The angle between parallel lines in space. Angle between crossed lines (2019). The relative position of two straight lines

AB and WITHD 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 criss-crossing corners: 3 and 5, 4 and 6;

external criss-crossing corners: 1 and 7, 2 and 8;

inner 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 what has been proved ∠ 4 = ∠ 6.

Therefore, ∠ 2 = ∠ 8.

3. Corresponding 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 inner 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 inner one-sided corners like corners vertical.

From the above justification, we obtain converse theorems.

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

1. The inner corners lying on the cross are the same;

or 2. The outer corners are the same;

or 3. The corresponding angles are the same;

or 4. The sum of the inner one-sided angles is 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 straight lines AB and CD are called parallel if they lie in the same plane and do not intersect, no matter how many of them continue (AB || CD). The angle between parallel lines is zero.

The length of the segment of the perpendicular, enclosed between two parallel straight lines, - distance between them.

Axiom: through a point not lying on a given straight line, only one straight line parallel to this straight line can be drawn.

Parallel Lines Properties:

1. If two lines are parallel to the 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 of the 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) internal criss-cross corners (4 and 5; 3 and 6 ); they pairwise equal;

3) external criss-crossing corners(1 and 8; 2 and 7 ); they are pairwise equal;

4) internal one-sided corners (3 and 5; 4 and 6 ); the sum of one-sided corners 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 the corner intersect with parallel straight lines(fig. 16) the sides of the corner are divided into proportional segments:

Similar triangles.

The two triangles are called like if their angles are respectively equal and the sides of one triangle are proportional to the similar sides of the other. Similar the sides of such 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 the similar sides of the triangle is called similarity coefficient.

Signs of similarity:

1. If two corners of one triangle respectively equal to two angles the other, then the tracks are similar.

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

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

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

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

Entrepreneurship as a self-organizing system exists and develops under the influence of a system of factors. In the late 70s. XX century such researchers as T. Bachkai, D. Mesena, D. Miko and others, studying the effect of risk factors, indicated that they are all interrelated. Along with the "natural" ...

RISK VALUE ASSESSMENT AND SOLUTION CRITERIA

Risk management is impossible without assessing its magnitude. The assessment method depends on the type of risk. Given the variety of risks and the complexity of their management tasks, in practice, three types of assessments are used: qualitative, axiological and quantitative. Qualitative risk assessment is widely used and allows you to quickly, ...
(Risks in accounting)

Intersecting straight lines

If straight lines intersect, then their projections of the same name intersect at a point that is the projection of the point of intersection of these straight lines. Indeed (Figure 2.30), if the point TO belongs to both direct AB and CD, then the projection of this point must be the intersection point ...
(Engineering graphics)

Crossed straight lines

Crossed straight lines do not intersect or parallel to each other. Figure 2.32 shows two intersecting straight lines in general position: although the projections of the same name intersect each other, their intersection points cannot be connected by a link line parallel to the link lines L "L" and...
(Engineering graphics)

DISTANCE BETWEEN CROSSING LINE

Distance between crossed lines a and B determined by the length of the perpendicular segment KM, intersecting both lines (a _1_ KM; Y.KM) (fig. 349, b, c). The problem is solved simply if one of the lines is projecting. Let, for example, α ± π, then the required segment KM...
(Descriptive geometry)

The relative position of a straight line and a plane, two planes

Signs of the relative position of a straight line and a plane, two planes Let us recall the signs of the mutual position of a straight line and a plane, as well as two planes, familiar from stereometry. 1. If a straight line and a plane have one common point, then the straight line and the plane intersect (Fig. 3.6a). 2. If the line and the plane ...
(Fundamentals of Engineering Graphics)

Signs of the relative position of a straight line and a plane, two planes

Let us recall the signs of the mutual position of a straight line and a plane, as well as two planes, familiar from stereometry. 1. If a straight line and a plane have one common point, then the straight line and the plane intersect (Fig. 3.6a). 2. If the straight line and the plane have two common points, then the straight line lies in the plane (Fig. 3.66) ....
(Fundamentals of Engineering Graphics)

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

Topic: Parallelism of lines and planes

Lesson: Corners with Co-Directed Sides. Angle between two straight lines

Any straight line, for example OO 1(Fig. 1.), cuts the plane into two half-planes. If the rays OA and О 1 А 1 are parallel and lie in the same half-plane, then they are called co-directed.

Beams О 2 А 2 and OA are not codirectional (Fig. 1.). They are parallel but do not lie in the same half-plane.

