illustration:
If the angle between the two vectors is not obtuse, then it will be the angle we need between the intersecting straight lines a and b. If it is obtuse, then the sought angle will be equal to the angle adjacent to the angle a →, b → ^. Thus, α \u003d a →, b → ^ if a →, b → ^ ≤ 90 °, and α \u003d 180 ° - a →, b → ^ if a →, b → ^\u003e 90 °.
Based on the fact that the cosines of equal angles are equal, we can rewrite the resulting equalities as follows: cos α \u003d cos a →, b → ^, if a →, b → ^ ≤ 90 °; cos α \u003d cos 180 ° - a →, b → ^ \u003d - cos a →, b → ^, if a →, b → ^\u003e 90 °.
In the second case, reduction formulas were used. Thus,
cos α cos a →, b → ^, cos a →, b → ^ ≥ 0 - cos a →, b → ^, cos a →, b → ^< 0 ⇔ cos α = cos a → , b → ^
Let's write the last formula in words:
Definition 3
The cosine of the angle formed by two intersecting straight lines will be equal to the modulus of the cosine of the angle between its direction vectors.
The general view of the formula for the cosine of the angle between two vectors a → \u003d (a x, a y) and b → \u003d (b x, b y) looks like this:
cos a →, b → ^ \u003d a →, b → ^ a → b → \u003d a x b x + a y + b y a x 2 + a y 2 b x 2 + b y 2
From it we can derive the formula for the cosine of the angle between two given straight lines:
cos α \u003d a x b x + a y + b y a x 2 + a y 2 b x 2 + b y 2 \u003d a x b x + a y + b y a x 2 + a y 2 b x 2 + b y 2
Then the angle itself can be found using the following formula:
α \u003d a r c cos a x b x + a y + b y a x 2 + a y 2 b x 2 + b y 2
Here a → \u003d (a x, a y) and b → \u003d (b x, b y) are direction vectors of given lines.
Let's give an example of solving the problem.
Example 1
In a rectangular coordinate system on a plane, two intersecting straight lines a and b are given. They can be described by the parametric equations x \u003d 1 + 4 · λ y \u003d 2 + λ λ ∈ R and x 5 \u003d y - 6 - 3. Calculate the angle between these lines.
Decision
We have a parametric equation in the condition, which means that for this straight line we can immediately write down the coordinates of its direction vector. To do this, we need to take the values \u200b\u200bof the coefficients at the parameter, i.e. the line x \u003d 1 + 4 λ y \u003d 2 + λ λ ∈ R will have a direction vector a → \u003d (4, 1).
The second straight line is described using the canonical equation x 5 \u003d y - 6 - 3. Here we can take the coordinates from the denominators. Thus, this line has a direction vector b → \u003d (5, - 3).
Next, we go directly to finding the angle. To do this, we simply substitute the available coordinates of the two vectors into the above formula α \u003d a r c cos a x b x + a y + b y a x 2 + a y 2 b x 2 + b y 2. We get the following:
α \u003d a r c cos 4 5 + 1 (- 3) 4 2 + 1 2 5 2 + (- 3) 2 \u003d a r c cos 17 17 34 \u003d a r c cos 1 2 \u003d 45 °
Answer: These straight lines form an angle of 45 degrees.
We can solve a similar problem by finding the angle between the normal vectors. If we have a straight line a with a normal vector na → \u003d (nax, nay) and a straight line b with a normal vector nb → \u003d (nbx, nby), then the angle between them will be equal to the angle between na → and nb → or the angle, which will be adjacent to na →, nb → ^. This method is shown in the picture:
The formulas for calculating the cosine of the angle between intersecting straight lines and this angle itself using the coordinates of normal vectors look like this:
cos α \u003d cos n a →, n b → ^ \u003d n a x n b x + n a y + n b y n a x 2 + n a y 2 n b x 2 + n b y 2 α \u003d a r c cos n a x n b x + n a y + n b y n a x 2 + n a y 2 n b x 2 + n b y 2
Here n a → and n b → denote the normal vectors of two given lines.
Example 2
In a rectangular coordinate system, two straight lines are given using the equations 3 x + 5 y - 30 \u003d 0 and x + 4 y - 17 \u003d 0. Find the sine, cosine of the angle between them and the value of this angle itself.
