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

Yandex.RTB R-A-339285-1 Definition 1

Parallel lines on a plane- two straight lines on a plane that do not have common points.

Definition 2

Parallel lines in three-dimensional space- two straight lines in three-dimensional space, lying in the same plane and having no common points.

It should be noted that in order to define parallel straight lines in space, it is extremely important to clarify "lying in the same plane": 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 intersecting.

To indicate parallelism of 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. In words, the parallelism of straight lines is denoted as follows: straight lines a and b are parallel, or straight line a is parallel to straight line b, or straight line b is parallel to straight line a.

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

Axiom

The only straight line parallel to the given one passes through a point that does not belong to the given line. This statement cannot be proved on the basis of the known axioms of planimetry.

In the case when we are talking about space, the theorem is true:

Theorem 1

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

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

The parallelism criterion is a sufficient condition under which the parallelism of straight lines is 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 straight lines on the plane and in space. Let us explain: necessary means that condition, the fulfillment of which is necessary for the parallelism of straight lines; if it is not fulfilled, the lines are not parallel.

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

Before giving an exact formulation of the necessary and sufficient condition, let us recall a few more additional concepts.

Definition 3

Secant line- a straight line intersecting each of two specified non-coinciding straight lines.

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

Theorem 2

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

Let us illustrate graphically a necessary and sufficient condition for the parallelism of straight lines on a 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, given that the two lines and the secant line belong to the same plane.

Let us point out a few more theorems that are often used in the proof of the fact of parallelism of lines.

Theorem 3

On the plane, two straight lines parallel to the third are parallel to each other. This criterion is proved on the basis of the axiom of parallelism indicated above.

Theorem 4

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

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

Let us give an illustration of these theorems:

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

Theorem 5

On the plane, two straight lines perpendicular to the third are parallel to each other.

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

Theorem 6

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

Let's illustrate:

All the above theorems, criteria, and conditions make it possible to conveniently prove the parallelism of straight lines by methods of geometry. That is, in order to prove the parallelism of straight lines, one can show that the corresponding angles are equal, or demonstrate the fact that two given straight lines are perpendicular to the third, etc. But note that it is often more convenient to use the coordinate method to prove the parallelism of straight lines on 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 in the plane of one of possible types... So 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 down the necessary and sufficient conditions for the parallelism of straight lines in a rectangular coordinate system depending on the type of equation describing the given straight lines.

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

Theorem 7

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

It becomes obvious that the condition for parallelism of straight lines on a 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 direction vectors of straight lines a and b;

and nb → = (nbx, nby) are normal vectors of lines a and b, then the above necessary and sufficient condition can be written as follows: a → = t b → ⇔ ax = t bxay = t by or na → = t nb → ⇔ nax = t nbxnay = t nby or a →, nb → = 0 ⇔ ax nbx + ay nby = 0, where t is some real number. The coordinates of the directional or straight vectors are determined by the given equations of the straight lines. Let's take a look at some of the main examples.

1. The straight line a in a rectangular coordinate system is determined by the general equation of the straight 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. The parallelism condition is written as follows:

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

1. The straight line a is described by the equation of the straight line with the slope of the form y = k 1 x + b 1. Line b - y = 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 the parallelism condition is written as follows:

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

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

1. Straight lines a and b in a rectangular coordinate system are given by the canonical equations of a straight line on a plane: x - x 1 ax = y - y 1 ay and x - x 2 bx = y - y 2 by or by parametric equations of a straight line on a plane: x = x 1 + λ axy = y 1 + λ ay and x = x 2 + λ bxy = y 2 + λ by.

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

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

Let's look at some examples.

Example 1

Two straight lines are given: 2 x - 3 y + 1 = 0 and x 1 2 + y 5 = 1. It is necessary 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, since 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 for the parallelism of straight lines on the plane is not satisfied, which means that the given straight lines are not parallel.

Answer: the given lines are not parallel.

Example 2

The straight lines y = 2 x + 1 and x 1 = y - 4 2 are given. Are they parallel?

Solution

Convert the canonical equation of the line x 1 = y - 4 2 to the equation of the line with the 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, let's check if the given lines coincide. We use any point of the straight line y = 2 x + 1, for example, (0, 1), the coordinates of this point do not correspond to the equation of the straight line x 1 = y - 4 2, and therefore the lines do not coincide.

