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1 definition of parallel lines, signs of parallel lines. Properties of parallel lines

Parallel lines. Properties and signs of parallel lines

1. Axiom of parallels. Through this point You can draw at most one straight line parallel to a 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 intersect with a third, then the internal crosswise angles formed are equal; corresponding angles are equal; internal one-sided angles add up to 180°.

5. If, when two straight lines intersect a third, equal internal crosswise angles are formed, then the straight lines are parallel.

6. If, when two straight lines intersect a third, equal corresponding angles are formed, then the straight lines are parallel.

7. If, when two straight lines intersect a third, the sum of the internal one-sided angles is equal to 180°, then the straight lines are parallel.

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

Proportional segment theorem. Parallel lines intersecting the sides of an angle cut out 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 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 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 sides.

2. Along the leg and hypotenuse.

3. By hypotenuse and acute angle.

4. Along the leg and acute angle.

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. External corner of the triangle equal to the sum two internal angles not adjacent to it.

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

4. The sum of the external angles of a he-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. Bisectors of internal one-sided angles with parallel lines and a transversal are perpendicular.

Basic properties and features 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. B isosceles triangle the median, bisector and height drawn to the base are the same.

4. If any pair of segments from the triple coincides in a triangle - median, bisector, altitude, 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 polyline 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. Opposite the larger side of the triangle lies the larger angle.

5. Hypotenuse right triangle more leg.

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

1) the perpendicular is shorter than the inclined ones;

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

The middle line of the triangle.

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

Triangle Midline Theorem.

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

Theorems on medians of a triangle

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

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

3. Median of a right triangle drawn from a vertex right angle, is equal to half 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.

Triangle bisector property. 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 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 product of the sides enclosing these angles.

In a right triangle

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

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

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

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

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

Pythagorean theorem and theorem, converse of the theorem Pythagoras

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 two other sides, then the triangle is right-angled.

Proportional means in a right triangle.

The height of a right triangle drawn from the vertex of a 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 twice the product of these sides by the cosine of the angle between them.

2. Corollary to 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. Theorem of sines. The sides of a triangle are proportional to the sines of the opposite angles.

5. Generalized theorem of sines. 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.

Triangle area formulas

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 and the sine of the angle between them.

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

4. The area of a triangle is equal to the product of its three sides divided by quadruple the radius of the circumcircle.

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

Elements 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 parallel in pairs.

Properties and signs of a parallelogram.

1. A diagonal divides a 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 a parallelogram intersect and are bisected by the intersection point.

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 point of intersection, 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 equal to half the area of the quadrilateral.

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

Properties and characteristics 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 whose sides are all equal.

Rhombus. A rhombus is a quadrilateral whose sides are all equal.

Properties and signs of a rhombus.

1. The diagonals of a rhombus are perpendicular.

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

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

4. If the diagonals of a parallelogram bisect its angles, then this parallelogram is a rhombus.

Trapezoid. A trapezoid is a quadrilateral whose only two opposite sides (bases) are parallel. The midline of a trapezoid is a segment connecting the midpoints of non-parallel sides (sides).

1. The midline 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 half the difference of the bases.

A remarkable property of a trapezoid. The point of intersection of the diagonals of a trapezoid, the point of intersection of the extensions of the sides and the middle of the bases lie on the same 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 a 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 half the 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 equal to 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 = , 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 geometric locus of points in the plane that are distant from a given point, called the center of the circle, by the same positive distance.

Basic properties of a circle

1. A diameter perpendicular to the chord divides the chord and the arcs subtended by it in half.

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

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

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

5. Chords of a circle that are equal distances from the center are equal.

6. A circle is symmetrical relative to any of its diameters.

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

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

9. Diameter is the largest chord of a circle.

Tangent to a circle. A straight line having a unique relationship with a circle common point, is called a tangent to the circle.

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

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

3. If straight lines passing through point M touch the circle at points A and B, then MA = MB and ﮮAMO = ﮮBMO, 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.

Tangent circles. Two circles are said to touch if they have a single common point (point of contact).

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 = 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. A certain straight line touches these circles at various points A and B and intersects the common tangent passing through point K at point C. Then ﮮAK B = 90° and ﮮO 1 CO 2 = 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 size of the arc of a circle is equal to the size of the central angle resting on it.

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

3. Inscribed angles subtending the same arc are equal.

4. The angle between intersecting chords is equal to half the 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 point of contact is equal to half the angular value of the arc cut out 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 segments of chords AB and CD of a circle intersecting at point E are equal, that is, AE EB = CE ED.

Inscribed and circumscribed circles

1. Centers of inscribed and circumscribed circles regular triangle match up.

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

3. If a circle can be inscribed in a quadrilateral, then its sums 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 drawn around it.

