Presentation for the lesson "adjacent and vertical angles" presentation for the lesson in geometry (grade 7) on the topic. Geometry presentation on "adjacent and vertical corners" Presentation adjacent corners frozen

Let's remember!

What is an angle?

Use a protractor to measure angles. .

What tool can be used to measure angles?

Show a right angle on a square.

What are the other corners called? (not straight)

Are they larger or smaller than the right angle?

What kinds of angles do you know?

Deployed

B i s sect r i s a

What is called the bisector of an angle?

Adjacent corners

Two corners in which one side is common and the other two are extensions of one another are called adjacent.

In Figure 1,  AOB and  BOC are adjacent. Since the beams OA and OS form an unfolded angle, then  AOB +  BOC \u003d 180 0

Thus, the sum of adjacent angles is 180 0.

This is a property of adjacent corners !!!

1.One side of the corner to continue

behind its top.

2.The resulting AOC angle

is adjacent to angle AOB.

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I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I

A corner adjacent to a sharp corner is obtuse .

1. Continue one side of the corner beyond its top.

2. The resulting angle AOC is adjacent to the angle AOB.

Angle adjacent to obtuse angle is acute .

Continue one side of the corner beyond its top.
The resulting angle AOC is adjacent to the angle AOB

An angle adjacent to a right angle is right

Solve the drawing problem

(by property of adjacent corners)

Vertical corners

Two corners are called vertical if the sides of one corner are extensions of the sides of the other.

In Figure 2,  1 and  3, and  2 and  4 are vertical.

 2 is adjacent to both  1 and  3. By the property of adjacent angles  1 +  2 \u003d 180 0 and  3 +  2 \u003d 180 0. From this we get that

 1 \u003d 180 0   2,  3 \u003d 180 0   2. Thus, the degree measures  1 and  3 are equal. Hence it follows that the angles themselves are equal.

So the vertical angles are equal.

This is a property of vertical angles !!!

Find the vertical corners.

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Construct corner.

2. Extend each side of the corner past its vertex.

Solve the drawing problem

(by property of vertical angles)

 MOF Given: F M Find:  FOK,  KOP,  POM,  MOF. O Solution: Let the measure  MOF \u003d x, then  FOK \u003d 2x. By the property of adjacent angles x + 2x \u003d 180 °, then x \u003d 60 °, and 2x \u003d 120 °. The corresponding vertical angles are 60 ° and 120 °. P K Answer: 60 0, 120 0, 60 0, 120 0 "width \u003d" 640 "

An example of a solution to the problem

One of the four corners formed by the intersection of two straight lines is twice the size of the other. Find the measure of each of the angles.

MK  PF \u003d O

 MOF \u003d  KOP (vertical)

 MOF,  FOK - adjacent,

 FOK 2 times  MOF

 FOK,  KOP,  POM,  MOF.

Let the measure  MOF \u003d x, then  FOK \u003d 2x. By the property of adjacent angles x + 2x \u003d 180 °, then x \u003d 60 °, and 2x \u003d 120 °. The corresponding vertical angles are 60 ° and 120 °.

Answer: 60 0, 120 0, 60 0, 120 0

In the figure  COA \u003d 40 O

OM - bisector  COB

 MOV -?

FROM

ABOUT

Solve the problems.

Given two adjacent angles ABC and CBD. ABC is 20 degrees greater than CBD). Find these corners.

Two adjacent corners PQR and RQS are given. RQS is 0.8 times the PQR. Find these corners.

Complete the sentence

If one of the adjacent angles is 50 °, then the other is ...
An angle adjacent to a right one ...
If one of the vertical angles is straight, then the second ...
Angle adjacent to acute ...
If one of the vertical angles is 25 °, then the second angle is ...

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Slide captions:

Lesson topic: Adjacent and vertical corners. School 291 Grade 7

Lesson objectives: To acquaint students with the concepts of adjacent and vertical angles, to consider their properties; To teach how to build an angle adjacent to a given angle, depict vertical angles, find vertical and adjacent angles in a drawing.

Let's remember! What is an angle?

AOB O V BOA A O Beam OA Beam OV How are angles designated?