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

The normal vector of the straight line y = 2 x + 1 is the vector n a → = (2, - 1), and the directing vector of the second given straight 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 parallelism of the original straight lines. Those. the given straight lines are parallel.

Answer: data lines are parallel.

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

Theorem 8

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

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

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

Example 3

The straight 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 set the canonical equations of one straight line in space and the parametric equations of the other straight line in space. Direction vectors a → and b → the 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 →.

Consequently, the necessary and sufficient condition for the parallelism of lines in space is satisfied.

Answer: the parallelism of the given lines is proved.

If you notice an error in the text, please select it and press Ctrl + Enter

Lesson objectives: In this lesson, you will get acquainted with the concept of "parallel lines", you will learn how to make sure that straight lines are parallel, and also what properties the angles formed by parallel lines and a secant have.

Parallel lines

You know that the concept of "straight line" is one of the so-called undefined concepts of geometry.

You already know that two straight lines can coincide, that is, have all common points, can intersect, that is, have one common point. Straight underneath intersect different angles, while the angle between the straight lines is considered the smallest of the angles that are formed by them. A special case of intersection can be considered the case of perpendicularity, when the angle formed by the straight lines is 90 0.

But two straight lines may not have common points, that is, they do not intersect. Such straight lines are called parallel.

Work with an electronic educational resource « ».

To get acquainted with the concept of "parallel lines", work in the video tutorial materials

Thus, you now know the definition of parallel lines.

From the materials of the video tutorial fragment, you learned about different types angles that are formed when two straight lines intersect the third.

Pairs of angles 1 and 4; 3 and 2 are called inner one-sided corners(they lie between the straight lines a and b).

Pairs of corners 5 and 8; 7 and 6 call external one-sided corners(they lie outside the straight lines a and b).

Pairs of angles 1 and 8; 3 and 6; 5 and 4; 7 and 2 are called one-sided corners for straight lines a and b and secant c... As you can see, out of a pair of corresponding angles, one lies between the right a and b and the other is outside of them.

Signs of parallelism of straight lines

Obviously, using the definition, it is impossible to draw a conclusion about the parallelism of two straight lines. Therefore, in order to conclude that two lines are parallel, use signs.

You can already formulate one of them by reading the materials of the first part of the video lesson:

Theorem 1... Two straight lines, perpendicular to the third, do not intersect, that is, they are parallel.

You will get acquainted with other signs of parallelism of straight lines based on the equality of certain pairs of angles by working with the materials of the second part of the video lesson"Signs of parallelism of straight lines".

Thus, you should know three more signs of parallelism of straight lines.

Theorem 2 (the first criterion for parallelism of lines)... If at the intersection of two intersecting straight lines, the lying angles are equal, then the straight lines are parallel.

Rice. 2. Illustration for the first sign parallelism of straight lines

Once again, repeat the first sign of parallelism of straight lines by working with an electronic educational resource « ».

Thus, when proving the first criterion for the parallelism of straight lines, the criterion for the equality of triangles (along two sides and the angle between them) is used, as well as the criterion for the parallelism of straight lines as perpendicular to one straight line.

Exercise 1.

Write down the formulation of the first criterion for parallelism of straight lines and its proof in your notebooks.

Theorem 3 (second criterion for parallelism of lines)... If at the intersection of two straight secant the corresponding angles are equal, then the straight lines are parallel.

Once again, repeat the second sign of parallelism of straight lines by working with an electronic educational resource « ».

When proving the second criterion for parallelism of straight lines, the property of vertical angles and the first criterion for parallelism of straight lines are used.

Task 2.

Write down the formulation of the second criterion for parallelism of straight lines and its proof in your notebooks.

Theorem 4 (third criterion for parallelism of lines)... If, at the intersection of two straight secant lines, the sum of one-sided angles is 180 0, then the straight lines are parallel.

Once again, repeat the third sign of parallelism of straight lines by working with an electronic educational resource « ».

Thus, in the proof of the first criterion for parallelism of straight lines, we use the property of adjacent angles and the first criterion for parallelism of straight lines.

Task 3.

Write down the formulation of the third criterion for parallelism of straight lines and its proof in your notebooks.

In order to practice solving the simplest tasks, work with the materials of the electronic educational resource « ».