Decision
The original straight lines are given using normal straight line equations of the form A x + B y + C \u003d 0. The normal vector is denoted by n → \u003d (A, B). Let's find the coordinates of the first normal vector for one straight line and write them down: n a → \u003d (3, 5). For the second straight line x + 4 y - 17 \u003d 0, the normal vector will have coordinates n b → \u003d (1, 4). Now let's add the obtained values \u200b\u200bto the formula and calculate the total:
cos α \u003d cos n a →, n b → ^ \u003d 3 1 + 5 4 3 2 + 5 2 1 2 + 4 2 \u003d 23 34 17 \u003d 23 2 34
If we know the cosine of an angle, then we can calculate its sine using the basic trigonometric identity. Since the angle α formed by straight lines is not obtuse, then sin α \u003d 1 - cos 2 α \u003d 1 - 23 2 34 2 \u003d 7 2 34.
In this case, α \u003d a r c cos 23 2 34 \u003d a r c sin 7 2 34.
Answer: cos α \u003d 23 2 34, sin α \u003d 7 2 34, α \u003d a r c cos 23 2 34 \u003d a r c sin 7 2 34
Let us examine the last case - finding the angle between straight lines, if we know the coordinates of the direction vector of one straight line and the normal vector of the other.
Suppose that line a has a direction vector a → \u003d (a x, a y), and line b is a normal vector n b → \u003d (n b x, n b y). We need to postpone these vectors from the intersection point and consider all the options for their relative position. See the picture:
If the angle between the given vectors is no more than 90 degrees, it turns out that it will complement the angle between a and b to a right angle.
a →, n b → ^ \u003d 90 ° - α if a →, n b → ^ ≤ 90 °.
If it is less than 90 degrees, then we get the following:
a →, n b → ^\u003e 90 °, then a →, n b → ^ \u003d 90 ° + α
Using the rule of equality of cosines of equal angles, we write:
cos a →, n b → ^ \u003d cos (90 ° - α) \u003d sin α as a →, n b → ^ ≤ 90 °.
cos a →, n b → ^ \u003d cos 90 ° + α \u003d - sin α as a →, n b → ^\u003e 90 °.
Thus,
sin α \u003d cos a →, nb → ^, a →, nb → ^ ≤ 90 ° - cos a →, nb → ^, a →, nb → ^\u003e 90 ° ⇔ sin α \u003d cos a →, nb → ^, a →, nb → ^\u003e 0 - cos a →, nb → ^, a →, nb → ^< 0 ⇔ ⇔ sin α = cos a → , n b → ^
Let us formulate a conclusion.
Definition 4
To find the sine of the angle between two straight lines intersecting on the plane, you need to calculate the modulus of the cosine of the angle between the direction vector of the first line and the normal vector of the second.
Let's write down the necessary formulas. Finding the sine of an angle:
sin α \u003d cos a →, n b → ^ \u003d a x n b x + a y n b y a x 2 + a y 2 n b x 2 + n b y 2
Finding the corner itself:
α \u003d a r c sin \u003d a x n b x + a y n b y a x 2 + a y 2 n b x 2 + n b y 2
Here a → is the direction vector of the first line, and n b → is the normal vector of the second.
Example 3
Two intersecting straight lines are given by the equations x - 5 \u003d y - 6 3 and x + 4 y - 17 \u003d 0. Find the angle of intersection.
Decision
We take the coordinates of the direction and normal vectors from the given equations. It turns out a → \u003d (- 5, 3) and n → b \u003d (1, 4). We take the formula α \u003d a r c sin \u003d a x n b x + a y n b y a x 2 + a y 2 n b x 2 + n b y 2 and consider:
α \u003d a r c sin \u003d - 5 1 + 3 4 (- 5) 2 + 3 2 1 2 + 4 2 \u003d a r c sin 7 2 34
Note that we took the equations from the previous problem and got exactly the same result, but in a different way.
Answer: α \u003d a r c sin 7 2 34
Here is another way to find the desired angle using the slopes of the given straight lines.
We have a line a, which is given in a rectangular coordinate system using the equation y \u003d k 1 x + b 1, and a line b, which is defined as y \u003d k 2 x + b 2. These are equations of straight lines with a slope. To find the angle of intersection, use the formula:
α \u003d a r c cos k 1 k 2 + 1 k 1 2 + 1 k 2 2 + 1, where k 1 and k 2 are the slopes of the given lines. To obtain this record, the formulas for determining the angle through the coordinates of normal vectors were used.
Example 4
There are two intersecting straight lines on the plane, given by the equations y \u003d - 3 5 x + 6 and y \u003d - 1 4 x + 17 4. Calculate the angle of intersection.