A protractor is used to measure angles. What tool can be used to measure angles? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

10 20 50 60 70 80 90 100 110 120 130 140 150 160 170 180 180 170 160 150 140 130 120 110 100 80 0 10 20 30 40 50 60 70 0 40 30 A B i s c ect r i s I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 А OB \u003d 70 0 What is called the bisector of an angle? B O

Angle units Total 18 0 parts. 1 part is 1 degree. 1/60 part of a degree is called a minute, denoted by "'" 1/60 part of a minute is called a second, denoted by "″"

Kinds of angles SHARP ANGLE Angle name Figure Degree measure RIGHT ANGLE BLUNT ANGLE REVERSED less than 90 ˚ 90 ˚\u003e 90 ˚, but

What angle does the crow's beak form when: "The crow held cheese in its mouth?" And when "The crow croaked at the top of the crow's throat?"

Sharp Dull

In the tale of the corners of a square, the brother-circle chopped off his corners. What did they become after that?

Two more views will be added to your knowledge of corners today: Adjacent and vertical corners.

1 2 A B C O Draw a flat AOC corner. Draw an arbitrary ray O B between the sides of the flattened corner.

Definition of adjacent corners Definition. Two corners are called adjacent if they have one side in common, and the other sides of these corners are opposite rays. A O B S  BOA and  BOS adjacent A O B S A O V S A O V S A O V S A O V S A O V S A O V S

Are  AOD and  BOD  AO C and  DO C  AO C and  DO B  AO C,  DO C, and и BOD adjacent?

Draw adjacent corners

A О В С The adjacent corner for an acute angle is obtuse. 1. Continue one side of the corner beyond its top. (2) The resulting angle AOC is adjacent to the angle AOB. I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1. Continue one side of the corner beyond its top. 2. The resulting angle AOC is adjacent to the angle AOB. А В С О The angle adjacent to an obtuse angle is acute.

Continue one side of the corner beyond its top. The resulting angle AOC is adjacent to the angle AOB A B O C The angle adjacent to a right angle is right

Theorem. The sum of adjacent angles is 180 0 Given:  AOC and  BOC are adjacent. Prove:  AOC +  BOC \u003d 180 . Evidence. 1) Since  AOC and  BOC are adjacent, the rays OA and OB are opposite, that is,  AOB is expanded, therefore,  AOB \u003d 180 . 2) Beam OC passes between sides  AOB, so  AOC +  BOC \u003d  AOB \u003d 180  C О A B C Property of adjacent angles 1. How many angles are shown in the picture? What are these angles? 2. Is there any relationship between these angles? (Remember the axiom of adding angles).

130 0? Decision:

Draw a random  AOB. Construct rays OC and OD opposite to its sides. B C A O D Definition. Two corners are called vertical if the sides of one corner are opposite rays to the sides of the other.

A D B C O Find the vertical corners. M N D C B A B A C D O B A C D M D C B A M D C B A

Drawing vertical corners

A O B I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I IIII I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 CD Draw a corner. 2. Extend each side of the corner past its vertex.

Property of vertical angles A O D B C Theorem. The vertical angles are equal. Given:  AOD and  COB are vertical. Prove:  AOD \u003d  COB Proof. Each of углов AOD and  COB is adjacent to  AOB. By the property of adjacent corners:  AOD +  AOB \u003d 180  and  CO B +  AOB \u003d 180 . We have:  AOD \u003d 180  -  AOB and  COB \u003d 180  -  AOB, so  AOD \u003d  COB

Solve the problem according to the drawing Solution:

Finish the sentence If one of the adjacent angles is 50 °, then the other is ... An angle adjacent to a right one, ... If one of the vertical angles is a straight line, then the second ... An angle adjacent to an acute one ... If one of the vertical angles is 25 °, then the second the angle is ... 130 ° straight line obtuse 25 °

50 °? 1 2 1 _ 2 \u003d 70 ° 79 °? 1 + 2 \u003d 90 ° 2 1 Tasks for self-test Identify from the pictures: Find  1 and  2 1 Find  1 and  2

Given:  \u003d 3 . Find:  and . OS- bisector Find  BOC Find  BOC

T E C T on the topic "Vertical and adjacent corners"