The signs of parallelism of straight lines are used when solving problems.

Now consider examples of solving problems for signs of parallelism of straight lines, having worked with the materials of the video lesson“Solving problems on the topic“ Signs of parallelism of straight lines ”.

Now test yourself by completing the tasks of the control electronic educational resource « ».

Anyone who wants to work with the solution of more complex problems can work with the materials of the video lesson "Problems on the signs of parallelism of straight lines."

Parallel Line Properties

Parallel lines have a number of properties.

You will find out what these properties are by working with the materials of the video tutorial. "Properties of Parallel Lines".

Thus, important fact which you should know is the axiom of parallelism.

Parallelism axiom... Through a point that does not lie on a given straight line, you can draw a straight line parallel to the given one, and, moreover, only one.

As you learned from the materials of the video lesson, based on this axiom, two consequences can be formulated.

Corollary 1. If a line intersects one of the parallel lines, then it also intersects the other parallel line.

Corollary 2. If two lines are parallel to the third, then they are parallel to each other.

Task 4.

Write down the wording of the formulated consequences and their proofs in your notebooks.

The properties of the angles formed by parallel lines and a secant are theorems opposite to the corresponding criteria.

So, from the materials of the video tutorial, you learned the property of criss-crossing corners.

Theorem 5 (theorem converse to the first criterion for parallelism of lines)... When two parallel intersecting straight lines intersect, the lying angles are equal.

Task 5.

Once again repeat the first property of parallel straight lines after working with the electronic educational resource « ».

Theorem 6 (theorem converse to the second criterion for parallelism of lines)... When two parallel straight lines intersect, the corresponding angles are equal.

Task 6.

Write down the statement of this theorem and its proof in your notebooks.

Once again, repeat the second property of parallel straight lines by working with an electronic educational resource « ».

Theorem 7 (theorem converse to the third criterion for parallelism of lines)... When two parallel straight lines intersect, the sum of the one-sided angles is 180 0.

Task 7.

Write down the statement of this theorem and its proof in your notebooks.

Repeat the third property of parallel straight lines once again by working with an electronic educational resource « ».

All properties of parallel lines are also used in solving problems.

Consider common problem solving examples from the video tutorial "Parallel lines and problems on the angles between them and the secant."

Parallel lines. Properties and features of parallel lines

1. Axiom of parallel. No more than one straight line parallel to the given one can be drawn through a given point.

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

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

4. If two parallel straight lines intersect a third, then the internal intersecting angles formed in this case are equal; the corresponding angles are equal; internal one-sided angles add up to 180 °.

5. If at the intersection of two straight lines the third is formed equal internal angles lying cross, then the straight lines are parallel.

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

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

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

Proportional line segment theorem... Parallel straight lines intersecting the sides of the corner cut proportional segments on them.

Triangle. Equality tests for 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 equal.

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

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

Equality tests for right-angled triangles

1. On two legs.

2. On the leg and hypotenuse.

3. By hypotenuse and acute angle.

4. Along the leg and sharp corner.

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

1. Amount inner corners triangle is 180 °.

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

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

4. Amount outer corners an ha-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 the inner one-sided angles with parallel straight lines and the secant are perpendicular.

Basic properties and signs of an isosceles triangle

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

2. If the 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 coincide.

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

Inequality of the triangle and its consequences

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

2. The sum of the links of the polyline is greater than the line segment connecting the origin

the first link with the end of the last one.

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

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

5. Hypotenuse right triangle more leg.

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

1) the perpendicular is shorter than the inclined;

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

The middle line of the triangle.

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

Centerline theorem of a triangle.

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

Median Triangle Theorems

1. The medians of the 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. Median of a right-angled triangle, drawn from the vertex right angle, is equal to half of the hypotenuse.

The property of the center perpendiculars to the sides of a triangle... The mid-perpendiculars to the sides of the triangle intersect at one point, which is the center of the circle described around the triangle.

The theorem on the heights of a triangle... Lines containing the heights of the triangle intersect at one point.

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

Property of the bisector 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 the two sides of one triangle are respectively proportional to the two sides of the other, and the angles 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 the other, then the triangles are similar.

The 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 referred to as the products of the sides enclosing these angles.

