Instruction

Before starting the proof, make sure that the lines lie in the same plane and can be drawn on it. The simplest method of proof is the method of measuring with a ruler. To do this, use a ruler to measure the distance between the straight lines in several places as far apart as possible. If the distance remains the same, the given lines are parallel. But this method is not accurate enough, so it is better to use other methods.

Draw a third line so that it intersects both parallel lines. It forms four outer and four inner corners with them. Consider interior corners. Those that lie through the secant line are called cross-lying. Those that lie on one side are called one-sided. Using a protractor, measure the two inner diagonal corners. If they are equal, then the lines will be parallel. If in doubt, measure one-sided interior angles and add up the resulting values. Lines will be parallel if the sum of one-sided interior angles is equal to 180º.

If you don't have a protractor, use a 90º square. Use it to construct a perpendicular to one of the lines. After that, continue this perpendicular in such a way that it intersects another line. Using the same square, check at what angle this perpendicular intersects it. If this angle is also equal to 90º, then the lines are parallel to each other.

In the event that the lines are given in the Cartesian coordinate system, find their guides or normal vectors. If these vectors are, respectively, collinear with each other, then the lines are parallel. Bring the equation of lines to a general form and find the coordinates of the normal vector of each of the lines. Its coordinates are equal to the coefficients A and B. In the event that the ratio of the corresponding coordinates of the normal vectors is the same, they are collinear, and the lines are parallel.

For example, straight lines are given by the equations 4x-2y+1=0 and x/1=(y-4)/2. The first equation is of general form, the second is canonical. Bring the second equation to a general form. Use the proportion conversion rule for this, and you'll end up with 2x=y-4. After reduction to a general form, get 2x-y + 4 = 0. Since the general equation for any line is written Ax + Vy + C = 0, then for the first line: A = 4, B = 2, and for the second line A = 2, B = 1. For the first direct coordinate of the normal vector (4;2), and for the second - (2;1). Find the ratio of the corresponding coordinates of the normal vectors 4/2=2 and 2/1=2. These numbers are equal, which means the vectors are collinear. Since the vectors are collinear, the lines are parallel.

This article is about parallel lines and about parallel lines. First, the definition of parallel lines in the plane and in space is given, notation is introduced, examples and graphic illustrations of parallel lines are given. Further, the signs and conditions of parallelism of straight lines are analyzed. In conclusion, solutions are shown for typical problems of proving the parallelism of straight lines, which are given by some equations of a straight line in a rectangular coordinate system on a plane and in three-dimensional space.

Page navigation.

Parallel lines - basic information.

Definition.

Two lines in a plane are called parallel if they do not have common points.

Definition.

Two lines in three dimensions are called parallel if they lie in the same plane and have no common points.

Note that the "if they lie in the same plane" clause in the definition of parallel lines in space is very important. Let's clarify this point: two straight lines in three-dimensional space that do not have common points and do not lie in the same plane are not parallel, but are skew.

Here are some examples of parallel lines. The opposite edges of the notebook sheet lie on parallel lines. The straight lines along which the plane of the wall of the house intersects the planes of the ceiling and floor are parallel. Railroad tracks on level ground can also be thought of as parallel lines.

The symbol "" is used to denote parallel lines. That is, if the lines a and b are parallel, then you can briefly write a b.

Note that if lines a and b are parallel, then we can say that line a is parallel to line b, and also that line b is parallel to line a.

Let us voice a statement that plays an important role in the study of parallel lines in the plane: through a point not lying on a given line, there passes the only line parallel to the given one. This statement is accepted as a fact (it cannot be proved on the basis of the known axioms of planimetry), and it is called the axiom of parallel lines.

For the case in space, the theorem is true: through any point in space that does not lie on a given line, there passes a single line parallel to the given one. This theorem can be easily proved using the axiom of parallel lines given above (you can find its proof in the geometry textbook for grades 10-11, which is listed at the end of the article in the bibliography).

For the case in space, the theorem is true: through any point in space that does not lie on a given line, there passes a single line parallel to the given one. This theorem is easily proved using the axiom of parallel lines given above.

Parallelism of lines - signs and conditions of parallelism.

A sign of parallel lines is a sufficient condition for parallel lines, that is, such a condition, the fulfillment of which guarantees parallel lines. In other words, the fulfillment of this condition is sufficient to state the fact that the lines are parallel.

There are also necessary and sufficient conditions for parallel lines in the plane and in three-dimensional space.

Let us explain the meaning of the phrase "necessary and sufficient condition for parallel lines".

We have already dealt with the sufficient condition for parallel lines. And what is the "necessary condition for parallel lines"? By the name "necessary" it is clear that the fulfillment of this condition is necessary for the lines to be parallel. In other words, if the necessary condition for parallel lines is not satisfied, then the lines are not parallel. In this way, necessary and sufficient condition for lines to be parallel is a condition, the fulfillment of which is both necessary and sufficient for parallel lines. That is, on the one hand, this is a sign of parallel lines, and on the other hand, this is a property that parallel lines have.

