Inverted corner. Straight angle in geometry. What angles are called adjacent? What is their sum?
Let's start by defining what an angle is. Firstly, it is Secondly, it is formed by two rays, which are called the sides of the angle. Thirdly, the latter emerge from one point, which is called the vertex of the angle. Based on these features, we can create a definition: an angle is a geometric figure that consists of two rays (sides) emerging from one point (vertex).
They are classified by degree value, by location relative to each other and relative to the circle. Let's start with the types of angles according to their magnitude.
There are several varieties of them. Let's take a closer look at each type.
There are only four main types of angles - straight, obtuse, acute and straight angles.
Straight
It looks like this:
Its degree measure is always 90 o, in other words, a right angle is an angle of 90 degrees. Only such quadrilaterals as square and rectangle have them.
Blunt
It looks like this:
The degree measure is always more than 90 o, but less than 180 o. It can be found in quadrilaterals such as a rhombus, an arbitrary parallelogram, and in polygons.
Spicy
It looks like this:
The degree measure of an acute angle is always less than 90°. It is found in all quadrilaterals except the square and any parallelogram.
Expanded
The unfolded angle looks like this:
It does not occur in polygons, but is no less important than all the others. A straight angle is a geometric figure whose degree measure is always 180º. You can build on it by drawing one or more rays from its top in any direction.
There are several other minor types of angles. They are not studied in schools, but it is necessary to at least know about their existence. There are only five secondary types of angles:
1. Zero
It looks like this:
The name of the angle itself already indicates its size. Its internal area is 0°, and the sides lie on top of each other as shown in the figure.
2. Oblique
An oblique angle can be a straight angle, an obtuse angle, an acute angle, or a straight angle. Its main condition is that it should not be equal to 0 o, 90 o, 180 o, 270 o.
3. Convex
Convex angles are zero, straight, obtuse, acute and straight angles. As you already understood, the degree measure of a convex angle is from 0° to 180°.
4. Non-convex
Angles with degree measures from 181° to 359° inclusive are non-convex.
5. Full
A complete angle is 360 degrees.
These are all types of angles according to their magnitude. Now let's look at their types according to their location on the plane relative to each other.
1. Additional
These are two acute angles forming one straight line, i.e. their sum is 90 o.
2. Adjacent
Adjacent angles are formed if a ray is passed through the unfolded angle, or rather through its vertex, in any direction. Their sum is 180 o.
3. Vertical
Vertical angles are formed when two straight lines intersect. Their degree measures are equal.
Now let's move on to the types of angles located relative to the circle. There are only two of them: central and inscribed.
1. Central
A central angle is an angle with its vertex at the center of the circle. Its degree measure is equal to the degree measure of the smaller arc subtended by the sides.
2. Inscribed
An inscribed angle is an angle whose vertex lies on a circle and whose sides intersect it. Its degree measure is equal to half the arc on which it rests.
That's it for the angles. Now you know that in addition to the most famous ones - acute, obtuse, straight and deployed - there are many other types of them in geometry.
An acute angle is an angle whose degree measure is up to 90 degrees.
A right angle is an angle whose degree measure is 90 degrees.
An obtuse angle is an angle whose degree measure is greater than 90 degrees. An acute angle is an angle less than 90°. An obtuse angle is an angle greater than 90° but less than 180°. A right angle is an angle = 90°.
20. What angles are called adjacent? What is their sum?
Adjacent angles- two angles with a common vertex, one of whose sides is common, and the remaining sides lie on the same straight line (not coinciding). The sum of adjacent angles is 180°. Or
Two angles are called adjacent, if they have one side in common, and the other sides are additional rays. the sum of adjacent angles is 180°. Each of these angles complements the other to the full angle.
21. What angles are called vertical? What property do they have?
Vertical angles - two angles in which the sides of one are continuations of the sides of the other. Vertical angles are equal. ( Angles are called vertical formed by intersecting straight lines and not adjacent to each other, that is, they do not have a common side, but vertical angles have a vertex at one point. Vertical angles are equal to each other).