Decision
The slopes of our lines are k 1 \u003d - 3 5 and k 2 \u003d - 1 4. Add them to the formula α \u003d a r c cos k 1 k 2 + 1 k 1 2 + 1 k 2 2 + 1 and calculate:
α \u003d a r c cos - 3 5 - 1 4 + 1 - 3 5 2 + 1 - 1 4 2 + 1 \u003d a r c cos 23 20 34 24 17 16 \u003d a r c cos 23 2 34
Answer: α \u003d a r c cos 23 2 34
In the conclusions of this paragraph, it should be noted that the formulas for finding the angle given here do not have to be learned by heart. To do this, it is enough to know the coordinates of the guides and / or normal vectors of the given straight lines and be able to determine them using different types of equations. But it is better to remember or write down the formulas for calculating the cosine of an angle.
How to calculate the angle between intersecting lines in space
The calculation of such an angle can be reduced to calculating the coordinates of the direction vectors and determining the value of the angle formed by these vectors. For such examples, the same reasoning is used that we gave before.
Let's say that we have a rectangular coordinate system located in 3D space. It contains two lines a and b with an intersection point M. To calculate the coordinates of the direction vectors, we need to know the equations of these lines. We denote the direction vectors a → \u003d (a x, a y, a z) and b → \u003d (b x, b y, b z). To calculate the cosine of the angle between them, we use the formula:
cos α \u003d cos a →, b → ^ \u003d a →, b → a → b → \u003d a x b x + a y b y + a z b z a x 2 + a y 2 + a z 2 b x 2 + b y 2 + b z 2
To find the angle itself, we need this formula:
α \u003d a r c cos a x b x + a y b y + a z b z a x 2 + a y 2 + a z 2 b x 2 + b y 2 + b z 2
Example 5
We have a straight line defined in three-dimensional space using the equation x 1 \u003d y - 3 \u003d z + 3 - 2. It is known that it intersects with the O z axis. Calculate the angle of intersection and the cosine of that angle.
Decision
Let us denote the angle to be calculated by the letter α. Let's write down the coordinates of the direction vector for the first straight line - a → \u003d (1, - 3, - 2). For the applicate axis, we can take the coordinate vector k → \u003d (0, 0, 1) as a direction. We have received the necessary data and can add it to the desired formula:
cos α \u003d cos a →, k → ^ \u003d a →, k → a → k → \u003d 1 0 - 3 0 - 2 1 1 2 + (- 3) 2 + (- 2) 2 0 2 + 0 2 + 1 2 \u003d 2 8 \u003d 1 2
As a result, we got that the angle we need will be equal to a r c cos 1 2 \u003d 45 °.
Answer: cos α \u003d 1 2, α \u003d 45 °.
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Definition
A geometric figure consisting of all points of the plane enclosed between two rays emanating from one point is called flat angle.
Definition
The angle between the twointersecting straight is the value of the smallest plane angle at the intersection of these straight lines. If two straight lines are parallel, then the angle between them is taken to be zero.
The value of the angle between two intersecting straight lines (if you measure planar angles in radians) can range from zero to $ \\ dfrac (\\ pi) (2) $.
Definition
The angle between two crossed lines called the value equal to the angle between two intersecting straight lines parallel to the intersecting. The angle between the straight lines $ a $ and $ b $ is denoted $ \\ angle (a, b) $.
The correctness of the introduced definition follows from the following theorem.
Plane Angle Theorem with Parallel Sides
The values \u200b\u200bof two convex plane angles with respectively parallel and equally directed sides are equal.
Evidence
If the corners are unfolded, then they are both equal to $ \\ pi $. If they are not unfolded, then set aside equal segments $ ON \u003d O_1ON_1 $ and $ OM \u003d O_1M_1 $ on the corresponding sides of the angles $ \\ angle AOB $ and $ \\ angle A_1O_1B_1 $.
The quadrilateral $ O_1N_1NO $ is a parallelogram since its opposite sides $ ON $ and $ O_1N_1 $ are equal and parallel. Similarly, the quadrilateral $ O_1M_1MO $ is a parallelogram. Hence, $ NN_1 \u003d OO_1 \u003d MM_1 $ and $ NN_1 \\ parallel OO_1 \\ parallel MM_1 $, therefore, $ NN_1 \u003d MM_1 $ and $ NN_1 \\ parallel MM_1 $ by transitivity. The quadrilateral $ N_1M_1MN $ is a parallelogram, since its opposite sides are equal and parallel. This means that the segments $ NM $ and $ N_1M_1 $ are equal. Triangles $ ONM $ and $ O_1N_1M_1 $ are equal in the third sign of equality of triangles, which means that the corresponding angles $ \\ angle NOM $ and $ \\ angle N_1O_1M_1 $ are equal.