Before stating the necessary and sufficient condition for lines to be parallel, it is useful to recall a few auxiliary definitions.

secant line is a line that intersects each of the two given non-coincident lines.

At the intersection of two lines of a secant, eight non-deployed ones are formed. The so-called lying crosswise, corresponding and one-sided corners. Let's show them on the drawing.

Theorem.

If two straight lines on a plane are crossed by a secant, then for their parallelism it is necessary and sufficient that the crosswise lying angles are equal, or the corresponding angles are equal, or the sum of one-sided angles is equal to 180 degrees.

Let us show a graphical illustration of this necessary and sufficient condition for parallel lines in the plane.

You can find proofs of these conditions for parallel lines in geometry textbooks for grades 7-9.

Note that these conditions can also be used in three-dimensional space - the main thing is that the two lines and the secant lie in the same plane.

Here are a few more theorems that are often used in proving the parallelism of lines.

Theorem.

If two lines in a plane are parallel to a third line, then they are parallel. The proof of this feature follows from the axiom of parallel lines.

There is a similar condition for parallel lines in three-dimensional space.

Theorem.

If two lines in space are parallel to a third line, then they are parallel. The proof of this feature is considered in the geometry lessons in grade 10.

Let us illustrate the voiced theorems.

Let us give one more theorem that allows us to prove the parallelism of lines in the plane.

Theorem.

If two lines in a plane are perpendicular to a third line, then they are parallel.

There is a similar theorem for lines in space.

Theorem.

If two lines in three-dimensional space are perpendicular to the same plane, then they are parallel.

Let us draw pictures corresponding to these theorems.

All the theorems formulated above, signs and necessary and sufficient conditions are perfectly suitable for proving the parallelism of straight lines by methods of geometry. That is, to prove the parallelism of two given lines, it is necessary to show that they are parallel to the third line, or to show the equality of cross-lying angles, etc. Many of these problems are solved in geometry lessons in high school. However, it should be noted that in many cases it is convenient to use the method of coordinates to prove the parallelism of lines in a plane or in three-dimensional space. Let us formulate the necessary and sufficient conditions for the parallelism of lines that are given in a rectangular coordinate system.

Parallelism of lines in a rectangular coordinate system.

In this section of the article, we will formulate necessary and sufficient conditions for parallel lines in a rectangular coordinate system, depending on the type of equations that determine these lines, and we will also give detailed solutions to typical problems.

Let's start with the condition of parallelism of two lines on the plane in the rectangular coordinate system Oxy . His proof is based on the definition of the directing vector of the line and the definition of the normal vector of the line on the plane.

Theorem.

For two non-coincident lines to be parallel in a plane, it is necessary and sufficient that the direction vectors of these lines are collinear, or the normal vectors of these lines are collinear, or the direction vector of one line is perpendicular to the normal vector of the second line.

Obviously, the condition of parallelism of two lines in the plane reduces to (direction vectors of lines or normal vectors of lines) or to (direction vector of one line and normal vector of the second line). Thus, if and are the direction vectors of the lines a and b, and and are the normal vectors of lines a and b, respectively, then the necessary and sufficient condition for parallel lines a and b can be written as , or , or , where t is some real number. In turn, the coordinates of the directing and (or) normal vectors of the straight lines a and b are found from the known equations of the straight lines.

In particular, if the line a in the rectangular coordinate system Oxy on the plane defines the general equation of the line of the form , and the straight line b - , then the normal vectors of these lines have coordinates and respectively, and the condition of parallelism of lines a and b will be written as .

If the straight line a corresponds to the equation of the straight line with the slope coefficient of the form . Therefore, if straight lines on a plane in a rectangular coordinate system are parallel and can be given by equations of straight lines with slope coefficients, then the slope coefficients of the lines will be equal. And vice versa: if non-coinciding straight lines on a plane in a rectangular coordinate system can be given by the equations of a straight line with equal slope coefficients, then such straight lines are parallel.

If the line a and the line b in a rectangular coordinate system define the canonical equations of the line on the plane of the form and , or parametric equations of a straight line on a plane of the form and respectively, then the direction vectors of these lines have coordinates and , and the parallelism condition for lines a and b is written as .

Let's take a look at a few examples.

Example.

Are the lines parallel? and ?

Solution.

We rewrite the equation of a straight line in segments in the form of a general equation of a straight line: . Now we can see that is the normal vector of the straight line , and is the normal vector of the straight line. These vectors are not collinear, since there is no real number t for which the equality ( ). Consequently, the necessary and sufficient condition for the parallelism of lines on the plane is not satisfied, therefore, the given lines are not parallel.

Answer:

No, the lines are not parallel.

Example.

Are lines and parallels?

Solution.

We bring the canonical equation of a straight line to the equation of a straight line with a slope: . Obviously, the equations of the lines and are not the same (in this case, the given lines would be the same) and the slopes of the lines are equal, therefore, the original lines are parallel.

