Let two non-zero vectors and be given on a plane or in three-dimensional space. Let's postpone from an arbitrary point O vectors and . Then the following definition is valid.
Definition.
Angle between vectors and the angle between the rays is called O.A. And O.B..
The angle between the vectors and will be denoted as .
The angle between the vectors can take values from 0 to or, which is the same thing, from to.
When the vectors are both co-directed, when the vectors are and oppositely directed.
Definition.
Vectors are called perpendicular, if the angle between them is equal to (radians).
If at least one of the vectors is zero, then the angle is not defined.
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 scalar product of vectors, or using the cosine theorem for a triangle built on the vectors and .
Let's look at 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 non-zero vectors: . This formula can be used if the lengths of the vectors and their scalar 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 scalar product is equal to -9 .
Solution.
The problem statement contains all the quantities necessary to apply the formula. We calculate the cosine of the angle between the vectors and: .
Now we find the angle between the vectors: .
Answer:
There are problems where vectors are specified by coordinates in a rectangular coordinate system on a plane or in space. In these cases, to find the cosine of the angle between 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 vectors given in a rectangular coordinate system.
Solution.
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 scalar product over the coordinates:
Answer:
The problem is reduced to the previous case when the coordinates of three points are given (for example A, IN And WITH) in a rectangular coordinate system and you need to find some angle (for example, ).
Indeed, the angle is equal to the angle between the vectors and . The coordinates of these vectors are calculated as the difference between the corresponding coordinates of the end and beginning points of the vector.
Example.
On a plane, the coordinates of three points are given in the Cartesian coordinate system. Find the cosine of the angle between the vectors and .
Solution.
Let's determine the coordinates of the vectors and the coordinates of the given points:
Now let’s use the formula to find the cosine of the angle between vectors on a plane in coordinates:
Answer:
The angle between the vectors and can also be calculated by cosine theorem. If we postpone from the point O vectors and , then by the cosine theorem in a triangle OAV we can write, which is equivalent to the equality, from which 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 can easily be found from 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 using the formula.
Calculation of orthogonal projection (own projection):
The projection of the vector onto the l axis is equal to the product of the vector modulus and the cosine of the angle φ between the vector and the axis, i.e. pr cosφ.
Doc: If φ=< , то пр l =+ = *cos φ.
If φ> (φ≤ ), then pr l =- =- * cos( -φ) = cosφ (see Fig.10)
If φ= , then pr l = 0 = cos φ.
Consequence: The projection of a vector onto an axis is positive (negative) if the vector forms an acute (obtuse) angle with the axis, and is equal to zero if this angle is right.
Consequence: Projections of equal vectors onto the same axis are equal to each other.
Calculation of the orthogonal projection of the sum of vectors (projection property):
The projection of the sum of several vectors onto the same axis is equal to the sum of their projections onto this axis.
Doc: Let, for example, = + + . We have pr l =+ =+ + - , i.e. pr l ( + + ) = pr l + pr l + pr l (see Fig.11)
RICE. eleven
Calculation of the product of a vector and a number:
When a vector is multiplied by a number λ, its projection onto the axis is also multiplied by this number, i.e. pr l (λ* )= λ* pr l .
Proof: For λ > 0 we have pr l (λ* )= *cos φ = λ* φ = λ*pr l
When λl (λ* )= *cos( -φ)=- * (-cosφ) = * cosφ= λ *pr l .
The property is also valid when
Thus, linear operations on vectors lead to corresponding linear operations on the projections of these vectors.
Consisting of two different rays emanating from one point. The rays are called sides of the U., and their common beginning is the top of the U. Let [ VA),[Sun) -
sides of the corner, IN - its vertex is a plane defined by the sides U. The figure divides the plane into two figures. The figure 
i==l, 2, also called U. or flat angle, called. the inner region of the flat U.
The two corners are called equal (or congruent) if they can be aligned so that their corresponding sides and vertices coincide. From any ray on a plane, in a given direction from it, a single axis equal to the given axis can be plotted. Comparison of the axis is carried out in two ways. If the beam is considered as a pair of rays with a common origin, then to clarify the question of which of the two beams is larger, it is necessary to combine the vertices of the beam and one pair of their sides in one plane (see Fig. 1). If the second side of one U. turns out to be located inside another U., then they say that the first U. is smaller than the second. The second method of comparing U. is based on comparing each U. with a certain number. Equal U. will correspond to the same degrees or (see below), a larger U. will correspond to a larger number, and a smaller one will correspond to a smaller number.