1. The sum of adjacent angles is…. 360 0 90 0 180 0 A B C

2. What is the name of the angle less than 180 0, but more than 90 0 acute obtuse straight line A B C

3. What is the angle if the adjacent angle is 47 0? 133 0 47 0 43 0 C B A

4. What angle do the hour and minute hands of the watch make when they show 6 o'clock? obtuse unfolded straight C B A

5. Find

6. Find

7. Find adjacent corners if one is twice the size of the other. 60 0 and 120 0 90 0 and 100 0 40 0 \u200b\u200band 80 0 C B A

8. The angle is 72 0. What is its vertical angle? 72 0 108 0 18 0 C B A

9. What angle do the hour and minute hands of the watch make when they show three o'clock? sharp blunt straight C B A

Check yourself. 1. C 2. B 3. A 4. B 5. B 6. B 7. B 8. C 9. C

An example of how to solve the problem When two straight lines intersect, four corners are formed. One of them is 43 0. Find the values \u200b\u200bfor the remaining angles. M O F P K 43 0 Given: Find: Solution: Answer: 137 0, 43 0, 137 0 MK  PF \u003d O  MO F \u003d 43 °  FOK,  KOP,  POM.  MO F and  KOP are vertical, which means, by the property of vertical angles,  MO F \u003d  KOP,  KOP \u003d 43 °  MO F +  FOK \u003d 180 °, since they are adjacent. Hence  FOK \u003d 180 ° - 43 ° \u003d 137 °  FOK and  POM are vertical, so  FOK \u003d  POM,  POM \u003d 137 °

Problem 1. Find the angles obtained at the intersection of two straight lines, if one of the angles is 102 0. Problem 2. Find the values \u200b\u200bof adjacent angles if one of them is 5 times smaller than the other. Problem 3. What are the adjacent angles if one of them is 30 0 greater than the other? Problem 4. Find the value of each of the two vertical angles if their sum is 98 0.

Self-Study A C B D 2. Draw the IOC corner. Construct adjacent to it: a) angle KO N; b) angle MOR. 3. Write down the pairs of adjacent corners shown in the picture: E A D C B F 4. Write down the pairs of vertical angles shown in the picture: D B A M C N 1. The figure shows straight lines AC and B D, intersecting at point O. Add entries:  BOC and . ... ... - vertical,  VOS and . ... ... - adjacent,  СО D and . ... ... - vertical,  CO D and . ... ... - adjacent. o

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Objectives:

introduce the concept of adjacent and vertical angles, find out through the system of exercises what properties they have;
consider the proof of the theorems on adjacent and vertical angles;
show their application in solving problems;

Two corners with one side in common and

the other two are extensions of one

the other are called adjacent.

FROM

OS beam divides

How many angles are shown

on the image?

FROM

3 corners:

Is there any relationship

between these corners?

How else can you write

given equality?

FROM

Yes:

As ° - expanded angle,

then °

Adjacent corners property:

FROM

The sum of adjacent angles is 180 °.

The two corners are called vertical if the sides of one corner are complementary half-lines of the sides of the other.

b 2

a 1

a 2

b 1

1 b 1 ) and 2 b 2 ) - vertical

Drawing vertical corners

Name the vertical corners

shown in the drawing

FROM

The vertical angles are equal

Name the vertical corners

shown in the drawing

Calculate the degree measures of the angles shown in the drawing if one of the angles is 50 0 more than another.

FROM

Decision

x + 50 °

Let the smaller angle x °,

then a larger angle

x + 50 (°)

If °

Since the sum of adjacent angles is 180 °, then we will compose the equation

x + x + 50 ° \u003d 180 °

2x \u003d 130 °

x \u003d 130 °: 2

2x + 50 ° \u003d 180 °

x \u003d 65 °

2x \u003d 180 ° - 50 °

° then ° + 50 ° \u003d 115 °

AC ∩ BE \u003d M, the sum of two angles - 50 0

Given:

these angles are?

To find:

Decision:

FROM

Since the sum of two angles is 50 0 then it can be only vertical corners.

° : 2 = 25 °

One of the adjacent corners at 32 0 more than another. Find the size of each corner.

Given:

 AOB and  VOS adjacent,

 AOB -  ВОС \u003d 32 °.

To find:

 AOB,  ВОС.