The second solution.

First, let's show that the original lines do not coincide: take any point of the line, for example, (0, 1) , the coordinates of this point do not satisfy the equation of the line, therefore, the lines do not coincide. Now let's check the fulfillment of the condition of parallelism of these lines. The normal vector of the line is the vector , and the direction vector of the line is the vector . Let's calculate and : . Consequently, the vectors and are perpendicular, which means that the necessary and sufficient condition for the parallelism of the given lines is satisfied. So the lines are parallel.

Answer:

The given lines are parallel.

To prove the parallelism of lines in a rectangular coordinate system in three-dimensional space, the following necessary and sufficient condition is used.

Theorem.

For non-coincident lines to be parallel in three-dimensional space, it is necessary and sufficient that their direction vectors be collinear.

Thus, if the equations of lines in a rectangular coordinate system in three-dimensional space are known and you need to answer the question whether these lines are parallel or not, then you need to find the coordinates of the direction vectors of these lines and check the fulfillment of the condition of collinearity of the direction vectors. In other words, if and - direction vectors of straight lines a given lines have coordinates and . Because , then . Thus, the necessary and sufficient condition for two lines to be parallel in space is satisfied. This proves the parallelism of the lines and .

Bibliography.

• Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Poznyak E.G., Yudina I.I. Geometry. Grades 7 - 9: a textbook for educational institutions.
• Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Kiseleva L.S., Poznyak E.G. Geometry. Textbook for 10-11 grades of high school.
• Pogorelov A.V., Geometry. Textbook for grades 7-11 of educational institutions.
• Bugrov Ya.S., Nikolsky S.M. Higher Mathematics. Volume One: Elements of Linear Algebra and Analytic Geometry.
• Ilyin V.A., Poznyak E.G. Analytic geometry.

In this article, we will talk about parallel lines, give definitions, designate the signs and conditions of parallelism. For clarity of theoretical material, we will use illustrations and the solution of typical examples.

Definition 1

Parallel lines in the plane are two straight lines in the plane that do not have common points.

Definition 2

Parallel lines in 3D space- two straight lines in three-dimensional space that lie in the same plane and do not have common points.

It should be noted that in order to determine parallel lines in space, the clarification “lying in the same plane” is extremely important: two lines in three-dimensional space that do not have common points and do not lie in the same plane are not parallel, but intersecting.

To denote parallel lines, it is common to use the symbol ∥ . That is, if the given lines a and b are parallel, this condition should be briefly written as follows: a ‖ b . Verbally, the parallelism of lines is indicated as follows: lines a and b are parallel, or line a is parallel to line b, or line b is parallel to line a.

Let us formulate a statement that plays an important role in the topic under study.

Axiom

Through a point that does not belong to a given line, there is only one line parallel to the given line. This statement cannot be proved on the basis of the known axioms of planimetry.

In the case when it comes to space, the theorem is true:

Theorem 1

Through any point in space that does not belong to a given line, there will be only one line parallel to the given one.

This theorem is easy to prove on the basis of the above axiom (geometry program for grades 10-11).

The sign of parallelism is a sufficient condition under which parallel lines are guaranteed. In other words, the fulfillment of this condition is sufficient to confirm the fact of parallelism.

In particular, there are necessary and sufficient conditions for the parallelism of lines in the plane and in space. Let us explain: necessary means the condition, the fulfillment of which is necessary for parallel lines; if it is not satisfied, the lines are not parallel.

Summarizing, a necessary and sufficient condition for the parallelism of lines is such a condition, the observance of which is necessary and sufficient for the lines to be parallel to each other. On the one hand, this is a sign of parallelism, on the other hand, a property inherent in parallel lines.

Before giving a precise formulation of the necessary and sufficient conditions, we recall a few more additional concepts.

Definition 3

secant line is a line that intersects each of the two given non-coinciding lines.

Intersecting two straight lines, the secant forms eight non-expanded angles. To formulate the necessary and sufficient condition, we will use such types of angles as cross-lying, corresponding, and one-sided. Let's demonstrate them in the illustration:

Theorem 2

If two lines on a plane intersect a secant, then for the given lines to be parallel it is necessary and sufficient that the crosswise lying angles be equal, or the corresponding angles be equal, or the sum of one-sided angles be equal to 180 degrees.

Let us graphically illustrate the necessary and sufficient condition for parallel lines on the plane:

The proof of these conditions is present in the geometry program for grades 7-9.

In general, these conditions are also applicable for three-dimensional space, provided that the two lines and the secant belong to the same plane.

Let us point out a few more theorems that are often used in proving the fact that lines are parallel.

Theorem 3

In a plane, two lines parallel to a third are parallel to each other. This feature is proved on the basis of the axiom of parallelism mentioned above.

Theorem 4

In three-dimensional space, two lines parallel to a third are parallel to each other.

The proof of the attribute is studied in the 10th grade geometry program.

We give an illustration of these theorems:

Let us indicate one more pair of theorems that prove the parallelism of lines.