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. Vertical U. are equal to each other. U., whose sides form a straight line, called. expanded. Half of the expanded U. called. straight U. Direct U. can be equivalently defined differently: U. equal to its adjacent one, called. direct. The interior of a flat plane, not exceeding the unfolded one, is a convex region on the plane. The unit of measurement of U. is taken to be the 90th fraction of direct U., called. degree.
The so-called U measure is also used. The numerical value of the radian U measure is equal to the length of the arc cut by the sides of the U from the unit circle. One radian is assigned to the U corresponding to the arc, which is equal to its radius. The expanded U. is equal to radians.
When two straight lines lying in the same plane intersect with a third straight line, Us are formed (see Fig. 3): 1 and 5, 2 and 6, 4 and 8, 3 and 7 - the so-called. appropriate; 2 and 5, 3 and 8 - internal one-sided; 1 and 6, 4 and 7 - external one-sided; 3 and 5, 2 and 8 - internally lying crosswise; 1 and 7, 4 and 6 - lying crosswise on the outside.
In practice In problems, it is advisable to consider rotation as a measure of rotation of a fixed beam around its origin to a given position. Depending on the direction of rotation of the signals in this case, both positive and negative ones can be considered. Thus, U. in this sense can have any value. The rotation of a ray is considered in trigonometric theory. functions: for any values of the argument (U.), you can determine the values of trigonometric. functions. The concept of geometry in geometrics. system, which is based on the point-vector axiomatics, is fundamentally different from the definitions of U. as a figure - in this axiomatics, U. is understood as a certain metric. a quantity related to two vectors using the operation of scalar vector multiplication. Namely, each pair of vectors a and b defines a certain angle - a number associated with the vectors by the formula
Where ( a, b) -
scalar product of vectors.
The concept of U. as a flat figure and as a certain numerical value is used in various geometrics. problems in which U. is determined in a special way. Thus, by the shape between intersecting curves that have certain tangents at the point of intersection, we mean the shape formed by these tangents.
The angle between a straight line and a plane is taken to be the angle formed by the straight line and its rectangular projection onto the plane; it is measured in the range from 0
Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.
Synonyms:See what "ANGLE" is in other dictionaries:
ember- angle / yok / ... Morphemic-spelling dictionary
Husband. fracture, kink, knee, elbow, protrusion or crease (depression) on one side. Linear angle, any two opposing lines and their interval; angle plane or in planes, meeting of two planes or walls; the corner is thick, body, meeting in one... Dahl's Explanatory Dictionary
Angle, about an angle, on (in) an angle and (mat.) in an angle, m. 1. Part of a plane between two straight lines emanating from one point (mat.). Top of the corner. Sides of the corner. Measuring an angle in degrees. Right angle. (90°). Sharp corner. (less than 90°). Obtuse angle.… … Ushakov's Explanatory Dictionary
CORNER- (1) attack angle between the direction of the air flow flowing onto the aircraft wing and the chord of the wing section. The value of the lifting force depends on this angle. The angle at which the lift force is maximum is called the critical angle of attack. U... ... Big Polytechnic Encyclopedia
- (flat) geometric figure formed by two rays (sides of an angle) emerging from one point (vertex of the angle). Any angle with a vertex at the center of a certain circle (central angle) defines an arc AB on the circle, bounded by points... ... Big Encyclopedic Dictionary
The head of the corner, from around the corner, the bearish corner, the unfinished corner, in all corners... Dictionary of Russian synonyms and expressions similar in meaning. under. ed. N. Abramova, M.: Russian Dictionaries, 1999. angle apex, corner point; bearing, shelter, deviatina, rumb,... ... Synonym dictionary
corner- angle, rod. angle; sentence about coal, in (on) the corner and in the speech of mathematicians in coal; pl. corners, rod. corners In prepositional and stable combinations: around the corner and it is permissible to go around the corner (go in, turn, etc.), from corner to corner (move, position, etc.), corner... ... Dictionary of difficulties of pronunciation and stress in modern Russian language
ANGLE, corner, about the corner, on (in) the corner, husband. 1. (in the corner.). In geometry: a flat figure formed by two rays (in 3 digits) emanating from one point. Top of the corner. Direct y. (90°). Acute u. (less than 90°). Dumb u. (more than 90°). External and internal... ... Ozhegov's Explanatory Dictionary
corner- ANGLE, angle, m. A quarter of the bet, when announced, the edge of the card is folded. ◘ Ace and queen of spades with corner // Killed. A.I. Polezhaev. A day in Moscow, 1832. ◘ After dinner, he scatters chervonets on the table, shuffles the cards; punters crack their decks... ... 19th century card terminology and jargon
This material is devoted to such a concept as the angle between two intersecting lines. In the first paragraph we will explain what it is and show it in illustrations. Then we will look at the ways in which you can find the sine, cosine of this angle and the angle itself (we will separately consider cases with a plane and three-dimensional space), we will give the necessary formulas and show with examples exactly how they are used in practice.