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

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 point of contact divides the side.

8. If a circle can be inscribed in a polygon, then its area is equal to the product of the semi-perimeter 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 to a circle from one point, then the product of the entire secant and its outer part is equal to the square of the tangent.

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

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

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. Transitivity property 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 lying crosswise with them.

Because $c \perp a$, then the angles will be $90^(\circ)$.

Therefore, $c \perp b$.

The proof is complete.

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Question 1. Prove that two lines parallel to a third are parallel.
Answer. Theorem 4.1. Two lines parallel to a third are parallel.
Proof. Let lines a and b be parallel to line c. Let us assume that a and b are not parallel (Fig. 69). Then they do not intersect at some point C. This means that two lines pass through point C parallel to line c. But this is impossible, since through a point not lying on a given line, it is possible to draw at most one straight line parallel to the given one. The theorem has been proven.

Question 2. Explain which angles are called one-sided interior angles. What angles are called internal cross-lying ones?
Answer. Pairs of angles that are formed when lines AB and CD intersect with secant AC have special names.
If points B and D lie in the same half-plane relative to straight line AC, then angles BAC and DCA are called one-sided internal angles (Fig. 71, a).
If points B and D lie in different half-planes relative to straight line AC, then angles BAC and DCA are called internal cross-lying angles (Fig. 71, b).

Rice. 71

Question 3. Prove that if the interior angles of one pair are equal, then the interior angles of the other pair are also equal, and the sum of the interior angles of each pair is 180°.
Answer. The secant AC forms with the straight lines AB and CD two pairs of internal one-sided angles and two pairs of internal cross-lying angles. The internal crosswise angles of one pair, for example angle 1 and corner 2, are adjacent to the internal crosswise angles of another pair: angle 3 and angle 4 (Fig. 72).

Rice. 72

Therefore, if the interior angles of one pair are congruent, then the interior angles of the other pair are also equal.
A pair of internal cross-lying angles, for example angle 1 and angle 2, and a pair of internal one-sided angles, for example angle 2 and angle 3, have one angle in common - angle 2, and two other angles are adjacent: angle 1 and angle 3.
Therefore, if internal crosswise angles are equal, then the sum of the internal angles is 180°. And vice versa: if the sum of internal intersecting angles is equal to 180°, then intersecting internal angles are equal. Q.E.D.

Question 4. Prove a test for parallel lines.
Answer. Theorem 4.2 (test for parallel lines). If internal crosswise angles are equal or the sum of internal one-sided angles is equal to 180°, then the lines are parallel.
Proof. Let the straight lines a and b form equal internal crosswise angles with the secant AB (Fig. 73, a). Let's say that lines a and b are not parallel, which means they intersect at some point C (Fig. 73, b).

Rice. 73

The secant AB divides the plane into two half-planes. Point C lies in one of them. Let's construct a triangle BAC 1, equal to triangle ABC, with vertex C 1 in another half-plane. By condition, internal crosswise angles for parallel a, b and the secant AB are equal. Since the corresponding angles of triangles ABC and BAC 1 with vertices A and B are equal, they coincide with the internal angles lying crosswise. This means that line AC 1 coincides with line a, and line BC 1 coincides with line b. It turns out that two different straight lines a and b pass through points C and C 1. And this is impossible. This means that lines a and b are parallel.
If the lines a and b and the transversal AB have the sum of the internal one-sided angles equal to 180°, then, as we know, the internal angles lying crosswise are equal. This means, according to what was proven above, lines a and b are parallel. The theorem has been proven.

Question 5. Explain which angles are called corresponding angles. Prove that if internal crosswise angles are equal, then the corresponding angles are also equal, and vice versa.

Answer. If for a pair of internal crosswise angles one angle is replaced by a vertical one, then we get a pair of angles that are called the corresponding angles of these lines with a transversal. Which is what needed to be explained.
From the equality of interior angles lying crosswise follows the equality of the corresponding angles, and vice versa. Let's say we have two parallel lines (since by condition, internal angles lying across each other are equal) and a transversal, which form angles 1, 2, 3. Angles 1 and 2 are equal as internal angles lying across each other. And angles 2 and 3 are equal as vertical. We get: $\angle$1 = $\angle$2 and $\angle$2 = $\angle$3. By the property of transitivity of the equal sign it follows that $\angle$1 = $\angle$3. The converse statement can be proven in a similar way.
From this we get the sign that straight lines are parallel at the corresponding angles. Namely: straight lines are parallel if the corresponding angles are equal. Q.E.D.

Question 6. Prove that through a point not lying on a given line you can draw a line parallel to it. How many lines parallel to a given line can be drawn through a point not lying on this line?