Theorem 5

In a plane, two lines perpendicular to a third are parallel to each other.

Let us formulate a similar one for a three-dimensional space.

Theorem 6

In three-dimensional space, two lines perpendicular to a third are parallel to each other.

Let's illustrate:

All the above theorems, signs and conditions make it possible to conveniently prove the parallelism of lines by the methods of geometry. That is, to prove the parallelism of lines, one can show that the corresponding angles are equal, or demonstrate the fact that two given lines are perpendicular to the third, and so on. But we note that it is often more convenient to use the coordinate method to prove the parallelism of lines in a plane or in three-dimensional space.

Parallelism of lines in a rectangular coordinate system

In a given rectangular coordinate system, a straight line is determined by the equation of a straight line on a plane of one of the possible types. Similarly, a straight line given in a rectangular coordinate system in three-dimensional space corresponds to some equations of a straight line in space.

Let us write the necessary and sufficient conditions for the parallelism of lines in a rectangular coordinate system, depending on the type of equation describing the given lines.

Let's start with the condition of parallel lines in the plane. It is based on the definitions of the direction vector of the line and the normal vector of the line in the plane.

Theorem 7

For two non-coincident lines to be parallel on a plane, it is necessary and sufficient that the direction vectors of the given lines be collinear, or the normal vectors of the given lines are collinear, or the direction vector of one line is perpendicular to the normal vector of the other line.

It becomes obvious that the condition of parallel lines on the plane is based on the condition of collinear vectors or the condition of perpendicularity of two vectors. That is, if a → = (a x , a y) and b → = (b x , b y) are the direction vectors of lines a and b ;

and n b → = (n b x , n b y) are normal vectors of lines a and b , then we write the above necessary and sufficient condition as follows: a → = t b → ⇔ a x = t b x a y = t b y or n a → = t n b → ⇔ n a x = t n b x n a y = t n b y or a → , n b → = 0 ⇔ a x n b x + a y n b y = 0 , where t is some real number. The coordinates of the directing or direct vectors are determined by the given equations of the lines. Let's consider the main examples.

1. The line a in a rectangular coordinate system is determined by the general equation of the line: A 1 x + B 1 y + C 1 = 0 ; line b - A 2 x + B 2 y + C 2 = 0 . Then the normal vectors of the given lines will have coordinates (A 1 , B 1) and (A 2 , B 2) respectively. We write the condition of parallelism as follows:

A 1 = t A 2 B 1 = t B 2

1. The straight line a is described by the equation of a straight line with a slope of the form y = k 1 x + b 1 . Straight line b - y \u003d k 2 x + b 2. Then the normal vectors of the given lines will have coordinates (k 1 , - 1) and (k 2 , - 1), respectively, and we write the parallelism condition as follows:

k 1 = t k 2 - 1 = t (- 1) ⇔ k 1 = t k 2 t = 1 ⇔ k 1 = k 2

Thus, if parallel lines on a plane in a rectangular coordinate system are given by equations with slope coefficients, then the slope coefficients of the given lines will be equal. And the converse statement is true: if non-coinciding lines on a plane in a rectangular coordinate system are determined by the equations of a line with the same slope coefficients, then these given lines are parallel.

1. The lines a and b in a rectangular coordinate system are given by the canonical equations of the line on the plane: x - x 1 a x = y - y 1 a y and x - x 2 b x = y - y 2 b y or the parametric equations of the line on the plane: x = x 1 + λ a x y = y 1 + λ a y and x = x 2 + λ b x y = y 2 + λ b y .

Then the direction vectors of the given lines will be: a x , a y and b x , b y respectively, and we write the parallelism condition as follows:

a x = t b x a y = t b y

Let's look at examples.

Example 1

Given two lines: 2 x - 3 y + 1 = 0 and x 1 2 + y 5 = 1 . You need to determine if they are parallel.

Solution

We write the equation of a straight line in segments in the form of a general equation:

x 1 2 + y 5 = 1 ⇔ 2 x + 1 5 y - 1 = 0

We see that n a → = (2 , - 3) is the normal vector of the line 2 x - 3 y + 1 = 0 , and n b → = 2 , 1 5 is the normal vector of the line x 1 2 + y 5 = 1 .

The resulting vectors are not collinear, because there is no such value of t for which the equality will be true:

2 = t 2 - 3 = t 1 5 ⇔ t = 1 - 3 = t 1 5 ⇔ t = 1 - 3 = 1 5

Thus, the necessary and sufficient condition of parallelism of lines on the plane is not satisfied, which means that the given lines are not parallel.

Answer: given lines are not parallel.

Example 2

Given lines y = 2 x + 1 and x 1 = y - 4 2 . Are they parallel?

Solution

Let's transform the canonical equation of the straight line x 1 \u003d y - 4 2 to the equation of a straight line with a slope:

x 1 = y - 4 2 ⇔ 1 (y - 4) = 2 x ⇔ y = 2 x + 4

We see that the equations of the lines y = 2 x + 1 and y = 2 x + 4 are not the same (if it were otherwise, the lines would be the same) and the slopes of the lines are equal, which means that the given lines are parallel.

