Oh-oh-oh-oh-oh... well, it’s tough, as if he was reading out a sentence to himself =) However, relaxation will help later, especially since today I bought the appropriate accessories. Therefore, let's proceed to the first section, I hope that by the end of the article I will maintain a cheerful mood.
The relative position of two straight lines
This is the case when the audience sings along in chorus. Two straight lines can:
1) match;
2) be parallel: ;
3) or intersect at a single point: .
Help for dummies : Please remember the mathematical intersection sign, it will appear very often. The notation means that the line intersects with the line at point .
How to determine the relative position of two lines?
Let's start with the first case:
Two lines coincide if and only if their corresponding coefficients are proportional, that is, there is a number “lambda” such that the equalities are satisfied
Let's consider the straight lines and create three equations from the corresponding coefficients: . From each equation it follows that, therefore, these lines coincide.
Indeed, if all the coefficients of the equation 
multiply by –1 (change signs), and all coefficients of the equation 
cut by 2, you get the same equation: .
The second case, when the lines are parallel:
Two lines are parallel if and only if their coefficients of the variables are proportional: 
, But.
As an example, consider two straight lines. We check the proportionality of the corresponding coefficients for the variables: 
However, it is quite obvious that.
And the third case, when the lines intersect:
Two lines intersect if and only if their coefficients of the variables are NOT proportional, that is, there is NO such value of “lambda” that the equalities are satisfied 
So, for straight lines we will create a system: 
From the first equation it follows that , and from the second equation: , which means the system is inconsistent(no solutions). Thus, the coefficients of the variables are not proportional.
Conclusion: lines intersect
In practical problems, you can use the solution scheme just discussed. By the way, it is very reminiscent of the algorithm for checking vectors for collinearity, which we looked at in class The concept of linear (in)dependence of vectors. Basis of vectors. But there is a more civilized packaging:
Example 1
Find out the relative position of the lines: 
Solution based on the study of directing vectors of straight lines:
a) From the equations we find the direction vectors of the lines: 
.
, which means that the vectors are not collinear and the lines intersect.
Just in case, I’ll put a stone with signs at the crossroads:
The rest jump over the stone and follow further, straight to Kashchei the Immortal =)
b) Find the direction vectors of the lines: 
The lines have the same direction vector, which means they are either parallel or coincident. There is no need to count the determinant here.
It is obvious that the coefficients of the unknowns are proportional, and .
Let's find out whether the equality is true: 
Thus,
c) Find the direction vectors of the lines: 
Let's calculate the determinant made up of the coordinates of these vectors: 
, therefore, the direction vectors are collinear. The lines are either parallel or coincident.
The proportionality coefficient “lambda” is easy to see directly from the ratio of collinear direction vectors. However, it can also be found through the coefficients of the equations themselves: 
.
Now let's find out whether the equality is true. Both free terms are zero, so:
The resulting value satisfies this equation (any number in general satisfies it).
Thus, the lines coincide.
Answer:
Very soon you will learn (or even have already learned) to solve the problem discussed verbally literally in a matter of seconds. In this regard, I see no point in offering anything for independent decision, it’s better to lay another important brick in the geometric foundation:
How to construct a line parallel to a given one?
For ignorance of this simplest task Nightingale the Robber severely punishes.
Example 2
The straight line is given by the equation. Write an equation for a parallel line that passes through the point.
Solution: Let's denote the unknown line by the letter . What does the condition say about her? The straight line passes through the point. And if the lines are parallel, then it is obvious that the direction vector of the straight line “tse” is also suitable for constructing the straight line “de”.
We take the direction vector out of the equation:
Answer:
The example geometry looks simple: 
Analytical testing consists of the following steps:
1) We check that the lines have the same direction vector (if the equation of the line is not simplified properly, then the vectors will be collinear).
2) Check whether the point satisfies the resulting equation.
In most cases, analytical testing can be easily performed orally. Look at the two equations, and many of you will quickly determine the parallelism of the lines without any drawing.
Examples for independent solutions today will be creative. Because you will still have to compete with Baba Yaga, and she, you know, is a lover of all sorts of riddles.