In a right triangle

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

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

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

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

The Pythagorean theorem and the converse theorem to the Pythagorean theorem

1. The square of the hypotenuse of a right-angled 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 rectangular.

Averages are proportional in a right-angled triangle.

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

Metric ratios in a triangle

1. Theorem of cosines. The square of the side of a triangle is equal to the sum of the squares of the other two sides without twice the product of these sides by 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. The theorem of sines. The sides of a triangle are proportional to the sines of the opposite angles.

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

Area formulas for a triangle

1. The area of ​​a triangle is equal to 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 by the sine of the angle between them.

3. The area of ​​a triangle is equal to the product of its semi-perimeter by the radius of the inscribed circle.

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

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

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

Parallelogram. A parallelogram is a quadrilateral, the opposite sides of which are pairwise parallel.

Properties and features of a parallelogram.

1. The diagonal splits 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 are halved by the intersection point.

5. If the opposite sides of the quadrilateral are pairwise equal, 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 the quadrilateral are halved 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, the area of ​​which is half the area of ​​the quadrilateral.

Rectangle. A rectangle is a right-angled parallelogram.

Properties and attributes of a rectangle.

1. The diagonals of the 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 called a quadrilateral, all sides of which are equal.

Properties and features of a rhombus.

1. The diagonals of the rhombus are perpendicular.

2. The diagonals of a rhombus divide its corners in half.

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.

Trapezium. A trapezoid is a quadrilateral in which only two opposite sides (bases) are parallel. The middle line of a trapezoid is the segment that connects the midpoints of the non-parallel sides (sides).

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

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

A wonderful property of the trapezoid... The point of intersection of the diagonals of the trapezoid, the point of intersection of the extensions of the lateral sides and the middle of the bases lie on one straight line.

Isosceles trapezoid... 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 the 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 equal to the half-sum of the bases.

Area formulas for a quadrilateral

1. The area of ​​the 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 by 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 equal to half the product of its diagonals.

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

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

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

S =, where a, b, c, d are the sides of this quadrilateral, p is the semiperimeter, and S is the area.

Similar figures

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

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 the radii of the inscribed and circumscribed circles. Then

Circle.

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

Basic properties of a circle

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

2. The diameter passing through the midpoint of a non-diameter chord is perpendicular to that chord.

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

4. Equal chords are equidistant from the center of the circle.

5. The chords of the circle at equal distances from the center are equal.

6. A circle is symmetrical about any of its diameters.

7. The arcs of a circle enclosed between parallel chords are equal.

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

9. The diameter is the largest chord of the circle.

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

1. The tangent line is perpendicular to the radius drawn to the tangent point.

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

3. If the lines passing through point M touch the circle at points A and B, then MA = MB and ﮮ AMO = ﮮ BMO, where 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.

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

1. The point of tangency of two circles lies on their center line.

2. Circles of radii r and R with centers O 1 and O 2 touch externally if and only if R + r = 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 ﮮ AK B = 90 ° and ﮮ О 1 СО 2 = 90 °.

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

Angles associated with a circle

1. The magnitude of the arc of a circle is equal to the magnitude of the central angle resting on it.

2. The inscribed angle is half angular magnitude the arc on which it rests.

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

4. The angle between the intersecting chords is equal to the half-sum of the 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 tangent point is equal to half the angular value of the arc cut out on the circle by this chord.

Properties of chords of a circle

1. The center line 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 = 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-angled triangle is the middle 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 the quadrangle is 180 °, then a circle can be described around it.

6. If a circle can be inscribed in the trapezoid, then the 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 tangency point divides the lateral side.

8. If a circle can be inscribed into a polygon, then its area is equal to the product of the polygon's half-perimeter by the radius of this circle.

The tangent and secant theorem and a consequence of it

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 equal to C = 2πR

The parallelism of two straight lines can be proved on the basis of the theorem, according to which, two drawn perpendiculars with respect to one straight line will be parallel. There are certain signs of parallelism of straight lines - there are three of them, and we will consider all of them more specifically.

The first sign of parallelism

Lines are parallel if at the intersection of their third line, the formed interior angles, lying in a cross, will be equal.

Suppose, at the intersection of straight lines AB and CD by a straight line EF, angles / 1 and / 2 were formed. They are equal, since the straight line EF runs at one slope in relation to the other two straight lines. At the intersection of the lines, we put the points Ki L - we got a segment of the secant EF. We find its middle and put the point O (Fig. 189).

