The parallelism of two lines can be proven based on the theorem, according to which two perpendiculars drawn in relation to one line will be parallel. There are certain signs of parallelism of lines - there are three of them, and we will consider all of them more specifically.
The first sign of parallelism
Lines are parallel if, when they intersect a third line, the internal angles formed, lying crosswise, will be equal.
Let's say that when straight lines AB and CD intersect with straight line EF, angles /1 and /2 were formed. They are equal, since the straight line EF runs at one slope with respect to the other two straight lines. Where the lines intersect, we put points Ki L - we have a secant segment EF. We find its middle and put point O (Fig. 189).
We drop a perpendicular from point O onto line AB. Let's call it OM. We continue the perpendicular until it intersects the line CD. As a result, the original straight line AB is strictly perpendicular to MN, which means that CD_|_MN is also, but this statement requires proof. As a result of drawing a perpendicular and an intersection line, we formed two triangles. One of them is MINE, the second is NOK. Let's look at them in more detail. signs of parallel lines grade 7
These triangles are equal, since, in accordance with the conditions of the theorem, /1 =/2, and in accordance with the construction of triangles, side OK = side OL. 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 triangle. Thus, triangle MOL = triangle NOK, and therefore angle LMO = angle KNO, but we know that /LMO is straight, which means that the corresponding angle KNO is also right. That is, we were able to prove that to the straight line MN, 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. Let's consider the remaining signs of parallelism of lines (grade 7), which differ from the first sign in the method of proof.
Second sign of parallelism
According to the second criterion for the parallelism of lines, we need to prove that the angles obtained in the process of intersection of parallel lines AB and CD of line EF will be equal. Thus, the signs of parallelism of two lines, both the first and the second, are based on the equality of the angles obtained when the third line intersects them. Let's assume that /3 = /2 and angle 1 = /3 since it is vertical to it. Thus, and /2 will be equal to angle 1, however, it should be taken into account that both angle 1 and angle 2 are internal, cross-lying angles. Therefore, all we have to do is apply our knowledge, namely, that two segments will be parallel if, when they intersect the third straight line, the crosswise angles formed are equal. Thus, we found out that AB || CD.
We were able to prove that, provided that two perpendiculars to one line are parallel, according to the corresponding theorem, the sign that the lines are parallel is obvious.
The third sign of parallelism
There is also a third sign of parallelism, which is proved by the sum of one-sided interior angles. This proof of the sign of parallelism of lines allows us to conclude that two lines will be parallel if, when they intersect the third line, the sum of the resulting one-sided interior angles will be equal to 2d. See Figure 192.
First, let's look at the difference between the concepts of sign, property and axiom.
Definition 1
Sign They call a certain fact by which the truth of a judgment about an object of interest can be determined.
Example 1
Lines are parallel if their transversal forms equal crosswise angles.
Definition 2
Property is formulated in the case when there is confidence in the fairness of the judgment.
Example 2
When parallel lines are parallel, their transversal forms equal crosswise angles.
Definition 3
Axiom they call a statement that does not require proof and is accepted as truth without it.
Each science has axioms on which subsequent judgments and their proofs are based.
Axiom of parallel lines
Sometimes the axiom of parallel lines is accepted as one of the properties of parallel lines, but at the same time other geometric proofs are based on its validity.
Theorem 1
Through a point that does not lie on a given line, only one straight line can be drawn on the plane, which will be parallel to the given one.
The axiom does not require proof.
Properties of parallel lines
Theorem 2
Property1. The property of transitivity of parallel lines:
When one of two parallel lines is parallel to the third, then the second line will be parallel to it.
Properties require proof.
Proof:
Let there be two parallel lines $a$ and $b$. Line $c$ is parallel to line $a$. Let us check whether in this case the straight line $c$ will also be parallel to the straight line $b$.
To prove this, we will use the opposite proposition:
Let us imagine that it is possible that line $c$ is parallel to one of the lines, for example, line $a$, and intersects the other line, line $b$, at some point $K$.