22. Which lines are called perpendicular? Two intersecting lines are called perpendicular(or mutually perpendicular) if they form four right angles. Or Perpendicular lines These are straight lines intersecting at an angle of 90 degrees. Or Two straight lines forming right angles when intersecting, are called perpendicular.
23. Explain what segment is called a perpendicular drawn from a given point to a given line. What is the base of a perpendicular? A line segment perpendicular to a given line is called, which has one of its ends at their intersection point. This end of the segment is called the base of the perpendicular. Perpendicular to a given line A line segment perpendicular to a given line is called, which has one of its ends at their intersection point. The end of a segment lying on a given line , is called the base of the perpendicular.
24. What is a theorem and proof of the theorem? In mathematics, a statement whose validity is established by reasoning is called a theorem, and the reasoning itself is called a proof of the theorem.
Theorem- a statement for which there is a proof (in other words, a conclusion) in the theory under consideration. Unlike the theorems, axioms are statements that, within the framework of a particular theory, are accepted as true without any evidence or justification. Proof is a statement explaining the theorem. Theorem - a hypothesis that needs to be proven; A hypothesis always requires proof. Proof - arguments confirming the effectiveness and correctness of the theorem.
1. Learn to identify acute and obtuse angles using the right angle model.
Educational:
1. Form an idea of flat geometric shapes as parts of a plane.
2. Continue working on the classification of geometric shapes.
Educational:
1. Cultivate accuracy and attentiveness.
Lesson type- introduction of new knowledge
Forms of student work - paired, individual, frontal work
Equipment: a circle with sectors, cards with geometric shapes, multi-level cards, wire, models of triangles, poems - reminders.
I Updating knowledge.
1. Organizational moment.
A student reads a poem.
There is a rumor about mathematics
That she puts her mind in order,
Because good words
People often talk about her.
Mathematics, you give us
Hardening is important for victory.
Young people study with you
Develop both will and ingenuity.
- So today in class we will continue to develop ingenuity, will, determination, accumulate knowledge, and practice skills.
During the lesson we will have to travel around the country of Mathematics. Here is our travel itinerary. There are 6 sectors on the map, 5 different areas of mathematics. Want to know them? Then let's open them in order. (Arithmetic, geometry, where we will get acquainted with a new topic, ecology and mathematics, folklore, logic.)
So, let's go! (Open the “Arithmetic” sector)
(Slide 1.)
A)
Game “Mathematical Basketball”.
Basketball- a team sports game, the goal of which is to throw a ball into a suspended basket with your hands.
Any of you will score a goal if you solve the example correctly. (Children solve examples in a chain).
8+ 7 9 + 5 12 – 4 6 + 5 13 – 7 14 – 6 8 – 8 5 + 7 15 – 9 9 + 9
b) Solve the problem in general form.
There are two expressions written on the board. Which expression is suitable for solving problems A+B A-B
- There was A candy on the plate, Masha ate B candy. How many candies are left?
- Olya solved A problems in mathematics, Misha solved B problems. How many problems did the guys solve in total?
- Lena has A pencils, and Olya B has pencils. How many more pencils does Lena have than Olya?
- There were A girls in the class, and there were B fewer boys. How many boys were there in the class?
c) Working with cards (image of geometric shapes)
What is shown on the leaves? (flat geometric shapes)
Divide them into groups, i.e. distribute into “bags” using colored pencils.
Let's check...
The first group included straight lines. Name them. Prove that these are straight lines.
The rays were separated into the second group. Name them. Prove that these are rays.
The third group was divided into segments. Name them. Prove it.
The fourth group is the corner.
II . “Discovery” of new knowledge by students
(Slide 2.)
1) - The crossword puzzle will tell you the topic of the lesson. Crossword “Geometric”.
1) Part of a line that has a beginning but no end. (Ray).
2) A geometric figure that has no corners. (Circle).
4) A geometric figure in the shape of an elongated circle. (Oval).
The topic of our lesson is hidden vertically. Find her. (Corner). (click, geometric shapes fly out).
