Angle between parallel lines in space. Angle between skew lines (2019). Mutual arrangement of two straight lines

AB And FROMD crossed by the third 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 cross-lying corners: 3 and 5, 4 and 6;

external cross-lying corners: 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 by the proven ∠ 4 = ∠ 6.

Therefore, ∠ 2 = ∠ 8.

3. Respective angles 2 and 6 are the same, since ∠ 2 = ∠ 4, and ∠ 4 = ∠ 6. We 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. 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 justification proved above, we obtain inverse theorems.

When, at the intersection of two lines of an arbitrary third line, we obtain that:

1. Internal cross lying angles are the same;

or 2. External cross lying angles are the same;

or 3. The corresponding angles are the same;

or 4. The sum of internal one-sided angles is equal to 2d = 180 0 ;

or 5. The sum of the outer one-sided is 2d = 180 0 ,

then the first two lines are parallel.

The two lines AB and CD are called parallel , if they lie in the same plane and do not intersect, no matter how long they are continued (AB|| CD). The angle between parallel lines is zero.

The length of a segment of a perpendicular enclosed between two parallel lines is distance between them.

Axiom: Through a point not on a given line, only one line can be drawn parallel to the given line.

Properties of parallel lines:

1. If two lines are parallel to a third line, then they are parallel to each other.

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

When crossing two parallel lines to a third line, eight corners are formed (Fig. 13), which are called in pairs:

1) corresponding angles (1 And 5; 2 And 6; 3 And 7; 4 And 8 );

corners pairwise equal: (https://pandia.ru/text/78/187/images/image003_66.gif" width="11" height="10 src="> 5; https://pandia.ru/text/78/187/images/image003_66.gif" width="11" height="10"> 6; https://pandia.ru/text/78/187/images/image003_66.gif" width="11" height="10"> 7; https://pandia.ru/text/78/187/images/image003_66.gif" width="11" height="10"> 8 );

2) domestic criss-cross corners (4 And 5; 3 And 6 ); they pairwise equal;

3) external cross-lying corners(1 And 8; 2 And 7 ); they are pairwise equal;

4) domestic one-sided corners (3 And 5; 4 And 6 ); the sum of one-sided angles is 180°

(https://pandia.ru/text/78/187/images/image003_66.gif" width="11" height="10"> 5 = 180°; 4 + 6 = 180°);

5) external one-sided corners (1 And 7; 2 And 8 ); their sum is 180° (https://pandia.ru/text/78/187/images/image003_66.gif" width="11" height="10"> 7 = 180°; 2 + 8 = 180°).

Thales' theorem. When the sides of an angle are intersected by parallel lines(fig.16) The sides of an angle are divided into proportional segments:

Similar triangles.

The two triangles are called similar, if their angles are respectively equal and the sides of one triangle are proportional to the similar sides of the other. Related the sides of similar triangles are the sides that lie opposite equal angles.

https://pandia.ru/text/78/187/images/image006_51.gif" alt="(!LANG: similar triangles" width="13" height="14">A = !} https://pandia.ru/text/78/187/images/image006_51.gif" alt="(!LANG: similar triangles" width="13" height="14">B = B1, С = С1 !} And Number k, equal to the ratio of similar sides of the triangle is called similarity coefficient.

Signs of similarity:

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

2. If two sides one triangle proportional to two sides of the other triangle and corners, concluded between these parties, equal, then the triangles are similar.

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

Consequences: 1. The areas of similar triangles are related as the square of the similarity coefficient:

2. Attitude perimeters similar triangles and bisector, medians, heights and perpendicular bisectors is equal to the similarity coefficient.

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In this lesson, we will define codirectional rays and prove the theorem on the equality of angles with codirectional sides. Next, we give the definition of the angle between intersecting lines and skew lines. Consider what the angle between two straight lines can be. At the end of the lesson, we will solve several problems for finding the angles between skew lines.

Topic: Parallelism of lines and planes

Lesson: Angles with codirectional sides. Angle between two lines

Any line, for example OO 1(Fig. 1.), cuts the plane into two half-planes. If the rays OA And O 1 A 1 are parallel and lie in the same half-plane, they are called co-directional.

Rays O 2 A 2 And OA are not co-directional (Fig. 1.). They are parallel, but do not lie in the same half-plane.

If the sides of two angles are codirectional, then such angles are equal.

Proof

Let us be given parallel rays OA And O 1 A 1 and parallel beams OV And About 1 in 1(Fig. 2.). That is, we have two corners AOB And A 1 O 1 B 1 whose sides lie on codirectional rays. We prove that these angles are equal.

On the side of the beam OA And O 1 A 1 select points BUT And A 1 so that the segments OA And O 1 A 1 were equal. Likewise, dots IN And IN 1 choose so that the segments OV And About 1 in 1 were equal.

Consider a quadrilateral A 1 O 1 OA(Fig. 3.) OA And O 1 A 1 A 1 O 1 OA A 1 O 1 OA OO 1 And AA 1 parallel and equal.

Consider a quadrilateral B 1 O 1 OB. In this quadrilateral side OV And About 1 in 1 parallel and equal. On the basis of a parallelogram, a quadrilateral B 1 O 1 OB is a parallelogram. Because B 1 O 1 OB- parallelogram, then sides OO 1 And BB 1 parallel and equal.