In order to understand what the angle formed when two lines intersect is, we need to remember the very definition of angle, perpendicularity and point of intersection.
Definition 1
We call two lines intersecting if they have one common point. This point is called the point of intersection of two lines.
Each straight line is divided by an intersection point into rays. Both straight lines form 4 angles, two of which are vertical, and two are adjacent. If we know the measure of one of them, then we can determine the remaining ones.
Let's say 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 equal to α. To find the remaining angles, we need to calculate the difference 180 ° - α. If α is equal to 90 degrees, then all angles will be right angles. 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 the measure of the smaller of the 4 angles that form these two lines.
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 lines are perpendicular, then the angle between them will in any case be equal to 90 degrees.
The ability to find the measure of the angle between two intersecting lines is useful for solving many practical problems. The solution method can be chosen from several options.
To begin with, we can take geometric methods. If we know something about complementary angles, then we can relate them to the angle we need using the properties of equal or similar figures. For example, if we know the sides of a triangle and need to calculate the angle between the lines on which these sides are located, then the cosine theorem is suitable for our solution. If we have a right triangle in our condition, then for calculations we will also need to know the sine, cosine and tangent of the angle.
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 straight lines are given. Let's denote them by letters a and b. The straight lines can be described using some equations. The original lines have an intersection point M. How to determine the required angle (let's denote it α) between these straight lines?
Let's start by formulating the basic principle of finding an angle under given conditions.
We know that the concept of a straight line is closely related to such concepts as a direction vector and a normal vector. If we have an equation of a certain line, we can take the coordinates of these vectors from it. We can do this for two intersecting lines at once.
The angle subtended by two intersecting lines can be found using:
- angle between direction vectors;
- angle between normal vectors;
- the angle between the normal vector of one line and the direction vector of the other.
Now let's look at each method separately.
1. Let us assume that we have a line a with a direction vector a → = (a x, a y) and a line b with a direction vector b → (b x, b y). Now let’s plot two vectors a → and b → from the intersection point. After this we will see that they will each be located on their own straight line. Then we have four options for their relative arrangement. See illustration:
If the angle between two vectors is not obtuse, then it will be the angle we need between the intersecting lines a and b. If it is obtuse, then the desired angle will be equal to the angle adjacent to the angle a →, b → ^. Thus, α = a → , b → ^ if a → , b → ^ ≤ 90 ° , and α = 180 ° - a → , b → ^ if a → , b → ^ > 90 ° .
Based on the fact that the cosines of equal angles are equal, we can rewrite the resulting equalities as follows: cos α = cos a →, b → ^, if a →, b → ^ ≤ 90 °; cos α = cos 180 ° - a →, b → ^ = - cos a →, b → ^, if a →, b → ^ > 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 form of the formula for the cosine of the angle between two vectors a → = (a x , a y) and b → = (b x , b y) looks like this:
cos a → , b → ^ = a → , b → ^ a → b → = 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 = a x b x + a y + b y a x 2 + a y 2 b x 2 + b y 2 = 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 = 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 → = (a x , a y) and b → = (b x , b y) are the direction vectors of the given lines.
Let's give an example of solving the problem.
Example 1
In a rectangular coordinate system on a plane, two intersecting lines a and b are given. They can be described by the parametric equations x = 1 + 4 · λ y = 2 + λ λ ∈ R and x 5 = y - 6 - 3. Calculate the angle between these lines.
Solution
We have a parametric equation in our condition, which means that for this line we can immediately write down the coordinates of its direction vector. To do this, we need to take the values of the coefficients for the parameter, i.e. the straight line x = 1 + 4 · λ y = 2 + λ λ ∈ R will have a direction vector a → = (4, 1).