We obtain a contradiction according to the axiom of parallel lines. This results in a situation in which two lines intersect at one point, moreover, parallel to the same line $a$. This situation is impossible; therefore, the lines $b$ and $c$ cannot intersect.
Thus, it has been proven that if one of two parallel lines is parallel to the third line, then the second line is parallel to the third line.
Theorem 3
Property 2.
If one of two parallel lines is intersected by a third, then the second line will also be intersected by it.
Proof:
Let there be two parallel lines $a$ and $b$. Also, let there be some line $c$ that intersects one of the parallel lines, for example, line $a$. It is necessary to show that line $c$ also intersects the second line, line $b$.
Let's construct a proof by contradiction.
Let's imagine that line $c$ does not intersect line $b$. Then two lines $a$ and $c$ pass through the point $K$, which do not intersect the line $b$, i.e., they are parallel to it. But this situation contradicts the axiom of parallel lines. This means that the assumption was incorrect and line $c$ will intersect line $b$.
The theorem has been proven.
Properties of corners, which form two parallel lines and a secant: opposite angles are equal, corresponding angles are equal, * the sum of one-sided angles is $180^(\circ)$.
Example 3
Given two parallel lines and a third line perpendicular to one of them. Prove that this line is perpendicular to another of the parallel lines.
Proof.
Let us have straight lines $a \parallel b$ and $c \perp a$.
Since line $c$ intersects line $a$, then, according to the property of parallel lines, it will also intersect line $b$.
The secant $c$, intersecting the parallel lines $a$ and $b$, forms equal internal angles with them.
Because $c \perp a$, then the angles will be $90^(\circ)$.
Therefore, $c \perp b$.
The proof is complete.
AB And WITHD crossed by the third straight line MN, then the angles formed in this case receive the following names in pairs:corresponding angles: 1 and 5, 4 and 8, 2 and 6, 3 and 7;
internal crosswise angles: 3 and 5, 4 and 6;
external crosswise angles: 1 and 7, 2 and 8;
internal one-sided corners: 3 and 6, 4 and 5;
external one-sided corners: 1 and 8, 2 and 7.
So, ∠ 2 = ∠ 4 and ∠ 8 = ∠ 6, but according to what has been proven, ∠ 4 = ∠ 6.
Therefore, ∠ 2 =∠ 8.
3. Corresponding angles 2 and 6 are the same, since ∠ 2 = ∠ 4, and ∠ 4 = ∠ 6. Let’s also make sure that the other corresponding angles are equal.
4. Sum internal one-sided corners 3 and 6 will be 2d because the sum adjacent corners 3 and 4 is equal to 2d = 180 0, and ∠ 4 can be replaced by the identical ∠ 6. We also make sure that sum of angles 4 and 5 is equal to 2d.
5. Sum external one-sided corners will be 2d because these angles are equal respectively internal one-sided corners like corners vertical.
From the above proven justification we obtain converse theorems.
When, at the intersection of two lines with an arbitrary third line, we obtain that:
1. Internal crosswise angles are the same;
or 2. External crosswise angles are identical;
or 3. Corresponding angles are equal;
or 4. The sum of internal one-sided angles is 2d = 180 0;
or 5. The sum of external one-sided ones is 2d = 180 0 ,
then the first two lines are parallel.
Signs of parallelism of two lines
Theorem 1. If, when two lines intersect with a secant:
crossed angles are equal, or
corresponding angles are equal, or
the sum of one-sided angles is 180°, then
lines are parallel(Fig. 1).
Proof. We limit ourselves to proving case 1.
Let the intersecting lines a and b be crosswise and the angles AB be equal. For example, ∠ 4 = ∠ 6. Let us prove that a || b.
