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инфоурок

школа

видеоуроки

геометрия

00:00:00

third criterion for the equality of triangles

00:00:05

theorem if three sides of one

00:00:09

triangle are respectively equal to three

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sides of another triangle then such

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triangles are equal proof

00:00:19

consider triangles a pc and 1 b 1 c 1

00:00:24

in which a b is equal to 1b 1d c is equal to b 1 c

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1 d a is equal to c11 we will prove that the triangle

00:00:36

a b c is equal to triangle 1 b 1 c 1

00:00:41

apply triangle a b c to

00:00:44

triangle 1 b 1 c 1 so that vertex

00:00:49

a coincides with vertex a1 and vertex b coincides with

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vertex b1 and vertices c&c are one

00:00:59

on opposite sides of the straight line

00:01:01

a1 and b1, three cases are possible c1 c

00:01:07

passes inside the angle a 1 c 1 d 1 2

00:01:14

case the ray c1 c coincides with one of the

00:01:19

sides of this angle and the third case the

00:01:23

ray c1 c passes outside the angle a 1 c 1 d 1

00:01:31

consider the first case and so since

00:01:34

by the conditions of the father's side theorem

00:01:38

and 1 c 1 d c and b 1 c 1 are equal

00:01:42

then triangle one c1 c and b 1 c 1 d are

00:01:49

isosceles by the theorem on the property of the

00:01:53

angles of an isosceles triangle angle

00:01:56

1 is equal to angle 2 angle 3 is equal to angle 4

00:02:02

therefore angle 1 cb-1 is equal to angle

00:02:08

1 c 1 d 1 means

00:02:12

a c is equal to 1 c 1 d c is equal to b 1 c 1 this is by

00:02:19

condition and we have just shown that

00:02:21

angle c is equal to angle c1 therefore

00:02:26

triangles about the father and 1 b 1 c 1 are equal

00:02:30

according to the first sign of equality of

00:02:33

triangles and so case 2 when we

00:02:39

get this drawing, that is, we

00:02:43

combined points a and a one b&b one and at

00:02:48

this point c & c 1 turned out to be located like this

00:02:52

in this case,

00:02:56

since according to the condition of the segment

00:03:00

a c and one c1 are equal,

00:03:04

then triangle c11 is isosceles, which means

00:03:11

angle c is equal to the angle c1

00:03:14

well, and since, according to the condition, the segment c b and

00:03:18

c 1 d 1 are equal, it means that the triangles

00:03:23

a c b and a 1 c 1 d 1 are equal according to the first

00:03:32

criterion because this side is equal to this

00:03:35

this is equal to this and the angles between these

00:03:39

sides are equal well and the third case when

00:03:44

we get such a drawing, here are

00:03:49

points a and a one b&b 1a to point c & c 1

00:03:54

here in this case it is similar

00:03:57

since c b is equal to c 1 d 1 then that means the

00:04:03

triangle cb-1 is isosceles and that means

00:04:09

this entire angle c is equal to this

00:04:13

angle c1,

00:04:15

then since according to the condition a c is equal to

00:04:19

one c1, then

00:04:22

these angles are equal, that is, some angle d

00:04:27

c a is equal to the angle to

00:04:32

c11, well, it follows that these

00:04:36

angles are equal, that is, the angle a c b is equal to the angle 1 c 1

00:04:42

d 1

00:04:44

now let's look at the triangle a b c and

00:04:48

1 b 1 c 1

00:04:50

they have these angles equal, we just

00:04:54

proved this, these sides are equal by

00:04:57

condition and these sides are equal by condition,

00:05:00

which means these triangles

00:05:03

will be equal by the first sign of

00:05:06

equality of triangles,

00:05:08

that is, by two sides and angle between

00:05:11

them and so the theorem is proven from 3

00:05:17

signs of equality of triangles

00:05:19

it follows that a triangle is a rigid

00:05:22

figure let’s explain what this means

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imagine two slats whose two

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ends are fastened with a nail such a

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structure is not rigid

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by moving or pushing apart the free ends

00:05:38

of the slats we can change the angle between them

00:05:42

now let's take another river and fasten

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its ends with the free ends of the first

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two slats, the resulting

00:05:52

triangle structure will already be rigid;

00:05:59

no two sides can be moved or moved apart, that is, not a

00:06:01

single angle can be changed;

00:06:04

indeed, if this were possible, then we would

00:06:07

get a new triangle

00:06:09

unequal to the original one, but this is impossible

00:06:13

since the new triangle must be equal to the

00:06:15

original one according to the third criterion of equality of

00:06:18

triangles;

00:06:19

this property of the rigidity of triangles is

00:06:23

widely used in practice;

00:06:25

in order to secure the pillar in a vertical

00:06:28

position, a

00:06:29

support is placed on it; the same principle

00:06:33

is used when installing a bracket

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