Example 3
Write an equation for a line passing through a point parallel to the line if
There is a rational and not so rational way to solve it. The shortest way is at the end of the lesson.
We worked a little with parallel lines and will return to them later. The case of coinciding lines is of little interest, so let’s consider a problem that is very familiar to you from the school curriculum:
How to find the point of intersection of two lines?
If straight 
intersect at point , then its coordinates are the solution systems of linear equations 
How to find the point of intersection of lines? Solve the system.
Here you go geometric meaning systems of two linear equations with two unknowns- these are two intersecting (most often) lines on a plane.
Example 4
Find the point of intersection of lines
Solution: There are two ways to solve - graphical and analytical.
The graphical method is to simply draw the given lines and find out the intersection point directly from the drawing: 
Here is our point: . To check, you should substitute its coordinates into each equation of the line; they should fit both there and there. In other words, the coordinates of a point are a solution to the system. Essentially, we looked at a graphical solution systems of linear equations with two equations, two unknowns.
The graphical method is, of course, not bad, but there are noticeable disadvantages. No, the point is not that seventh graders decide this way, the point is that it will take time to create a correct and ACCURATE drawing. In addition, some straight lines are not so easy to construct, and the point of intersection itself may be located somewhere in the thirtieth kingdom outside the notebook sheet.
Therefore, it is more expedient to search for the intersection point using the analytical method. Let's solve the system: 
To solve the system, the method of term-by-term addition of equations was used. To develop relevant skills, take a lesson How to solve a system of equations?
Answer:
The check is trivial - the coordinates of the intersection point must satisfy every equation of the system.
Example 5
Find the point of intersection of the lines if they intersect.
This is an example for you to solve on your own. It is convenient to split the task into several stages. Analysis of the condition suggests that it is necessary:
1) Write down the equation of the straight line.
2) Write down the equation of the straight line.
3) Find out the relative position of the lines.
4) If the lines intersect, then find the point of intersection.
The development of an action algorithm is typical for many geometric problems, and I will repeatedly focus on this.
Full solution and answer at the end of the lesson:
Not even a pair of shoes were worn out before we got to the second section of the lesson:
Perpendicular lines. Distance from a point to a line.
Angle between straight lines
Let's start with a typical and very important task. In the first part, we learned how to build a straight line parallel to this one, and now the hut on chicken legs will turn 90 degrees:
How to construct a line perpendicular to a given one?
Example 6
The straight line is given by the equation. Write an equation perpendicular to the line passing through the point.
Solution: By condition it is known that . It would be nice to find the directing vector of the line. Since the lines are perpendicular, the trick is simple:
From the equation we “remove” the normal vector: , which will be the directing vector of the straight line.
Let's compose the equation of a straight line using a point and a direction vector:
Answer: 
Let's expand the geometric sketch:
Hmmm... Orange sky, orange sea, orange camel.
Analytical verification of the solution:
1) We take out the direction vectors from the equations and use scalar product of vectors we come to the conclusion that the lines are indeed perpendicular: .
By the way, you can use normal vectors, it's even easier.
2) Check whether the point satisfies the resulting equation 
.
The test, again, is easy to perform orally.
Example 7
Find the point of intersection of perpendicular lines if the equation is known 
and period.
This is an example for you to solve on your own. The problem has several actions, so it is convenient to formulate the solution point by point.
Our exciting journey continues:
Distance from point to line
We have a straight strip of river in front of us and our task is to get to it by the shortest route. There are no obstacles, and the most optimal route will be to move perpendicularly. That is, the distance from a point to a line is the length of the perpendicular segment.
Distance in geometry is traditionally denoted by the Greek letter “rho”, for example: – the distance from the point “em” to the straight line “de”.
Distance from point to line 
expressed by the formula
Example 8
Find the distance from a point to a line 
Solution: all you need to do is carefully substitute the numbers into the formula and carry out the calculations:
Answer: 
Let's make the drawing: 
The found distance from the point to the line is exactly the length of the red segment. If you draw up a drawing on checkered paper on a scale of 1 unit. = 1 cm (2 cells), then the distance can be measured with an ordinary ruler.