Please formulate the topic of our lesson. (Sector “Geometry”)
Guys, why are we going to study angles?
Do you think this knowledge will be useful to you?
(Children's answers)
Angles surround us in everyday life. Give your own examples of where you can find angles around us.
Slide 3-4.
Look at the pictures: a connecting corner for pipes and a stationery corner for papers; carpenter's square and drafting square; corner table and corner sofa.
Guys, maybe someone knows what an angle is? (children's opinions are listened to)
We will check the correctness of our formulation a little later.
People of what professions are most likely to encounter angles? (constructor, engineer, designer, builder, architect, sailor, astronomer, architect, tailor, etc.)
Guys, now step back one cell from the red fields and place point O. Draw two rays from this point.
Draw point O (2) on the board in advance. I call 2 children to draw rays on the board.
What kind of figures did we get? (corner)
Look how different these angles are.
Guys, now try to define an angle.
Work in pairs.
(Conclusion: an angle is a geometric figure formed by two different rays
with a common beginning).
Guys, now look at the figure that I drew.
Is it an angle or not.
(The children say no, we return to the rule again, after which we conclude that this is also an angle - a reversed one)
Slide 6. (output by angle)
Poster on blackboard
Point O is the vertex of the angle. An angle can be called by one letter written near its vertex. Angle O. But there can be several angles that have the same vertex. What to do then? (There is a drawing of such angles on the board)
Children's answers.
In such cases, if you call different angles with the same letter, it will not be clear which angle you are talking about. If this does not happen, you can mark one point on each side of the angle, put a letter near it and designate the angle with three letters, while always writing in the middle the letter indicating the vertex of the angle. Angle AOB. Rays AO and OB are the sides of the angle.
Drawing on the board
Working with the textbook text in the orange frame p.52.
III . Primary consolidation.
Work in pairs. Task No. 2
- The angles are different. Here are different types of angles.
What is this angle called? (straight) How to prove that it is really straight?
- What are these angles called? (indirect)
- Today we will find out what they are called.
IV . Formulation of new knowledge.
(Slide 7 - 9)
It is not always convenient to determine a right angle by eye. To do this, use a ruler-square.
What color is used to highlight an angle greater than a right angle? (Blue).
Less direct? (Green).
Which of the three proposed angles is a straight line?
Why did you decide so? (The vertex and sides of the angle coincide with the right angle on the square ruler).
How to determine the type of angle?
CONCLUSION:
To determine the type of angle, you need to combine its vertex and side, respectively, with the vertex and side of the right angle on the square.
Each of the corners has its own name. An acute angle is an angle that is less than a right angle. An obtuse angle is an angle that is larger than a right angle.
(Tables with the names of the angles appear on the board)
Working with textbook text in an orange frame p. 53.
My mother took the piece of paper
And folded the corner
This is the angle for adults
It's called DIRECT.
If the corner is already SHARP,
If wider, then - DUMB.
V .Formulation of the topic and objectives of the lesson.
VI . Physical education minute.
How many mushrooms are there?
We squat a lot.
How many flowers are there?
We raise our hands.
We raise our hands,
We clear the clouds.
Brighter, sunshine, shine,
Ban the gloomy rain.
The long journey is over.
You can sit down and relax.
VII . Application of new knowledge.
Independent work. (Multi-level tasks)
Card No. 1.
1.Write the names of the angles
2.Divide the angles into groups:
Card No. 2
Circle all the figures for which the statement “The figure has an obtuse angle” is true.
Card No. 3
4.Write the names of acute, right and obtuse angles
Sharp corners: ___________________________________
Right angles:__________________________________________
Obtuse angles:__________________________________________
VIII. Mathematics and folklore.(Sector “Mathematics and folklore”)
- The creativity of the Russian people is closely connected with mathematics . People use the word with great pleasure corner in their proverbs and sayings. What proverbs and sayings did you find at home?
Now listen to my proverbs and sayings.
A house cannot be built without corners; speech cannot be said without a proverb.
The hut is red not in its corners, but in its pies.