Let's try to solve the problem differently. First, we check whether the given lines coincide. We use any point of the line y \u003d 2 x + 1, for example, (0, 1) , the coordinates of this point do not correspond to the equation of the line x 1 \u003d y - 4 2, which means that the lines do not coincide.

The next step is to determine the fulfillment of the parallelism condition for the given lines.

The normal vector of the line y = 2 x + 1 is the vector n a → = (2 , - 1) , and the direction vector of the second given line is b → = (1 , 2) . The scalar product of these vectors is zero:

n a → , b → = 2 1 + (- 1) 2 = 0

Thus, the vectors are perpendicular: this demonstrates to us the fulfillment of the necessary and sufficient condition for the original lines to be parallel. Those. given lines are parallel.

Answer: these lines are parallel.

To prove the parallelism of lines in a rectangular coordinate system of three-dimensional space, the following necessary and sufficient condition is used.

Theorem 8

For two non-coincident lines in three-dimensional space to be parallel, it is necessary and sufficient that the direction vectors of these lines be collinear.

Those. for given equations of lines in three-dimensional space, the answer to the question: are they parallel or not, is found by determining the coordinates of the direction vectors of the given lines, as well as checking the condition of their collinearity. In other words, if a → = (a x, a y, a z) and b → = (b x, b y, b z) are the direction vectors of the lines a and b, respectively, then in order for them to be parallel, the existence of such a real number t is necessary, so that equality holds:

a → = t b → ⇔ a x = t b x a y = t b y a z = t b z

Example 3

Given lines x 1 = y - 2 0 = z + 1 - 3 and x = 2 + 2 λ y = 1 z = - 3 - 6 λ . It is necessary to prove the parallelism of these lines.

Solution

The conditions of the problem are the canonical equations of one straight line in space and the parametric equations of another straight line in space. Direction vectors a → and b → given lines have coordinates: (1 , 0 , - 3) and (2 , 0 , - 6) .

1 = t 2 0 = t 0 - 3 = t - 6 ⇔ t = 1 2 , then a → = 1 2 b → .

Therefore, the necessary and sufficient condition for parallel lines in space is satisfied.

Answer: the parallelism of the given lines is proved.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

Parallel lines. Properties and signs of parallel lines

1. Axiom of parallel. Through a given point, at most one straight line can be drawn parallel to the given one.

2. If two lines are parallel to the same line, then they are parallel to each other.

3. Two lines perpendicular to the same line are parallel.

4. If two parallel lines are intersected by a third, then the internal cross-lying angles formed at the same time are equal; corresponding angles are equal; interior one-sided angles add up to 180°.

5. If at the intersection of two straight lines the third one forms equal interior crosswise lying angles, then the straight lines are parallel.

6. If at the intersection of two lines the third form equal corresponding angles, then the lines are parallel.

7. If at the intersection of two lines of the third, the sum of the internal one-sided angles is 180 °, then the lines are parallel.

Thales' theorem. If equal segments are laid out on one side of the angle and parallel straight lines are drawn through their ends, intersecting the second side of the angle, then equal segments will also be deposited on the second side of the angle.

Theorem on proportional segments. Parallel straight lines intersecting the sides of the angle cut proportional segments on them.

Triangle. Signs of equality of triangles.

1. If two sides and the angle between them of one triangle are respectively equal to two sides and the angle between them of another triangle, then the triangles are congruent.

2. If the side and two angles adjacent to it of one triangle are respectively equal to the side and two angles adjacent to it of another triangle, then the triangles are congruent.

3. If three sides of one triangle are respectively equal to three sides of another triangle, then the triangles are congruent.

Signs of equality of right triangles

1. On two legs.

2. Along the leg and hypotenuse.

3. By hypotenuse and acute angle.

4. Along the leg and an acute angle.

The theorem on the sum of the angles of a triangle and its consequences

1. The sum of the interior angles of a triangle is 180°.

2. The external angle of a triangle is equal to the sum of two internal angles not adjacent to it.

3. The sum of the interior angles of a convex n-gon is

4. The sum of the external angles of a ga-gon is 360°.

5. Angles with mutually perpendicular sides are equal if they are both acute or both obtuse.

6. The angle between the bisectors of adjacent angles is 90°.

7. The bisectors of internal one-sided angles with parallel lines and a secant are perpendicular.

The main properties and signs of an isosceles triangle

1. The angles at the base of an isosceles triangle are equal.

2. If two angles of a triangle are equal, then it is isosceles.

3. In an isosceles triangle, the median, bisector and height drawn to the base are the same.

4. If any pair of segments from the triple - median, bisector, height - coincides in a triangle, then it is isosceles.

The triangle inequality and its consequences

1. The sum of two sides of a triangle is greater than its third side.

2. The sum of the links of the broken line is greater than the segment connecting the beginning

the first link with the end of the last.