The second line is described using the canonical equation x 5 = y - 6 - 3. Here we can take the coordinates from the denominators. Thus, this line has a direction vector b → = (5 , - 3) .
Next, we move directly to finding the angle. To do this, simply substitute the existing coordinates of the two vectors into the above formula α = 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 = a r c cos 4 5 + 1 (- 3) 4 2 + 1 2 5 2 + (- 3) 2 = a r c cos 17 17 34 = a r c cos 1 2 = 45 °
Answer: These straight lines form an angle of 45 degrees.
We can solve a similar problem by finding the angle between normal vectors. If we have a line a with a normal vector n a → = (n a x , n a y) and a line b with a normal vector n b → = (n b x , n b y), then the angle between them will be equal to the angle between n a → and n b → or the angle that will be adjacent to n a →, n b → ^. This method is shown in the picture:
Formulas for calculating the cosine of the angle between intersecting lines and this angle itself using the coordinates of normal vectors look like this:
cos α = cos n a → , n b → ^ = 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 = 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 = 0 and x + 4 y - 17 = 0. Find the sine and cosine of the angle between them and the magnitude of this angle itself.
Solution
The original lines are specified using normal line equations of the form A x + B y + C = 0. We denote the normal vector as n → = (A, B). Let's find the coordinates of the first normal vector for one line and write them: n a → = (3, 5) . For the second line x + 4 y - 17 = 0, the normal vector will have coordinates n b → = (1, 4). Now let’s add the obtained values to the formula and calculate the total:
cos α = cos n a → , n b → ^ = 3 1 + 5 4 3 2 + 5 2 1 2 + 4 2 = 23 34 17 = 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 = 1 - cos 2 α = 1 - 23 2 34 2 = 7 2 34.
In this case, α = a r c cos 23 2 34 = a r c sin 7 2 34.
Answer: cos α = 23 2 34, sin α = 7 2 34, α = a r c cos 23 2 34 = a r c sin 7 2 34
Let us analyze 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.
Let us assume that straight line a has a direction vector a → = (a x , a y) , and straight line b has a normal vector n b → = (n b x , n b y) . We need to set these vectors aside from the intersection point and consider all options for their relative positions. See in 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 → ^ = 90 ° - α if a → , n b → ^ ≤ 90 ° .
If it is less than 90 degrees, then we get the following:
a → , n b → ^ > 90 ° , then a → , n b → ^ = 90 ° + α
Using the rule of equality of cosines of equal angles, we write:
cos a → , n b → ^ = cos (90 ° - α) = sin α for a → , n b → ^ ≤ 90 ° .
cos a → , n b → ^ = cos 90 ° + α = - sin α for a → , n b → ^ > 90 ° .
Thus,
sin α = cos a → , n b → ^ , a → , n b → ^ ≤ 90 ° - cos a → , n b → ^ , a → , n b → ^ > 90 ° ⇔ sin α = cos a → , n b → ^ , a → , n b → ^ > 0 - cos a → , n b → ^ , a → , n b → ^< 0 ⇔ ⇔ sin α = cos a → , n b → ^
Let us formulate a conclusion.
Definition 4
To find the sine of the angle between two lines intersecting on a 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 = cos a → , n b → ^ = 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 angle itself:
α = a r c sin = 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 lines are given by the equations x - 5 = y - 6 3 and x + 4 y - 17 = 0. Find the angle of intersection.
Solution
We take the coordinates of the guide and normal vector from the given equations. It turns out a → = (- 5, 3) and n → b = (1, 4). We take the formula α = a r c sin = 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 calculate:
α = a r c sin = - 5 1 + 3 4 (- 5) 2 + 3 2 1 2 + 4 2 = a r c sin 7 2 34
Please note that we took the equations from the previous problem and obtained exactly the same result, but in a different way.
Answer:α = a r c sin 7 2 34
Let us present another way to find the desired angle using the angular coefficients of given straight lines.
We have a line a, which is defined in a rectangular coordinate system using the equation y = k 1 x + b 1, and a line b, defined as y = k 2 x + b 2. These are equations of lines with slopes. To find the angle of intersection, we use the formula:
α = 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, formulas for determining the angle through the coordinates of normal vectors were used.
Example 4
There are two lines intersecting in a plane, given by the equations y = - 3 5 x + 6 and y = - 1 4 x + 17 4. Calculate the value of the intersection angle.
