Straight, obtuse, acute and straight angles. Straight angle in geometry 1 types of angles
“Little son came to his father and asked Tiny: “What are the angles?” But father, I forgot the answer. This is very bad!
In our article, we suggest remembering your math lessons and finding answers to Krochi’s questions.
What is an angle
What an angle is is of course easier to show than to explain. From elementary school we know that a plane angle:
- This is a geometric figure.
- It is formed by two sides - rays.
- The rays come out from one vertex - a point.
- Measured in degrees.
That is, if you put a point on any plane, and then draw two rays from this point (a ray is a straight line with a beginning but no end), then we get an angle, and not one, but two. This is because the rays divided the plane into two parts. We have formed two angles - internal and external.
Angle designation
An angle is denoted in mathematics by this symbol – “˪” and Greek letters: β, δ, φ. You can also designate angles in small or capital Latin letters. Lowercase (d, c, b) denote rays forming an angle, therefore, the name will consist of two letters and the icon - ˪ab. Large Latin letters indicate three points of the angle: two on the sides and one vertex (˪ DEF). Moreover, the letter of the vertex will always be in the middle of the name, but it makes no difference how to read DEF or FED.
Types of angles
Depending on the degrees (measurement), angles are divided into:
- Sharp (>90 degrees);
- Straight (exactly 90);
- Dumb (180);
- Expanded (equal to 180);
- Non-convex (more than 180, but less than 360);
- Full(360);
All angles that are not right or straight are called oblique.
Also, what are the angles?
- Adjacent - they have one side in common, while the others lie, not coinciding, on the same plane. The sum of such angles will always be equal to 180.
- Vertical - angles formed by two intersecting straight lines and they do not have common sides, but their rays come out from one point. That is, the side of one angle is a continuation of the other. These angles are equal.
- Central - an angle whose vertex is the center of the circle.
- Inscribed angle. Its vertex is on a circle, and the rays that form it intersect this circle.
Now you know which is a right angle, and you can also tell which angle is acute. It’s not difficult to remember, and other types of angles also have characteristic names.
An angle is a figure formed by two rays emanating from one point.
The rays forming an angle are called the sides of the angle, and the point from which they emerge is the vertex of the angle.
In my drawing, the rays OB and OS are the sides of the angle, the vertex is the point O, and the angle is designated as: BOS.
When writing an angle, write a letter in the middle to indicate its vertex. An angle can also be denoted by one letter - the name of its vertex, for example: angle O. The word “angle” is replaced by the sign “”.
For example: BOS = O
Like all geometric shapes, angles are compared using superimposition. If one angle is superimposed on another and they coincide, then these angles are equal.
For example: MRL= AKV
Of all the angles we can distinguish:
1. Acute (the magnitude of such angles is greater than 0, but less than 90).
2. Straight (the value of which is 90).
3. Obtuse (the magnitude of such angles is more than 90, but less than 180).
4. Expanded (the value of which is 180).
A protractor is used to measure angles.
The protractor scale is located on a semicircle. The center of this semicircle is marked on the protractor with a dash. The protractor scale lines divide the semicircle into 180 parts. The rays drawn from the center of the semicircle through these strokes form 180 angles, each of which is equal to a fraction of a developed angle. Such angles are called degrees. Degrees are indicated by a sign. Each division of the protractor scale is equal to 1. In addition to divisions of 1, the protractor also has divisions of 5 and 10.
The protractor is also used to construct angles.
Historical background
Since ancient times, people have been faced with the need to measure. The concept of a degree and the appearance of the first instruments for measuring angles are associated with the development of civilization in ancient Babylon, although the word degree itself is of Latin origin (degree - from the Latin gradus - “step, step”).
History has not preserved the name of the scientist who invented the protractor - perhaps in ancient times this instrument had a completely different name. The modern name comes from the French word “TRANSPORTER”, which means “to carry”.
But ancient scientists made measurements not only with a protractor - after all, this instrument was inconvenient for taking measurements on the ground and solving applied problems. Namely, applied problems were the main subject of interest of ancient geometers. The invention of the first instrument that allows measuring angles on the ground is associated with the name of the ancient Greek scientist Heron of Alexandria (1st century BC). He described the “diopter” tool, which allows one to measure angles on the ground and solve many applied problems.
Thus, we can talk about the emergence of geodesy - a system of sciences about determining the shape and size of the Earth and about measurements on the earth's surface to display it on plans and maps. Geodesy is related to astronomy, geophysics, astronautics, cartography, etc., and is widely used in the design and construction of structures, shipping canals, and roads.