Decision:

ABOUT

FROM

Let  BOC \u003d x, then  AOB \u003d 32 + x

By the property of adjacent angles, we compose the equation

x + (32  + x) \u003d 180 

2x \u003d 180  - 32 

2x \u003d 148 

x \u003d 74 

Means  BOC \u003d 74  , a  AOB \u003d 32  +74  =106 

Answer:  AOB \u003d 106  ,  BOC \u003d 74 

Test

"Vertical and Adjacent Corners"

1. The sum of adjacent angles is

360 0

90 0

180 0

2. What is the name of the angle less than 180 0 but more than 90 0

acute

stupid

straight

3. What is the angle if the adjacent angle is 47 0 ?

133 0

47 0

43 0

4. What angle do the hour and minute hands of the watch make when they show 6 o'clock?

stupid

deployed

straight

5. Find

77 0

103 0

3 0

6. Find

54 0

126 0

36 0

7. Find adjacent corners if one is twice the size of the other.

90 0 and 100 0

60 0 and 120 0

40 0 and 80 0

8. The angle is 72 0 ... What is its vertical angle?

18 0

108 0

72 0

9. What angle do the hour and minute hands of the watch make when they show three o'clock?

acute

stupid

straight

Self-test

1.C

2. B

3. A

4. B

5. B

6. B

7.B

8.C

9.C

Thank you for attention

Slide 2

Purpose: introduce the concept of adjacent and vertical angles, consider their properties

Slide 3

Repetition: the tree of knowledge

1.What is a ray? How is it indicated? 2. What shape is called an angle? 3. What angle is called unfolded? 4. How do you compare two angles? 5. Which ray is called the bisector of the angle? 6.What is the degree measure of the angle? 7. What angle is called acute? Direct? Dumb?

Slide 4

ADJACENT CORNERS

Practical task: 1. Construct an acute angle AOB; 2. Draw the OC beam, which is a continuation of the OA beam. A O B S AOB and BOS - adjacent corners

Slide 5

Definition:

Two corners in which one side is common and the other two are a continuation of one another are called adjacent corners. A O B S

Slide 6

Adjacent corners property

1. What is the angle of the AOB? 2. What is the degree measure of the angle? 3. At what angles does the OB beam divide this angle? 4. What is the sum of these angles? 1. AOC - deployed 2.180˚ 3. AOB and BOS 4.180˚

Slide 7

CONCLUSION:

AOB + The sum of adjacent angles is 180˚ BOS \u003d 180˚

Slide 8

Strengthening exercises

1. Draw three angles: sharp, straight, obtuse. For each of these corners, draw an adjacent corner. Decision:

Slide 9

2. One of the adjacent corners of a straight line. What (sharp, straight, obtuse) is the other angle?

Slide 10

3. Is it true: if adjacent angles are equal, then they are right?

Consider:

Slide 11

4. Find the corner adjacent to the corner if:

a) ASO \u003d 15˚ c) DSV \u003d 111˚ D S A O D S V A

Slide 12

VERTICAL CORNERS

Practical task: 1. build an acute angle; 2. select it with an arc and denote it by the number 1; 3. construct the continuation of the sides of corner 1; 4.mark with an arc the angle whose sides are a continuation of the sides of angle 1 and denote it by the number 2 1 2

Slide 13

Definition

Two corners are called vertical if the sides of one corner are a continuation of the sides of the other. 1 2 3 4 1 and 2 - vertical corners

Slide 14

Vertical Angle Property

Conclusion: The vertical angles are equal. 1 2 3 4 1 \u003d 35˚ Find: Given: 3, 4 Solution: 1, 3-adjacent 3 \u003d 180˚-35˚ \u003d 145˚ 1, 4-adjacent 4 \u003d 180˚-35˚ \u003d 145˚ 3 \u003d 4 \u003d 145˚, but 3 and 4 are vertical

Slide 15

Strengthening exercises

1. At the intersection of two straight lines a and b, the sum of some angles is 60˚. What are these angles? Answer: vertical corners, because the sum of adjacent angles is 180˚. 2. At the intersection of two straight lines a and b, the difference of some angles is equal to 30˚. What are these angles? Answer: adjacent, because the difference in vertical angles is 0˚