Solution
The angular coefficients of our lines are equal to k 1 = - 3 5 and k 2 = - 1 4. Let's add them to the formula α = a r c cos k 1 k 2 + 1 k 1 2 + 1 k 2 2 + 1 and calculate:
α = a r c cos - 3 5 · - 1 4 + 1 - 3 5 2 + 1 · - 1 4 2 + 1 = a r c cos 23 20 34 24 · 17 16 = a r c cos 23 2 34
Answer:α = 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 given lines and be able to determine them using different types of equations. But it’s 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 magnitude of the angle formed by these vectors. For such examples, the same reasoning that we gave before is used.
Let's assume that we have a rectangular coordinate system located in three-dimensional space. It contains two straight 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. Let us denote the direction vectors a → = (a x , a y , a z) and b → = (b x , b y , b z) . To calculate the cosine of the angle between them, we use the formula:
cos α = cos a → , b → ^ = a → , b → a → b → = 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 = 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 line defined in three-dimensional space using the equation x 1 = y - 3 = z + 3 - 2. It is known that it intersects with the O z axis. Calculate the intercept angle and the cosine of that angle.
Solution
Let us denote the angle that needs to be calculated by the letter α. Let's write down the coordinates of the direction vector for the first straight line – a → = (1, - 3, - 2) . For the applicate axis, we can take the coordinate vector k → = (0, 0, 1) as a guide. We have received the necessary data and can add it to the desired formula:
cos α = cos a → , k → ^ = a → , k → a → k → = 1 0 - 3 0 - 2 1 1 2 + (- 3) 2 + (- 2) 2 0 2 + 0 2 + 1 2 = 2 8 = 1 2
As a result, we found that the angle we need will be equal to a r c cos 1 2 = 45 °.
Answer: cos α = 1 2 , α = 45 ° .
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In this lesson we will give the definition of codirectional rays and prove the theorem about the equality of angles with codirectional sides. Next, we will give the definition of the angle between intersecting lines and skew lines. Let's consider what the angle between two straight lines can be. At the end of the lesson, we will solve several problems on finding angles between intersecting lines.
Topic: Parallelism of lines and planes
Lesson: Angles with aligned sides. Angle between two straight lines
Any straight line, for example OO 1(Fig. 1.), cuts the plane into two half-planes. If the rays OA And O 1 A 1 are parallel and lie in the same half-plane, then they are called co-directed.
Rays O 2 A 2 And OA are not co-directional (Fig. 1.). They are parallel, but do not lie in the same half-plane.
If the sides of two angles are aligned, then the angles are equal.
Proof
Let us be given parallel rays OA And O 1 A 1 and parallel rays OB And About 1 In 1(Fig. 2.). That is, we have two angles AOB And A 1 O 1 B 1, whose sides lie on codirectional rays. Let us prove that these angles are equal.
On the beam side OA And O 1 A 1 select points A And A 1 so that the segments OA And O 1 A 1 were equal. Likewise, points IN And IN 1 choose so that the segments OB And About 1 In 1 were equal.
Consider a quadrilateral A 1 O 1 OA(Fig. 3.) OA And O 1 A 1 A 1 O 1 OA A 1 O 1 OA OO 1 And AA 1 parallel and equal.
Consider a quadrilateral B 1 O 1 OV. This quadrilateral side OB And About 1 In 1 parallel and equal. Based on parallelogram, quadrilateral B 1 O 1 OV is a parallelogram. Because B 1 O 1 OV- parallelogram, then the sides OO 1 And BB 1 parallel and equal.
And straight AA 1 parallel to the line OO 1, and straight BB 1 parallel to the line OO 1, means straight AA 1 And BB 1 parallel.
Consider a quadrilateral B 1 A 1 AB. This quadrilateral side AA 1 And BB 1 parallel and equal. Based on parallelogram, quadrilateral B 1 A 1 AB is a parallelogram. Because B 1 A 1 AB- parallelogram, then the sides AB And A 1 B 1 parallel and equal.
Consider triangles AOB And A 1 O 1 B 1. Parties OA And O 1 A 1 equal in construction. Parties OB And About 1 In 1 are also equal in construction. And as we have proven, both sides AB And A 1 B 1 are also equal. So triangles AOB And A 1 O 1 B 1 equal on three sides. In equal triangles, equal angles lie opposite equal sides. So the angles AOB And A 1 O 1 B 1 are equal, as required to prove.
1) Intersecting lines.