In the 17th century, the level device was invented, and in the next century, the English mechanic Jesse Ramsden invented the theodolite. Today the theodolite is a complex device. Many works (including construction) require preliminary consultation with surveyors for measurements using a theodolite.
However, the improvement of angle measuring tools is not only related to construction work. Since ancient times, people have traveled, exploring the world around them. Travelers needed to be able to navigate in space. For many centuries, the stars became the main reference point for travelers. The first tool for travelers appeared - the astrolabe. Astrolabe (Greek astrolabion, from astron - “star” and labe - “grasping”; Latin astrolabium) is a goniometric device that served until the beginning of the 18th century to determine the positions of luminaries in the sky.
The sextant is the most advanced device for measuring the angular coordinates of celestial bodies of that time. Its invention is attributed to Isaac Newton. The sextant made it possible to measure both the latitude and longitude of the observation point, and with fairly high accuracy. Note that there are other units for measuring angles.
Artillerymen have to not only measure angles, but also quickly mentally convert the resulting angular values into linear ones and vice versa. Therefore, measuring angles in degrees and minutes is inconvenient for artillerymen. The artillerymen came up with a completely different measure of angles. This measure is “thousandths”, or, as it is otherwise called, “division of the protractor”. To get the thousandth, the circle is divided into 6000 parts.
In maritime navigation, it is customary to use the rhumb as the main unit of measurement. The nautical rhumb is determined by the central angle corresponding to an arc equal to 1/32 of the circle. Meteorology has its own rhumb, which is twice as large as the sea.
Students are introduced to the concept of angle in elementary school. But as a geometric figure that has certain properties, they begin to study it from the 7th grade in geometry. Seems, quite a simple figure, what can be said about her. But, acquiring new knowledge, schoolchildren increasingly understand that they can learn quite interesting facts about it.
When studied
The school geometry course is divided into two sections: planimetry and stereometry. In each of them there is considerable attention is given to the corners:
- In planimetry, their basic concept is given and an introduction is made to their types by size. The properties of each type of triangle are studied in more detail. New definitions are emerging for students - these are geometric figures formed by the intersection of two straight lines with each other and the intersection of several straight lines with transversals.
- In stereometry, spatial angles are studied - dihedral and trihedral.
Attention! This article discusses all types and properties of angles in planimetry.
Definition and measurement
When starting to study, first determine what is an angle in planimetry.
If we take a certain point on the plane and draw two arbitrary rays from it, we obtain a geometric figure - an angle, consisting of the following elements:
- vertex - the point from which the rays were drawn, denoted by a capital letter of the Latin alphabet;
- the sides are half-straight lines drawn from the vertex.
All the elements that form the figure we are considering divide the plane into two parts:
- internal - in planimetry does not exceed 180 degrees;
- external.
The principle of measuring angles in planimetry explained on an intuitive basis. To begin with, students are introduced to the concept of a turned angle.
Important! An angle is said to be developed if the half-lines emerging from its vertex form a straight line. The undeveloped angle is all other cases.
If it is divided into 180 equal parts, then it is customary to consider the measure of one part to be equal to 10. In this case, they say that the measurement is made in degrees, and the degree measure of such a figure is 180 degrees.
Main types
Types of angles are divided according to criteria such as degrees, the nature of their formation, and the categories presented below.
By size
Taking into account the magnitude, angles are divided into:
- expanded;
- direct;
- blunt;
- spicy.
Which angle is called unfolded was presented above. Let's define the concept of direct.
It can be obtained by dividing the expanded into two equal parts. In this case, it is easy to answer the question: how many degrees is a right angle?
Divide 180 degrees of unfolded by 2 and we get that a right angle is 90 degrees. This is a wonderful figure, since many facts in geometry are connected with it.
It also has its own characteristics in the designation. To show a right angle in the figure, it is denoted not by an arc, but by a square.
Angles that are obtained by dividing a straight line by an arbitrary ray are called acute. Logically, it follows that an acute angle is less than a right angle, but its measure is different from 0 degrees. That is, it has a value from 0 to 90 degrees.
An obtuse angle is larger than a right angle, but smaller than a straight angle. Its degree measure varies from 90 to 180 degrees.
This element can be divided into different types of figures under consideration, excluding the expanded one.
Regardless of how a non-rotated angle is divided, the basic axiom of planimetry is always used - “the basic property of measurement.”
At dividing an angle with one beam or several, the degree measure of a given figure is equal to the sum of the measures of the angles into which it is divided.
At the 7th grade level, the types of angles according to their size end there. But to increase erudition, we can add that there are other varieties that have a degree measure greater than 180 degrees. They are called convex.
Figures at the intersection of lines
The next types of angles that students are introduced to are elements formed by the intersection of two straight lines. Figures that are placed opposite each other are called vertical. Their distinctive feature is that they are equal.