If the sides of two corners are co-directed, then such angles are equal.

Proof

Let us be given parallel rays OA and О 1 А 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 co-directed rays. Let us prove that these angles are equal.

On the side of the beam OA and О 1 А 1 select points A and A 1 so that the line segments OA and О 1 А 1 were equal. Similarly, points V 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 О 1 А 1 A 1 O 1 OA A 1 O 1 OA OO 1 and AA 1 are parallel and equal.

Consider a quadrilateral В 1 О 1 ОВ... In this quadrangles side OV and About 1 In 1 are parallel and equal. Parallelogram, quadrilateral В 1 О 1 ОВ is a parallelogram. Because В 1 О 1 ОВ- parallelogram, then the sides OO 1 and BB 1 are parallel and equal.

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

Consider a quadrilateral B 1 A 1 AB... In this quadrangles side AA 1 and BB 1 are parallel and equal. Parallelogram, quadrilateral B 1 A 1 AB is a parallelogram. Because B 1 A 1 AB- parallelogram, then the sides AB and A 1 B 1 are parallel and equal.

Consider triangles AOB and A 1 O 1 B 1. Parties OA and О 1 А 1 are equal in structure. Parties OV and About 1 In 1 are also equal in construction. And as we have proved, both sides 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 to equal sides. So the angles AOB and A 1 O 1 B 1 are equal, as required.

1) Intersecting straight lines.

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

Rice. 4. The angle between two intersecting straight lines

2) Crossed straight lines

Let the straight lines a and b interbreeding. Choose an arbitrary point O... Through point O let's draw a straight line a 1 parallel to the straight line a, and straight b 1 parallel to the straight line b(Fig. 5.). Direct a 1 and b 1 intersect at the point O... The angle between two intersecting straight lines a 1 and b 1, the angle φ, and is called the angle between crossing lines.

Rice. 5. The angle between two crossed straight lines

Does the value of the angle depend on the selected point O? Let's choose a point About 1... Through point About 1 let's draw a straight line a 2 parallel to the straight line a, and straight b 2 parallel to the straight line b(Fig. 6.). Angle between intersecting straight lines a 2 and b 2 denote φ 1... Then the angles φ and φ 1 - corners with co-directed sides. As we have proved, such angles are equal to each other. Hence, the value of the angle between crossing lines does not depend on the choice of the point O.

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

1) ∠AOB= 40 °.

Let's choose a point WITH... Pass straight through it CD... We will carry out CA 1 parallel OA(Fig. 7.). Then the angle A 1 CD- the angle between crossing straight lines OA and CD... By the theorem on angles with codirectional sides, the angle A 1 CD equal to the angle AOB, that is, 40 °.

Rice. 7. Find the angle between two straight lines

2) ∠AOB= 135 °.

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

3) ∠AOB= 90 °.

Let's make the same construction (Fig. 9.). Then all the angles that are obtained at the intersection of straight lines CD and CA 1 are equal to 90 °. The desired 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 quadrangle ABCD. M,N,K,L- the middle of the ribs BD,AD,AC,BC respectively (Fig. 10.). It is necessary to prove that MNKL- parallelogram.

Consider a triangle ABD. МN МN parallel AB and is equal to its half.

Consider a triangle ABC. LK- middle line. By the property of the middle line, LK parallel AB and is equal to its half.

AND МN, and LK parallel AB... Means, МN parallel LK by the three parallel lines theorem.

We get that in the quadrangle MNKL- sides МN and LK are parallel and equal, since МN and LK equal to half AB... So, based on the parallelogram, the quadrilateral MNKL is a parallelogram, as required.

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

As we have already proved, МN parallel to the straight line AB. NK- the middle line of the triangle ACD, by property, NK parallel DC... Hence, through the point N there are two straight lines МN and NK that are parallel to crossing straight lines AB and DC respectively. Hence, the angle between the straight lines МN and NK is the angle between crossing lines AB and DC... We are given an obtuse angle MNK= 135 °. Angle between straight lines МN and NK- the smallest of the angles obtained at the intersection of these straight lines, that is, 45 °.

So, we considered the angles with co-directional sides and proved their equality. We considered the angles between intersecting and crossed lines and solved several problems to find the angle between two straight lines. In the next lesson, we will continue with problem solving and theory revision.

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

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

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

V) BC and D 1 IN 1.

Rice. 11. Find the angle between straight lines

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

Tasks 13, 14, 15 p. 54