If the lines intersect, then we have four different angles. Angle between two straight lines, is called the smallest angle between two straight lines. Angle between intersecting lines A And b let's denote α (Fig. 4.). The angle α is such that .
Rice. 4. Angle between two intersecting lines
2) Crossing lines
Let straight A And b interbreeding. Let's choose an arbitrary point ABOUT. Through the point ABOUT let's make a direct a 1, parallel to the line A, and straight b 1, parallel to the line b(Fig. 5.). Direct a 1 And b 1 intersect at a point ABOUT. Angle between two intersecting lines a 1 And b 1, angle φ, and is called the angle between intersecting lines.
Rice. 5. Angle between two intersecting lines
Does the size of the angle depend on the selected point O? Let's choose a point O 1. Through the point O 1 let's make a direct a 2, parallel to the line A, and straight b 2, parallel to the line b(Fig. 6.). Angle between intersecting lines a 2 And b 2 let's denote φ 1. Then the angles φ And φ 1 - corners with aligned sides. As we have proven, such angles are equal to each other. This means that the magnitude of the angle between intersecting lines does not depend on the choice of point ABOUT.
Direct OB And CD parallel, OA And CD interbreed. Find the angle between the lines OA And CD, If:
1) ∠AOB= 40°.
Let's choose a point WITH. Pass a straight line through it CD. Let's carry out CA 1 parallel OA(Fig. 7.). Then the angle A 1 CD- angle between intersecting lines OA And CD. According to the theorem about angles with concurrent sides, the angle A 1 CD equal to angle AOB, that is 40°.
Rice. 7. Find the angle between two straight lines
2) ∠AOB= 135°.
Let's do the same construction (Fig. 8.). Then the angle between the crossing lines OA And CD is equal to 45°, since it is the smallest of the angles that are obtained when straight lines intersect CD And CA 1.
3) ∠AOB= 90°.
Let's do the same construction (Fig. 9.). Then all the angles that are obtained when the lines intersect CD And CA 1 equal 90°. The required angle is 90°.
1) Prove that the midpoints of the sides of a spatial quadrilateral are the vertices of a parallelogram.
Proof
Let us be given a spatial quadrilateral ABCD. M,N,K,L- middle of ribs B.D.A.D.AC,B.C. accordingly (Fig. 10.). It is necessary to prove that MNKL- parallelogram.
Consider a triangle ABD. MN MN parallel AB and equals half of it.
Consider a triangle ABC. LК- middle line. According to the property of the midline, LК parallel AB and equals half of it.
AND MN, And LК parallel AB. Means, MN parallel LК by the theorem of three parallel lines.
We find that in a quadrilateral MNKL- sides MN And LК parallel and equal, since MN And LК equal to half AB. So, according to the parallelogram criterion, a quadrilateral MNKL- a parallelogram, which is what needed to be proven.
2) Find the angle between the lines AB And CD, if the angle MNK= 135°.
As we have already proven, MN parallel to the line AB. NK- middle line of the triangle ACD, by property, NK parallel DC. So, through the point N there are two straight lines MN And NK, which are parallel to skew lines AB And DC respectively. So, the angle between the lines MN And NK is the angle between intersecting lines AB And DC. We are given an obtuse angle MNK= 135°. Angle between straight lines MN And NK- the smallest of the angles obtained by intersecting these straight lines, that is, 45°.
So, we looked at angles with codirectional sides and proved their equality. We looked at the angles between intersecting and skewing lines and solved several problems on finding the angle between two lines. In the next lesson we will continue solving problems and reviewing theory.
1. Geometry. Grades 10-11: textbook for students of general education institutions (basic and specialized levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and expanded - M.: Mnemosyne, 2008. - 288 p. : ill.
2. Geometry. Grades 10-11: Textbook for general education institutions / Sharygin I.F. - M.: Bustard, 1999. - 208 pp.: ill.
3. Geometry. Grade 10: Textbook for general education institutions with in-depth and specialized study of mathematics /E. V. Potoskuev, L. I. Zvalich. - 6th edition, stereotype. - M.: Bustard, 008. - 233 p. :il.
IN) B.C. And D 1 IN 1.
Rice. 11. Find the angle between lines
4. Geometry. Grades 10-11: textbook for students of general education institutions (basic and specialized levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and expanded - M.: Mnemosyne, 2008. - 288 pp.: ill.
Tasks 13, 14, 15 p. 54