Answer. Problem (8). Given a line AB and a point C that does not lie on this line. Prove that through point C you can draw a line parallel to line AB.
Solution. Line AC divides the plane into two half-planes (Fig. 75). Point B lies in one of them. Let us add angle ACD from the half-line CA to another half-plane, equal to angle CAB. Then the lines AB and CD will be parallel. In fact, for these lines and the secant AC, the interior angles BAC and DCA lie crosswise. And since they are equal, the lines AB and CD are parallel. Q.E.D.
Comparing the statement of problem 8 and axiom IX (the main property of parallel lines), we come to an important conclusion: through a point not lying on a given line, it is possible to draw a line parallel to it, and only one.

Question 7. Prove that if two lines are intersected by a third line, then the intersecting interior angles are equal, and the sum of the interior one-sided angles is 180°.

Answer. Theorem 4.3(the converse of Theorem 4.2). If two parallel lines intersect with a third line, then the intersecting internal angles are equal, and the sum of the internal one-sided angles is 180°.
Proof. Let a and b be parallel lines and c be a line intersecting them at points A and B. Let us draw a line a 1 through point A so that the internal crosswise angles formed by the transversal c with the lines a 1 and b are equal (Fig. 76).
According to the principle of parallelism of lines, lines a 1 and b are parallel. And since only one line passes through point A, parallel to line b, then line a coincides with line a 1.
This means that internal crosswise angles formed by a transversal with
parallel lines a and b are equal. The theorem has been proven.

Question 8. Prove that two lines perpendicular to a third are parallel. If a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.
Answer. From Theorem 4.2 it follows that two lines perpendicular to a third are parallel.
Suppose that any two lines are perpendicular to a third line. This means that these lines intersect with the third line at an angle equal to 90°.
From the property of angles formed when parallel lines intersect with a transversal, it follows that if a line is perpendicular to one of the parallel lines, then it is also perpendicular to the other.

Question 9. Prove that the sum of the angles of a triangle is 180°.

Answer. Theorem 4.4. The sum of the angles of a triangle is 180°.
Proof. Let ABC be the given triangle. Let us draw a line through vertex B parallel to line AC. Let's mark point D on it so that points A and D lie on opposite sides of straight line BC (Fig. 78).
Angles DBC and ACB are congruent as internal cross-lying ones formed by the transversal BC with parallel lines AC and BD. Therefore, the sum of the angles of a triangle at vertices B and C is equal to angle ABD.
And the sum of all three angles of a triangle is equal to the sum of angles ABD and BAC. Since these are one-sided interior angles for parallel AC and BD and secant AB, their sum is 180°. The theorem has been proven.

Question 10. Prove that any triangle has at least two acute angles.
Answer. Indeed, let us assume that the triangle has only one acute angle or no acute angles at all. Then this triangle has two angles, each of which is at least 90°. The sum of these two angles is no less than 180°. But this is impossible, since the sum of all the angles of a triangle is 180°. Q.E.D.

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. Consequently, 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 of parallel lines 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.

CHAPTER III.
PARALLEL DIRECT

§ 35. SIGNS OF PARALLEL TWO 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 the parallelism of two lines.

1. The first sign of parallelism.

If, when two straight lines intersect a third, the internal angles lying crosswise are equal, then these lines are parallel.

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

Let us lower the perpendicular OM from point O onto the straight line AB and continue it until it intersects with the straight 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 according to the conditions of the theorem; ОK = ОL - by construction;
/ MOL = / NOK, like vertical angles. Thus, the side and two adjacent angles of one triangle are respectively equal to the side and two adjacent angles of another triangle; hence, /\ MOL = /\ NOK, and hence
/ LMO = / KNO, but / LMO is direct, which means / KNO is also straight. Thus, the lines AB and CD are perpendicular to the same line MN, therefore, they are parallel (§ 33), which was what needed to be proven.

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 whether straight lines AB and CD are parallel if, when they intersect the third straight line EF, the corresponding angles are equal.

Let some corresponding angles be equal, for example / 3 = / 2 (drawing 190);
/ 3 = / 1, as the angles are vertical; Means, / 2 will be equal / 1. But angles 2 and 1 are intersecting interior angles, and we already know that if when two straight lines intersect the third, the intersecting interior angles are equal, then these lines are parallel. Therefore AB || CD.

If, when two lines intersect a third, the corresponding angles are equal, then these two lines are parallel.

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

Let's 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 some other side of the triangle. These lines will be parallel.

3. The third sign of parallelism.

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

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

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

If, when two straight lines intersect a third, the sum of the internal one-sided angles is equal to 2 d, then these two lines are parallel.

Exercise.