3. Opposite the larger angle of the triangle lies the larger side.

4. Against the larger side of the triangle lies a larger angle.

5. The hypotenuse of a right triangle is greater than the leg.

6. If perpendicular and inclined are drawn from one point to a straight line, then

1) the perpendicular is shorter than the inclined ones;

2) a larger slope corresponds to a larger projection and vice versa.

The middle line of the triangle.

The line segment connecting the midpoints of the two sides of a triangle is called the midline of the triangle.

Triangle midline theorem.

The median line of the triangle is parallel to the side of the triangle and equal to half of it.

Triangle median theorems

1. The medians of a triangle intersect at one point and divide it in a ratio of 2: 1, counting from the top.

2. If the median of a triangle is equal to half of the side to which it is drawn, then the triangle is right-angled.

3. The median of a right triangle drawn from the vertex of the right angle is equal to half of the hypotenuse.

Property of perpendicular bisectors to the sides of a triangle. The perpendicular bisectors to the sides of the triangle intersect at one point, which is the center of the circle circumscribed about the triangle.

Triangle altitude theorem. The lines containing the altitudes of the triangle intersect at one point.

Triangle bisector theorem. The bisectors of a triangle intersect at one point, which is the center of the circle inscribed in the triangle.

Bisector property of a triangle. The bisector of a triangle divides its side into segments proportional to the other two sides.

Signs of similarity of triangles

1. If two angles of one triangle are respectively equal to two angles of another, then the triangles are similar.

2. If two sides of one triangle are respectively proportional to two sides of another, and the angles enclosed between these sides are equal, then the triangles are similar.

3. If the three sides of one triangle are respectively proportional to the three sides of another, then the triangles are similar.

Areas of Similar Triangles

1. The ratio of the areas of similar triangles is equal to the square of the similarity coefficient.

2. If two triangles have equal angles, then their areas are related as the products of the sides that enclose these angles.

In a right triangle

1. The leg of a right triangle is equal to the product of the hypotenuse and the sine of the opposite or the cosine of the acute angle adjacent to this leg.

2. The leg of a right triangle is equal to the other leg multiplied by the tangent of the opposite or the cotangent of the acute angle adjacent to this leg.

3. The leg of a right triangle lying opposite an angle of 30 ° is equal to half the hypotenuse.

4. If the leg of a right triangle is equal to half of the hypotenuse, then the angle opposite this leg is 30°.

5. R = ; g \u003d, where a, b are legs, and c is the hypotenuse of a right triangle; r and R are the radii of the inscribed and circumscribed circles, respectively.

The Pythagorean theorem and the converse of the Pythagorean theorem

1. The square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs.

2. If the square of a side of a triangle is equal to the sum of the squares of its other two sides, then the triangle is right-angled.

Mean proportionals in a right triangle.

The height of a right triangle, drawn from the vertex of the right angle, is the average proportional to the projections of the legs onto the hypotenuse, and each leg is the average proportional to the hypotenuse and its projection onto the hypotenuse.

Metric ratios in a triangle

1. Theorem of cosines. The square of a side of a triangle is equal to the sum of the squares of the other two sides without doubling the product of those sides times the cosine of the angle between them.

2. Corollary from the cosine theorem. The sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of all its sides.

3. Formula for the median of a triangle. If m is the median of the triangle drawn to side c, then m = where a and b are the remaining sides of the triangle.

4. Sine theorem. The sides of a triangle are proportional to the sines of the opposite angles.

5. Generalized sine theorem. The ratio of a side of a triangle to the sine of the opposite angle is equal to the diameter of the circle circumscribing the triangle.

Triangle area formulas

1. The area of ​​a triangle is half the product of the base and the height.

2. The area of ​​a triangle is equal to half the product of its two sides and the sine of the angle between them.

3. The area of ​​a triangle is equal to the product of its semiperimeter and the radius of the inscribed circle.

4. The area of ​​a triangle is equal to the product of its three sides divided by four times the radius of the circumscribed circle.

5. Heron's formula: S=, where p is the semiperimeter; a, b, c - sides of the triangle.

Elements of an equilateral triangle. Let h, S, r, R be the height, area, radii of the inscribed and circumscribed circles of an equilateral triangle with side a. Then
Quadrilaterals

Parallelogram. A parallelogram is a quadrilateral whose opposite sides are pairwise parallel.

Properties and features of a parallelogram.

1. The diagonal divides the parallelogram into two equal triangles.

2. Opposite sides of a parallelogram are equal in pairs.

3. Opposite angles of a parallelogram are equal in pairs.

4. The diagonals of the parallelogram intersect and bisect the point of intersection.

5. If the opposite sides of a quadrilateral are equal in pairs, then this quadrilateral is a parallelogram.

6. If two opposite sides of a quadrilateral are equal and parallel, then this quadrilateral is a parallelogram.

7. If the diagonals of a quadrilateral are bisected by the intersection point, then this quadrilateral is a parallelogram.