Let's consider another task based on the same drawing:
The task is to find the coordinates of a point that is symmetrical to the point relative to the straight line 
. I suggest performing the steps yourself, but I will outline a solution algorithm with intermediate results:
1) Find a line that is perpendicular to the line.
2) Find the point of intersection of the lines: 
.
Both actions are discussed in detail in this lesson.
3) The point is the midpoint of the segment. We know the coordinates of the middle and one of the ends. By formulas for the coordinates of the midpoint of a segment we find .
It would be a good idea to check that the distance is also 2.2 units.
Difficulties may arise in calculations here, but a microcalculator is a great help in the tower, allowing you to count common fractions. I have advised you many times and will recommend you again.
How to find the distance between two parallel lines?
Example 9
Find the distance between two parallel lines
This is another example for you to decide on your own. I’ll give you a little hint: there are infinitely many ways to solve this. Debriefing at the end of the lesson, but it’s better to try to guess for yourself, I think your ingenuity was well developed.
Angle between two straight lines
Every corner is a jamb: 
In geometry, the angle between two straight lines is taken to be the SMALLER angle, from which it automatically follows that it cannot be obtuse. In the figure, the angle indicated by the red arc is not considered the angle between intersecting lines. And his “green” neighbor or oppositely oriented"raspberry" corner.
If the lines are perpendicular, then any of the 4 angles can be taken as the angle between them.
How are the angles different? Orientation. Firstly, the direction in which the angle is “scrolled” is fundamentally important. Secondly, a negatively oriented angle is written with a minus sign, for example if .
Why did I tell you this? It seems that we can get by with the usual concept of an angle. The fact is that the formulas by which we will find angles can easily result in a negative result, and this should not take you by surprise. An angle with a minus sign is no worse, and has a very specific geometric meaning. In the drawing, for a negative angle, be sure to indicate its orientation with an arrow (clockwise).
How to find the angle between two straight lines? There are two working formulas:
Example 10
Find the angle between lines
Solution And Method one
Let's consider two straight lines defined by equations in general form: 
If straight not perpendicular, That oriented The angle between them can be calculated using the formula: 
Let us pay close attention to the denominator - this is exactly dot product directing vectors of straight lines:
If , then the denominator of the formula becomes zero, and the vectors will be orthogonal and the lines will be perpendicular. That is why a reservation was made about the non-perpendicularity of straight lines in the formulation.
Based on the above, it is convenient to formalize the solution in two steps:
1) Let's calculate the scalar product of the direction vectors of the lines:
, which means the lines are not perpendicular.
2) Find the angle between straight lines using the formula:
By using inverse function It's easy to find the corner itself. In this case, we use the oddness of the arctangent (see. Graphs and properties of elementary functions):
Answer: 
In the answer we indicate the exact value, as well as an approximate value (preferably in both degrees and radians), calculated using a calculator.
Well, minus, minus, no big deal. Here is a geometric illustration: 
It is not surprising that the angle turned out to have a negative orientation, because in the problem statement the first number is a straight line and the “unscrewing” of the angle began precisely with it.
If you really want to get a positive angle, you need to swap the lines, that is, take the coefficients from the second equation 
, and take the coefficients from the first equation. In short, you need to start with a direct 
.
Definition. Angle between crossing straight lines The angle between intersecting lines parallel to the given intersecting lines is called.
|
Example. Given a cube ABCDA 1 B 1 C 1 D 1 . Find the angle between intersecting lines A 1 B And C 1 D. On the verge CDD 1 C 1 draw a diagonal CD 1 ; CD 1 || B.A. 1 (A 1 B;C 1 D) = (CD 1 ;C 1 D) =90 0 (angle between the diagonals of the square). |
D 1 WITH 1 IN 1 A 1 |
. The angle between a straight line and a plane.
If a line is parallel to a plane or lies in it, then the angle between these lines and the plane is considered equal to 0 0.