Suppose that lines a and b are not parallel. Then they intersect at some point M and, therefore, one of the angles 4 or 6 will be the external angle of triangle ABM. For definiteness, let ∠ 4 be the external angle of the triangle ABM, and ∠ 6 the internal one. From the theorem on the external angle of a triangle it follows that ∠ 4 is greater than ∠ 6, and this contradicts the condition, which means that lines a and 6 cannot intersect, so they are parallel.
Corollary 1. Two different lines in a plane perpendicular to the same line are parallel(Fig. 2).
Comment. The way we just proved case 1 of Theorem 1 is called the method of proof by contradiction or reduction to absurdity. This method received its first name because at the beginning of the argument an assumption is made that is contrary (opposite) to what needs to be proven. It is called leading to absurdity due to the fact that, reasoning on the basis of the assumption made, we come to an absurd conclusion (to the absurd). Receiving such a conclusion forces us to reject the assumption made at the beginning and accept the one that needed to be proven.
Task 1. Construct a line passing through a given point M and parallel to a given line a, not passing through the point M.
Solution. We draw a straight line p through the point M perpendicular to the straight line a (Fig. 3).
Then we draw a line b through point M perpendicular to the line p. Line b is parallel to line a according to the corollary of Theorem 1.
An important conclusion follows from the problem considered:
through a point not lying on a given line, it is always possible to draw a line parallel to the given one.
The main property of parallel lines is as follows.
Axiom of parallel lines. Through a given point that does not lie on a given line, there passes only one line parallel to the given one.
Let us consider some properties of parallel lines that follow from this axiom.
1) If a line intersects one of two parallel lines, then it also intersects the other (Fig. 4).
2) If two different lines are parallel to a third line, then they are parallel (Fig. 5).
The following theorem is also true.
Theorem 2. If two parallel lines are intersected by a transversal, then:
crosswise angles are equal;
corresponding angles are equal;
the sum of one-sided angles is 180°.
Corollary 2. If a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other(see Fig. 2).
Comment. Theorem 2 is called the inverse of Theorem 1. The conclusion of Theorem 1 is the condition of Theorem 2. And the condition of Theorem 1 is the conclusion of Theorem 2. Not every theorem has an inverse, that is, if a given theorem is true, then the inverse theorem may be false.
Let us explain this using the example of the theorem on vertical angles. This theorem can be formulated as follows: if two angles are vertical, then they are equal. The converse theorem would be: if two angles are equal, then they are vertical. And this, of course, is not true. Two equal angles do not have to be vertical.
Example 1. Two parallel lines are crossed by a third. It is known that the difference between two internal one-sided angles is 30°. Find these angles.
Solution. Let Figure 6 meet the condition.
The video lesson “Signs for the parallelism of two lines” contains proof of theorems that describe the signs that indicate the parallelism of lines. At the same time, the video describes 1) the theorem on the parallelism of lines in which equal angles are created by a transversal, 2) a sign that means the parallelism of two straight lines - at equal formed corresponding angles, 3) a sign that means the parallelism of two lines in the case when, when they intersect with a secant one-sided angles add up to 180°. The purpose of this video lesson is to familiarize students with the signs that mean the parallelism of two lines, knowledge of which is necessary for solving many practical problems, to clearly present the proof of these theorems, and to develop skills in proving geometric statements.
The advantages of a video lesson are related to the fact that with the help of animation, voice accompaniment, and the ability to highlight with color, it provides a high degree of clarity and can serve as a replacement for the presentation of a standard block of new educational material by the teacher.