Property of the midpoints of the sides of a quadrilateral. The midpoints of the sides of any quadrilateral are the vertices of a parallelogram whose area is half the area of ​​the quadrilateral.

Rectangle. A rectangle is a parallelogram with a right angle.

Properties and signs of a rectangle.

1. The diagonals of a rectangle are equal.

2. If the diagonals of a parallelogram are equal, then this parallelogram is a rectangle.

Square. A square is a rectangle all sides of which are equal.

Rhombus. A rhombus is a quadrilateral all sides of which are equal.

Properties and signs of a rhombus.

1. The diagonals of the rhombus are perpendicular.

2. The diagonals of a rhombus bisect its corners.

3. If the diagonals of a parallelogram are perpendicular, then this parallelogram is a rhombus.

4. If the diagonals of a parallelogram divide its angles in half, then this parallelogram is a rhombus.

Trapeze. A trapezoid is a quadrilateral in which only two opposite sides (bases) are parallel. The median line of a trapezoid is a segment connecting the midpoints of non-parallel sides (lateral sides).

1. The median line of the trapezoid is parallel to the bases and equal to their half-sum.

2. The segment connecting the midpoints of the diagonals of the trapezoid is equal to the half-difference of the bases.

Remarkable property of a trapezoid. The point of intersection of the diagonals of the trapezoid, the point of intersection of the extensions of the sides and the midpoints of the bases lie on the same straight line.

Isosceles trapezium. A trapezoid is called isosceles if its sides are equal.

Properties and signs of an isosceles trapezoid.

1. The angles at the base of an isosceles trapezoid are equal.

2. The diagonals of an isosceles trapezoid are equal.

3. If the angles at the base of the trapezoid are equal, then it is isosceles.

4. If the diagonals of a trapezoid are equal, then it is isosceles.

5. The projection of the lateral side of an isosceles trapezoid onto the base is equal to the half-difference of the bases, and the projection of the diagonal is half the sum of the bases.

Formulas for the area of ​​a quadrilateral

1. The area of ​​a parallelogram is equal to the product of the base and the height.

2. The area of ​​a parallelogram is equal to the product of its adjacent sides and the sine of the angle between them.

3. The area of ​​a rectangle is equal to the product of its two adjacent sides.

4. The area of ​​a rhombus is half the product of its diagonals.

5. The area of ​​a trapezoid is equal to the product of half the sum of the bases and the height.

6. The area of ​​a quadrilateral is equal to half the product of its diagonals and the sine of the angle between them.

7. Heron's formula for a quadrilateral around which a circle can be described:

S \u003d, where a, b, c, d are the sides of this quadrilateral, p is the semi-perimeter, and S is the area.

Similar figures

1. The ratio of the corresponding linear elements of similar figures is equal to the similarity coefficient.

2. The ratio of the areas of similar figures is equal to the square of the similarity coefficient.

regular polygon.

Let a n be the side of a regular n-gon, and r n and R n be the radii of the inscribed and circumscribed circles. Then

Circle.

A circle is the locus of points in a plane that are at the same positive distance from a given point, called the center of the circle.

Basic properties of a circle

1. The diameter perpendicular to the chord divides the chord and the arcs it subtracts in half.

2. A diameter passing through the middle of a chord that is not a diameter is perpendicular to that chord.

3. The median perpendicular to the chord passes through the center of the circle.

4. Equal chords are removed from the center of the circle at equal distances.

5. The chords of a circle that are equidistant from the center are equal.

6. The circle is symmetrical with respect to any of its diameters.

7. Arcs of a circle enclosed between parallel chords are equal.

8. Of the two chords, the one that is less distant from the center is larger.

9. Diameter is the largest chord of a circle.

Tangent to circle. A line that has a single point in common with a circle is called a tangent to the circle.

1. The tangent is perpendicular to the radius drawn to the point of contact.

2. If the line a passing through a point on the circle is perpendicular to the radius drawn to this point, then the line a is tangent to the circle.

3. If the lines passing through the point M touch the circle at points A and B, then MA = MB and ﮮAMO = ﮮBMO, where the point O is the center of the circle.

4. The center of a circle inscribed in an angle lies on the bisector of this angle.

tangent circle. Two circles are said to touch if they have a single common point (tangent point).

1. The point of contact of two circles lies on their line of centers.

2. Circles of radii r and R with centers O 1 and O 2 touch externally if and only if R + r \u003d O 1 O 2.

3. Circles of radii r and R (r

4. Circles with centers O 1 and O 2 touch externally at point K. Some straight line touches these circles at different points A and B and intersects with a common tangent passing through point K at point C. Then ﮮAK B \u003d 90 ° and ﮮO 1 CO 2 \u003d 90 °.

5. The segment of the common external tangent to two tangent circles of radii r and R is equal to the segment of the common internal tangent enclosed between the common external ones. Both of these segments are equal.