Elements that are adjacent to the same line are called adjacent. The theorem reflecting their property says that adjacent angles add up to 180 degrees.
Elements in a triangle
If we consider a figure as an element in a triangle, then the angles are divided into internal and external. A triangle is bounded by three segments and consists of three vertices. The angles located inside the triangle at each vertex are called internal.
If we take any internal element at any vertex and extend any side, then the angle that is formed and is adjacent to the internal one is called external. This pair of elements has the following property: their sum is equal to 180 degrees.
Intersection of two straight lines
Intersection of lines
When two straight lines intersect with a transversal, angles are also formed., which are usually distributed in pairs. Each pair of elements has its own name. It looks like this:
- internal crosswise lying: ∟4 and ∟6, ∟3 and ∟5;
- internal one-sided: ∟4 and ∟5, ∟3 and ∟6;
- corresponding: ∟1 and ∟5, ∟2 and ∟6, ∟4 and ∟8, ∟3 and ∟7.
In the case when a secant intersects two lines, all these pairs of angles have certain properties:
- Internal crosswise lying and corresponding figures are equal to each other.
- Internal one-way elements add up to 180 degrees.
We study angles in geometry, their properties
Types of angles in mathematics
Conclusion
This article presents all the main types of angles that are found in planimetry and are studied in the seventh grade. In all subsequent courses, the properties relating to all the elements considered are the basis for further study of geometry. For example, when studying, you will need to remember all the properties of the angles formed when two parallel lines intersect with a transversal. When studying the features of triangles, it is necessary to remember what adjacent angles are. Moving to stereometry, all volumetric figures will be studied and constructed based on planimetric figures.
Each angle, depending on its size, has its own name:
| Angle type | Size in degrees | Example |
|---|---|---|
| Spicy | Less than 90° | |
| Direct | Equal to 90°. In a drawing, a right angle is usually denoted by a symbol drawn from one side of the angle to the other. |
 |
| Blunt | More than 90° but less than 180° |  |
| Expanded | Equal to 180° A straight angle is equal to the sum of two right angles, and a right angle is half of a straight angle. |
 |
| Convex | More than 180° but less than 360° |  |
| Full | Equal to 360° |  |
The two angles are called adjacent, if they have one side in common, and the other two sides form a straight line:
Angles MOP And PON adjacent, since the beam OP- the common side, and the other two sides - OM And ON make up a straight line.
The common side of adjacent angles is called oblique to straight, on which the other two sides lie, only in the case when adjacent angles are not equal to each other. If adjacent angles are equal, then their common side will be perpendicular.
The sum of adjacent angles is 180°.
The two angles are called vertical, if the sides of one angle complement the sides of the other angle to straight lines:
Angles 1 and 3, as well as angles 2 and 4, are vertical.
Vertical angles are equal.
Let us prove that the vertical angles are equal:
The sum of ∠1 and ∠2 is a straight angle. And the sum of ∠3 and ∠2 is a straight angle. So these two amounts are equal:
∠1 + ∠2 = ∠3 + ∠2.
In this equality, there is an identical term on the left and right - ∠2. Equality will not be violated if this term on the left and right is omitted. Then we get it.
The most widely known and easiest to use tool for measuring angles is the protractor. In order to measure a plane angle with it, you need to align the central hole of the protractor with the vertex of the angle, and the zero division with one of its sides. The division value that the second side of the angle will intersect will be the magnitude of the angle. This way you can measure angles up to 180 degrees. If you need to measure an angle greater than 180 degrees, it is enough to measure the angle, its sides and vertex and its complement to 360 degrees (full angle), and then subtract the measured value from 360 degrees. The resulting value will be the value of the desired angle.Rulers. Bradis tables
To measure the value of a plane angle, it is enough to add another side to the angle so that a right triangle is formed. By measuring the sides of the resulting triangle, you can obtain the value of any trigonometric function of the angle whose value you need to know. Knowing the value of the sine, cosine, tangent or cotangent of an angle, you can use the Bradis table to find out the size of the angle.There are certain known angles that can be measured using a school square ruler. They produce two types of such rulers, both types are right-angled triangles made of wood, plastic or metal. The first type of square is an isosceles right triangle, two angles of which measure 45 degrees. The second type is a right triangle, one of the angles of which is 30 degrees, and the second is 60 degrees, respectively. By aligning one of the vertices of the square with the vertex of the angle - with the side of the angle, when the other side of the angle coincides with the adjacent side of the square, you can find the corresponding value of the angle. Thus, using rulers-gons you can find angles of 30, 45, 60 and 90 degrees.