The video lesson begins with the title displayed on the screen. Before describing the signs of parallel lines, students are introduced to the concept of a secant. A secant is defined as a line that intersects other lines. The screen shows two straight lines a and b, which intersect with straight line c. The constructed line c is highlighted in blue, emphasizing the fact that it is a secant of the given lines a and b. In order to consider the signs of parallelism of lines, it is necessary to become more familiar with the area of intersection of lines. The secant at the points of intersection with the lines forms 8 angles ∠1, ∠2, ∠3, ∠4, ∠5, ∠6, ∠7, ∠8, by analyzing the relationships of which it is possible to derive signs of the parallelism of these lines. It is noted that angles ∠3 and ∠5, as well as ∠2 and ∠4 are called crosswise. A detailed explanation is given using animation of the arrangement of crosswise angles as angles that lie between parallel straight lines and adjoin straight lines, lying crosswise. Then the concept of one-sided angles is introduced, which include the pairs ∠4 and ∠5, as well as ∠3 and ∠6. Pairs of corresponding angles are also indicated, of which there are 4 pairs in the constructed image - ∠1-∠5, ∠4-∠8, ∠2-∠6, ∠3-∠7.
The next part of the video lesson examines three signs of parallelism of any two lines. The first description appears on the screen. The theorem states that if the crosswise angles formed by the transversal are equal, the given lines will be parallel. The statement is accompanied by a drawing showing two straight lines a and b and a secant AB. It is noted that the lying angles ∠1 and ∠2 formed crosswise are equal to each other. This statement requires proof.
The most easily proven special case is when the given crosswise angles are right angles. This means that the secant is perpendicular to the lines, and according to the theorem already proven, in this case the lines a and b will not intersect, that is, they are parallel. The proof for this particular case is described using the example of an image constructed next to the first figure, highlighting important details of the proof using animation.
To prove this in the general case, it is necessary to draw an additional perpendicular from the middle of the segment AB to straight line a. Next, segment BH 1 equal to segment AN is laid out on straight line b. From the resulting point H 1, a segment is drawn connecting points O and H 1. Next, we consider two triangles ΔОНА and ΔОВН 1, the equality of which is proved by the first sign of equality of two triangles. The sides OA and OB are equal in construction, since point O was marked as the middle of the segment AB. The sides HA and H 1 B are also equal in construction, since we laid off the segment H 1 B, equal to HA. And the angles are ∠1=∠2 according to the conditions of the problem. Since the formed triangles are equal to each other, the corresponding remaining pairs of angles and sides are also equal to each other. It follows from this that the segment OH 1 is a continuation of the segment OH, constituting one segment HH 1. It is noted that since the constructed segment OH is perpendicular to straight line a, then, accordingly, segment HH 1 is perpendicular to straight lines a and b. This fact means, using the theorem on the parallelism of lines to which one perpendicular is constructed, that the given lines a and b are parallel.
The next theorem that requires proof is a sign of the equality of parallel lines by the equality of the corresponding angles formed when intersecting a transversal. The statement of this theorem is displayed on the screen and can be proposed by students for recording. The proof begins with the construction on the screen of two parallel lines a and b, to which the secant c is constructed. Highlighted in blue in the picture. The secant forms the corresponding angles ∠1 and ∠2, which by condition are equal to each other. The adjacent angles ∠3 and ∠4 are also marked. ∠2 in relation to angle ∠3 is a vertical angle. And vertical angles are always equal. In addition, angles ∠1 and ∠3 are crosswise lying between each other - their equality (according to the already proven statement) means that lines a and b are parallel. The theorem has been proven.
The last part of the video lesson is devoted to proving the statement that if the sum of one-sided angles that are formed when two lines intersect with a transversal line equals 180°, in this case these lines will be parallel to each other. The proof is demonstrated using a figure showing lines a and b intersecting a secant c. The angles formed by the intersection are marked similarly to the previous proof. By condition, the sum of angles ∠1 and ∠4 is equal to 180°. Moreover, it is known that the sum of angles ∠3 and ∠4 is equal to 180°, since they are adjacent. This means that angles ∠1 and ∠3 are equal to each other. This conclusion gives the right to assert that lines a and b are parallel. The theorem has been proven.
The video lesson “Signs of parallelism of two lines” can be used by the teacher as an independent block demonstrating the proofs of these theorems, replacing the teacher’s explanation or accompanying it. A detailed explanation makes it possible for students to use the material for independent study and will help in explaining the material during distance learning.