Angles associated with a circle

1. The value of the arc of a circle is equal to the value of the central angle based on it.

2. An inscribed angle is equal to half the angular magnitude of the arc on which it rests.

3. Inscribed angles based on the same arc are equal.

4. The angle between intersecting chords is equal to half the sum of opposite arcs cut by the chords.

5. The angle between two secants intersecting outside the circle is equal to the half-difference of the arcs cut by the secants on the circle.

6. The angle between the tangent and the chord drawn from the point of contact is equal to half the angular value of the arc cut on the circle by this chord.

Properties of circle chords

1. The line of centers of two intersecting circles is perpendicular to their common chord.

2. The products of the lengths of the segments of the chords AB and CD of the circle intersecting at the point E are equal, that is, AE EB \u003d CE ED.

Inscribed and circumscribed circles

1. The centers of the inscribed and circumscribed circles of a regular triangle coincide.

2. The center of a circle circumscribed about a right triangle is the midpoint of the hypotenuse.

3. If a circle can be inscribed in a quadrilateral, then the sums of its opposite sides are equal.

4. If a quadrilateral can be inscribed in a circle, then the sum of its opposite angles is 180°.

5. If the sum of the opposite angles of a quadrilateral is 180°, then a circle can be circumscribed around it.

6. If a circle can be inscribed in a trapezoid, then the lateral side of the trapezoid is visible from the center of the circle at a right angle.

7. If a circle can be inscribed in a trapezoid, then the radius of the circle is the average proportional to the segments into which the tangent point divides the lateral side.

8. If a circle can be inscribed in a polygon, then its area is equal to the product of the semiperimeter of the polygon and the radius of this circle.

The tangent and secant theorem and its corollary

1. If a tangent and a secant are drawn from one point to the circle, then the product of the entire secant by its outer part is equal to the square of the tangent.

2. The product of the entire secant by its outer part for a given point and a given circle is constant.

The circumference of a circle of radius R is C= 2πR

CHAPTER III.
PARALLEL LINES

§ 35. SIGNS OF PARALLELITY OF TWO DIRECT LINES.

The theorem that two perpendiculars to one line are parallel (§ 33) gives a sign that two lines are parallel. It is possible to derive more general signs of parallelism of two lines.

1. The first sign of parallelism.

If, at the intersection of two lines with a third, the interior angles lying across are equal, then these lines are parallel.

Let lines AB and CD intersect line EF and / 1 = / 2. Take the point O - the middle of the segment KL of the secant EF (Fig. 189).

Let us drop the perpendicular OM from the point O to the line AB and continue it until it intersects with the line CD, AB_|_MN. Let us prove that CD_|_MN.
To do this, consider two triangles: MOE and NOK. These triangles are equal to each other. Indeed: / 1 = / 2 by the condition of the theorem; OK = OL - by construction;
/ MOL = / NOK as vertical corners. Thus, the side and two angles adjacent to it of one triangle are respectively equal to the side and two angles adjacent to it of another triangle; Consequently, /\ MOL = /\ NOK, and hence
/ LMO = / kno but / LMO is direct, hence, and / KNO is also direct. Thus, the lines AB and CD are perpendicular to the same line MN, hence they are parallel (§ 33), which was to be proved.

Note. The intersection of the lines MO and CD can be established by rotating the triangle MOL around the point O by 180°.

2. The second sign of parallelism.

Let's see if the lines AB and CD are parallel if, at the intersection of their third line EF, the corresponding angles are equal.

Let some corresponding angles be equal, for example / 3 = / 2 (dev. 190);
/ 3 = / 1, as the corners are vertical; means, / 2 will be equal / 1. But angles 2 and 1 are internal crosswise angles, and we already know that if at the intersection of two straight lines by a third, the internal crosswise lying angles are equal, then these lines are parallel. Therefore, AB || CD.

If at the intersection of two lines of the third the corresponding angles are equal, then these two lines are parallel.

The construction of parallel lines with the help of a ruler and a drawing triangle is based on this property. This is done as follows.

Let us attach the triangle to the ruler as shown in drawing 191. We will move the triangle so that one of its sides slides along the ruler, and draw several straight lines along any other side of the triangle. These lines will be parallel.

3. The third sign of parallelism.

Let us know that at the intersection of two lines AB and CD by the third line, the sum of any internal one-sided angles is equal to 2 d(or 180°). Will the lines AB and CD be parallel in this case (Fig. 192).

Let / 1 and / 2 interior one-sided angles and add up to 2 d.
But / 3 + / 2 = 2d as adjacent angles. Consequently, / 1 + / 2 = / 3+ / 2.

From here / 1 = / 3, and these corners are internally lying crosswise. Therefore, AB || CD.

If at the intersection of two lines by a third, the sum of the interior one-sided angles is equal to 2 d, then the two lines are parallel.

An exercise.

Prove that the lines are parallel:
a) if the external cross-lying angles are equal (Fig. 193);
b) if the sum of external unilateral angles is 2 d(devil 194).