If you say it from ear to ear, they will know from corner to corner.
Threshing - so from the edge, and at the table - so he climbed into the corner.
IX . Mathematics and ecology.(Sector “Mathematics and Ecology”)
Solving the problem. (Solve in different ways).
For the project “Mushrooms of the Bryansk Forest,” the children made 12 dummies of mushrooms. 4 of them were milk mushrooms, 5 were chanterelles, and the rest were porcini mushrooms. How many dummies of porcini mushrooms have the children made?
X . Logics.(Sector “Logic”)
The children put into boxes dummies of mushrooms brought to create a corner of the Bryansk forest. Find out where each mushroom is located if all the labels on the boxes are false.
Here Here Here
milk mushroom there is no russula. boletus
XI . Lesson summary. Reflection.
There is wire on your desks. Make a right angle out of it and test it with a square, then make it sharp and obtuse.
(Slide 10.)
Tell me, using a diagram, what did you learn from today's math lesson?
XII. Homework.(Sector “DZ”)
P. 53, No. 6, No. 7 – optional
June 26, 2013Let's start by defining what an angle is. Firstly, it is a geometric figure. Secondly, it is formed by two rays, which are called the sides of the angle. Thirdly, the latter emerge from one point, which is called the vertex of the angle. Based on these features, we can create a definition: an angle is a geometric figure that consists of two rays (sides) emerging from one point (vertex).
They are classified by degree value, by location relative to each other and relative to the circle. Let's start with the types of angles according to their magnitude.
There are several varieties of them. Let's take a closer look at each type.
There are only four main types of angles - straight, obtuse, acute and straight angles.
Straight
It looks like this:
Its degree measure is always 90 o, in other words, a right angle is an angle of 90 degrees. Only such quadrilaterals as square and rectangle have them.
Blunt
It looks like this:
The degree measure of an obtuse angle is always greater than 90°, but less than 180°. It can be found in quadrilaterals such as a rhombus, an arbitrary parallelogram, and in polygons.
Spicy
It looks like this:
The degree measure of an acute angle is always less than 90°. It is found in all quadrilaterals except the square and any parallelogram.
Expanded
The unfolded angle looks like this:
It does not occur in polygons, but is no less important than all the others. A straight angle is a geometric figure whose degree measure is always 180º. You can construct adjacent angles on it by drawing one or more rays from its vertex in any direction.
There are several other minor types of angles. They are not studied in schools, but it is necessary to at least know about their existence. There are only five secondary types of angles:
1. Zero
It looks like this:
The name of the angle itself already indicates its size. Its internal area is 0°, and the sides lie on top of each other as shown in the figure.
2. Oblique
An oblique angle can be a straight angle, an obtuse angle, an acute angle, or a straight angle. Its main condition is that it should not be equal to 0 o, 90 o, 180 o, 270 o.
3. Convex
Convex angles are zero, straight, obtuse, acute and straight angles. As you already understood, the degree measure of a convex angle is from 0° to 180°.
4. Non-convex
Angles with degree measures from 181° to 359° inclusive are non-convex.
5. Full
A complete angle is 360 degrees.
These are all types of angles according to their magnitude. Now let's look at their types according to their location on the plane relative to each other.
1. Additional
These are two acute angles forming one straight line, i.e. their sum is 90 o.
2. Adjacent
Adjacent angles are formed if a ray is passed through the unfolded angle, or rather through its vertex, in any direction. Their sum is 180 o.
3. Vertical
Vertical angles are formed when two straight lines intersect. Their degree measures are equal.
Now let's move on to the types of angles located relative to the circle. There are only two of them: central and inscribed.
1. Central
A central angle is an angle with its vertex at the center of the circle. Its degree measure is equal to the degree measure of the smaller arc subtended by the sides.
2. Inscribed
An inscribed angle is an angle whose vertex lies on a circle and whose sides intersect it. Its degree measure is equal to half the arc on which it rests.
That's it for the angles. Now you know that in addition to the most famous ones - acute, obtuse, straight and deployed - there are many other types of them in geometry.
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