On the line AB we drop the perpendicular from the point O. Let's call it OM. We continue the perpendicular until it intersects with the straight line CD. As a result, the original line AB is strictly perpendicular to МN, which means that СD_ | _МN, but this statement requires proof. As a result of drawing a perpendicular and a line of intersection, we have formed two triangles. One of them is MY, the second is NOK. Let's consider them in more detail. signs of parallelism of straight lines 7 grade

These triangles are equal, since, in accordance with the conditions of the theorem, / 1 = / 2, and in accordance with the construction of triangles, side ОK = side ОL. Angle MOL = / NOK, since these are vertical angles. It follows from this that the side and two angles adjacent to it of one of the triangles are respectively equal to the side and two angles adjacent to it of the other of the triangles. Thus, the triangle MOL = triangle NOK, and hence the angle LMO = angle KNO, but we know that / LMO is straight, which means that the corresponding angle KNO is also right. That is, we managed to prove that to the straight line МN, both the straight line AB and the straight line CD are perpendicular. That is, AB and CD are parallel to each other. This is what we needed to prove. Consider the rest of the criteria for parallelism of straight lines (grade 7), which differ from the first criterion in the way of proof.

The second sign of parallelism

According to the second criterion of parallelism of straight lines, we need to prove that the angles obtained in the process of intersection of parallel straight lines AB and CD of straight line EF will be equal. Thus, the signs of parallelism of two straight lines, both the first and the second, are based on the equality of the angles obtained when the third line intersects them. We assume that / 3 = / 2, and the angle 1 = / 3, since it is vertical to it. Thus, u / 2 will be equal to angle 1, however, it should be borne in mind that both angle 1 and angle 2 are internal, criss-crossing angles. Therefore, it remains for us to apply our knowledge, namely, that two segments will be parallel if, when they intersect the third straight line, the angles formed crosswise will be equal. Thus, we found out that AB || CD.

We managed to prove that, provided that two perpendiculars are parallel to one straight line, according to the corresponding theorem, the criterion for the parallelism of straight lines is obvious.

The third sign of parallelism

There is also a third sign of parallelism, which is proved by means of the sum of one-sided interior angles. Such a proof of the criterion of parallelism of straight lines allows us to conclude that two straight lines will be parallel if, when their third straight line intersects, the sum of the obtained one-sided interior angles equals 2d. See Figure 192.

CHAPTER III.
PARALLEL LINE

§ 35. SIGNS OF PARALLELITY OF TWO LINE.

The theorem that two perpendiculars to one straight line are parallel (§ 33) gives a criterion for the parallelism of two straight lines. You can withdraw more common features parallelism of two lines.

1. The first sign of parallelism.

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

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

Let us drop the perpendicular ОМ from the point O to the line AB and continue it to the intersection with the line CD, AB_ | _МN. Let us prove that СD_ | _МN.
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; ОK = ОL - by construction;
/ MOL = / NOK like vertical corners. Thus, a side and two adjacent angles of one triangle are respectively equal to a side and two adjacent angles of another triangle; hence, /\ MOL = /\ NOK, and hence
/ LMO = / KNO, but / LMO is straight, hence / KNO is also straightforward. Thus, lines AB and CD are perpendicular to the same line MN, therefore, they are parallel (§ 33), as required.

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

2. The second sign of parallelism.

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

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

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

This property is based on the construction of parallel lines using a ruler and a drawing triangle. This is done as follows.

Let us apply the triangle to the ruler as shown in drawing 191. We will move the triangle so that one side of it slides along the ruler, and on some other side of the triangle we draw several straight lines. These lines will be parallel.

3. The third sign of parallelism.

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

Let be / 1 and / 2 are internal one-sided corners and add up to 2 d.
But / 3 + / 2 = 2d as the corners are adjacent. Hence, / 1 + / 2 = / 3+ / 2.

From here / 1 = / 3, and these internal angles lie crosswise. Therefore, AB || CD.

If at the intersection of two straight lines the third, the sum of the inner one-sided angles is 2 d, then these two lines are parallel.

The exercise.

Prove that the lines are parallel:
a) if the outer cross lying angles are equal (Fig. 193);
b) if the sum of the outer one-sided corners is 2 d(Fig. 194).