Definition. The straight line is said to be perpendicular to the plane , if it is perpendicular to any line lying in this plane. In this case, the angle between the straight line and the plane is considered equal to 90 0.
|
Definition. A straight line is called an oblique line to a certain plane if it intersects this plane but is not perpendicular to it. MK MN– inclined to KN – projection MN to |
|
Definition. The angle between the inclined plane and this plane The angle between the inclined plane and its projection onto a given plane is called.
(MN;) = (MN;KN) = MNK=
Theorem 7 (about three perpendiculars ) . An inclined line to a plane is perpendicular to a line lying in the plane if and only if the projection of this inclined line onto this plane is perpendicular to the given straight line.
|
MK MN– inclined to KN – projection MN to m MNm KNm |
|
. Distances in space.
Definition. Distance from a point to a line, not containing this point is the length of the perpendicular segment drawn from this point to the given plane.
Definition. Distance from point to plane , not containing this point, is the length of the perpendicular drawn from this point to the given plane.
Distance between parallel lines is equal to the distance from any point of one of these lines to the other line.
Distance between parallel planes equal to the distance from an arbitrary point on one of the planes to the other plane.
Distance between a straight line and a plane parallel to it equal to the distance from any point on this line to the plane.
Definition. Distance between two crossing lines is called the length of their common perpendicular.
Distance between crossing lines is equal to the distance from any point of one of these lines to the plane passing through the second line parallel to the first line (in other words: the distance between two parallel planes containing these lines).
V. Angle between planes. Dihedral angle.
If the planes are parallel, then the angle between them is considered equal to 0 0.
Definition. Dihedral angle is a geometric figure formed by two half-planes with a common boundary that do not lie in the same plane. Half-planes are called edges , and their common boundary dihedral edge .
Definition. Linear dihedral angle is the angle obtained when a given dihedral angle is intersected by a plane perpendicular to its edge. All linear angles of a given dihedral angle are equal to each other. The magnitude of a dihedral angle is equal to the magnitude of its linear angle.
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Example. Given a pyramid MABCD , the base of which is a square ABCD with side 2, M.A.ABC, M.A. = 2. Find the angle of the face MBC base plane. (based on the perpendicularity of a straight line and a plane). So the plane MAB intersects a dihedral angle with an edge B.C. and perpendicular to it. Therefore, by definition of linear angle: MBA– linear angle of a given dihedral angle. |
It will be useful for every student who is preparing for the Unified State Exam in mathematics to repeat the topic “Finding an angle between straight lines.” As statistics show, when passing the certification test, tasks in this section of stereometry cause difficulties for a large number of students. At the same time, tasks that require finding the angle between straight lines are found in the Unified State Exam at both the basic and specialized levels. This means that everyone should be able to solve them.
Highlights
There are 4 types in space relative position straight They can coincide, intersect, be parallel or intersecting. The angle between them can be acute or straight.
To find the angle between lines in the Unified State Exam or, for example, in solving, schoolchildren in Moscow and other cities can use several ways to solve problems in this section of stereometry. You can complete the task using classical constructions. To do this, it is worth learning the basic axioms and theorems of stereometry. The student needs to be able to reason logically and create drawings in order to bring the task to a planimetric problem.
You can also use the coordinate vector method using simple formulas, rules and algorithms. The main thing in this case is to perform all calculations correctly. Hone your skills in solving problems in stereometry and other areas school course will help you educational project"Shkolkovo".
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 B 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 and 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 and 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 against equal sides the angles are equal. 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 straight 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. LK- middle line. According to the property of the midline, LK parallel AB and equals half of it.
AND MN, And LK parallel AB. Means, MN parallel LK by the theorem of three parallel lines.
We find that in a quadrilateral MNKL- sides MN And LK parallel and equal, since MN And LK 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 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 educational institutions(basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and expanded - M.: Mnemosyne, 2008. - 288 p. : ill.
2. Geometry. 10-11 grade: Textbook for general education educational institutions/ Sharygin I.F. - M.: Bustard, 1999. - 208 p.: 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 B 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
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