And straight AA 1 parallel to a straight line OO 1, and a straight line BB 1 parallel to a straight line OO 1, means straight AA 1 And BB 1 are parallel.

Consider a quadrilateral B 1 A 1 AB. In this quadrilateral side AA 1 And BB 1 parallel and equal. On the basis of a parallelogram, a quadrilateral B 1 A 1 AB is a parallelogram. Because B 1 A 1 AB- parallelogram, then sides AB And A 1 B 1 parallel and equal.

Consider triangles AOB And A 1 O 1 B 1. Parties OA And O 1 A 1 are equal in construction. Parties OV And About 1 in 1 are also equal in construction. And as we proved, the parties AB And A 1 B 1 are also equal. So the triangles AOB And A 1 O 1 B 1 equal on three sides. Equal triangles have equal angles opposite equal sides. So the corners AOB And A 1 O 1 B 1 are equal, which was to be proved.

1) Intersecting lines.

If the lines intersect, then we have four different angles. Angle between two lines, is the smallest of the angles between two lines. Angle between intersecting lines but And b denote α (Fig. 4.). The angle α is such that .

Rice. 4. Angle between two intersecting lines

2) Intersecting lines

Let straight but And b crossing. Pick an arbitrary point ABOUT. Through the dot ABOUT let's draw a straight line a 1, parallel to the line but, and direct b 1, parallel to the line b(Fig. 5.). Direct a 1 And b 1 intersect at a point ABOUT. Angle between two intersecting lines a 1 And b 1, the angle φ, and is called the angle between skew lines.

Rice. 5. Angle between two intersecting lines

Does the value of the angle depend on the chosen point O? Pick a point About 1. Through the dot About 1 let's draw a straight line a 2, parallel to the line but, and direct b 2, parallel to the line b(Fig. 6.). Angle between intersecting lines a 2 And b 2 denote φ 1. Then the angles φ And φ 1 - corners with concurrent sides. As we have shown, such angles are equal to each other. This means that the angle between the skew lines does not depend on the choice of the point ABOUT.

Direct OV And CD parallel, OA And CD interbreed. Find the angle between the lines OA And CD, if:

1) ∠AOB= 40°.

Pick a point FROM. Go straight through it CD. Let's spend SA 1 parallel OA(Fig. 7.). Then the angle A 1 CD- angle between intersecting lines OA And CD. By the angle theorem with codirectional sides, the angle A 1 CD equal to the angle AOB, i.e. 40°.

Rice. 7. Find the angle between two lines

2) ∠AOB= 135°.

Let's make the same construction (Fig. 8.). Then the angle between the skew lines OA And CD is equal to 45 °, since it is the smallest of the angles that are obtained by crossing lines CD And SA 1.

3) ∠AOB= 90°.

Let's make the same construction (Fig. 9.). Then all the angles that are obtained when the lines intersect CD And SA 1 are equal to 90°. The required angle is 90°.

1) Prove that the midpoints of the sides of a spatial quadrilateral are the vertices of a parallelogram.

Proof

Let us be given a spatial quadrilateral ABCD. M,N,K,L- the middle of the ribs BD,AD,AC,BC respectively (Fig. 10.). We need to prove that MNKL- parallelogram.

Consider a triangle ABD. MN MN parallel AB and equals half of it.

Consider a triangle ABC. LC- middle line. According to the property of the midline, LC parallel AB and equals half of it.

AND MN, And LC are parallel AB. Means, MN parallel LC by the three parallel lines theorem.

We get that in a quadrilateral MNKL- sides MN And LC are parallel and equal because MN And LC equal to half AB. So, by the criterion of a parallelogram, a quadrilateral MNKL is a parallelogram, as required.

2) Find the angle between the lines AB And CD if the angle MNK= 135°.

As we have already proven, MN parallel to a straight line AB. NK- middle line of the triangle ACD, by property, NK parallel DC. So through the point N go through two straight lines MN And NK, which are parallel to the skew lines AB And DC respectively. So the angle between the lines MN And NK is the angle between the skew lines AB And DC. We are given an obtuse angle MNK= 135°. Angle between lines MN And NK- the smallest of the angles obtained at the intersection of these lines, that is, 45 °.

So, we have considered angles with codirectional sides and proved their equality. We considered the angles between intersecting and crossing lines and solved several problems on finding the angle between two lines. In the next lesson, we will continue to solve problems and repeat the theory.

1. Geometry. Grade 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and supplemented - M .: Mnemosyne, 2008. - 288 p. : ill.

2. Geometry. Grade 10-11: A textbook for general educational institutions / Sharygin I. F. - M .: Bustard, 1999. - 208 p.: ill.

3. Geometry. Grade 10: Textbook for general educational institutions with in-depth and profile study of mathematics / E. V. Potoskuev, L. I. Zvalich. - 6th edition, stereotype. - M. : Bustard, 008. - 233 p. :ill.

IN) BC And D 1 IN 1.

Rice. 11. Find the angle between lines

4. Geometry. Grade 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and supplemented - M.: Mnemozina, 2008. - 288 p.: ill.

Tasks 13, 14, 15 p. 54