Types of angles for measuring angles. Corner

Measure angle- means finding its magnitude. The magnitude of the angle shows how many times the angle chosen for the unit of measurement fits into a given angle.

Typically, the unit of measurement for angles is a degree. Degree- this is an angle equal to part of a straight angle. To indicate degrees in the text, the ° sign is used, which is placed in the upper right corner of the number indicating the number of degrees (for example, 60°).

Measuring angles with a protractor

To measure angles, a special device is used - protractor:

The protractor has two scales - internal and external. The reference point for the internal and external scales is located on different sides. To get the correct measurement result, the degree count must start from the correct side.

Angles are measured as follows: the protractor is placed on the angle so that the top of the angle coincides with the center of the protractor, and one of the sides of the angle passes through the zero division on the scale. Then the other side of the angle will indicate the size of the angle in degrees:

They say: corner BOC equals 60 degrees, angle MON is equal to 120 degrees and write: ∠ BOC= 60°, ∠ MON= 120°.

For more precise measurement Angles use fractions of a degree: minutes and seconds. Minute is an angle equal to part of a degree. Second is an angle equal to a fraction of a minute. Minutes are indicated by " , a seconds - sign "" . The minutes and seconds sign is placed in the upper right corner of the number. For example, if the angle is 50 degrees 34 minutes and 19 seconds, then write:

50°34 " 19""

Angle Measurement Properties

If a ray divides a given angle into two parts (two angles), then the value of this angle is equal to the sum of the values ​​of the two resulting angles.

The most widely known and easiest to use tool for measuring angles is the protractor. In order to use it to measure a plane angle, it is necessary 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 flat angle, it is enough to add one more side to the angle so that a right triangle. By measuring the sides of the resulting triangle, you can get the value of any trigonometric function the angle whose value needs to be known. 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.

Theodolite

The tools listed in the previous paragraphs are used to measure angles on a plane. In practice - in construction, topography - a special device is used to measure the so-called horizontal and vertical angles called a theodolite. The main measuring elements of a theodolite are special cylindrical rings (limbs), on which degree markings are evenly applied. Installed using a special stand at the top of the corner, the device is aimed using a telescope, first at a point located on one side of the corner where the measurement is made, then on the other side of the corner, and the measurement is taken again. The difference in measurements determines the angle in the first half-step. Then the second half-reception is performed - in the opposite direction. The arithmetic mean of the values ​​obtained in two half-steps is the value of the measured angle.

Angle between straight lines. Perpendicular lines.

Adjacent and vertical angles and their properties. Angle bisector.

Angle is geometric figure formed by two rays emanating from one point.

Angle units: radian and degree. A degree is an angle equal to 1/360 of a full angle. One degree is divided into 60 minutes (symbol: 1 0 = 60"); one minute is divided into 60 seconds (symbol: 1" = 60").

An angle of 90 0 is called a right angle; an angle less than 90 0 is called acute; An angle greater than 90 0 is called obtuse.

Two lines are called mutually perpendicular if they form a right angle when they intersect. If lines AB and MK are perpendicular, then this is denoted: AB MK.

Two angles are called adjacent if they have one side in common, and the other two sides are continuations of one another. Thus, the sum of adjacent angles is 180 0.

Two angles with a common vertex, in which the sides of one are continuations of the sides of the other, are called vertical. Vertical angles are equal.

The bisector of an angle is the ray that bisects the angle.

Angle bisector property: Each point of an angle bisector is the same distance from the sides of that angle.

Z struggling with a solution.

1. Find the values ​​of adjacent angles if one of them is 20 0 greater than the other.

Let's designate one of the corners as X, then the second one will be equal X+20 0. Since the angles are adjacent, their sum is 180 0.

We get the equation X+(X+20 0)= 180 0. Then 2 X=160 0 , X=80 0 .

80 0 +20 0 =100 0

Answer: 80 0 and 100 0

2. . Two adjacent angles are given. one of these angles and the other add up to a right angle. Find these adjacent angles.

Let's designate one of the corners as X. Since the angles are adjacent, their sum is 180 0. Then the second angle will be equal to 180 0 - X.

Let's make an equation: X + (180 0 – X) =90 0 .

Let's multiply both sides of the equation by the common denominator of the fractions 28.

We get: 16 X +7(180 0 – X)= 28·90 0

16X+ 7·180 0 – 7 X= 28·90 0

9X= -7 180 0 + 28 90 0 Divide both sides of the equation by 9.

X= –7·20 0 + 28·10 0

X= –140 0 + 280 0

X= 140 0 - the first angle, then the second is equal to 180 0 – 140 0 =40 0.

Answer: 140 0 and 40 0

3. The sum of three angles formed by the intersection of two straight lines is 280 0 greater than the fourth angle. Find these four corners.

When two straight lines intersect, two pairs of vertical angles are formed. Vertical angles are equal to each other.

Let X– the size of one of the angles. Then the angle adjacent to it will be equal to 180 0 - X. We have four corners: X, X, 180 0 – X, 180 0 – X.

Let's make an equation: X+ X+ (180 0 – X) = (180 0 – X)+ 280 0 .

We get 2 X=280 0 , X=140 0 , 180 0 – 140 0 =40 0

Answer: 140 0, 40 0, 140 0 and 40 0

4. The angle between lines a and b is equal to 17 0, and the angle between lines a and c is equal to 33 0

Find the angle between lines b and c.

A)

then (b,^c)= 17 0 + 33 0 =50 0

b)

then (b,^c)= 33 0 - 17 0 =16 0

Answer: 50 0 or 16 0

5. Segments MR and OK intersect at point E. One of the angles at vertex E is equal to 110º. Find the angle KES, where EN is the bisector of the angle REC.

There are two possible options for the problem:

A)

then Ð KES= 70 0: 2 =35 0

b)

When two beam (A.O. And O.B.) come from one point, then the figure formed by these rays (together with the part of the plane limited by them) is called angle

The rays forming an angle are called parties. The point from which they originate is top corner.

Sides of the corner should be imagined as infinitely extended from the top.

Corner usually denoted by three letters, of which the middle one is placed at peaks, and the extreme ones are at some points of the sides. For example, they say “angle AOB or angle SAI" But you can denote an angle with one letter placed at the vertex, if there are no other angles at this vertex. We will sometimes denote an angle by a number placed inside the angle at the vertex. The word “angle” in writing is often replaced by the sign / .

When two rays come from one points, then strictly they say that they form not one angle, but two angles.

These two angles are equal to each other only if the rays A.O. And O.B. constitute one direct .

This angle is called turned angle.

Two angles count equal angles , if when superimposed they can be combined.

We take it as obvious that inside any angle, from its vertex it is possible to draw a ray (and only one) that divides this angle in half. Such a beam is called angle bisector .

Two corners ( A.O.B And BOC) are called adjacent, if they have one side in common, and the other two sides are straight line.

Damn 1. Damn 2

When two adjacent angle are equal (Fig. 2), then common side their O.B. called perpendicular to a straight line A.C., on which the other sides lie.

If adjacent angles are unequal (Fig. 1), then the common side O.B. called inclined To A.C..

In both cases, point O called basis(perpendicular or oblique).

From any point on a straight line you can, on either side of this straight line, restore to it perpendicular and only one at that .

Each of the equal adjacent angles is called direct. A right angle is constant a value equal to 90 0 (it is usually denoted by the sign d, i.e. the initial letter of the French word “droit” - straight). As a result, ordinary angles are compared in size to a right angle.

Any expanded angle is 2 d= 180°.

Every corner ( AOC), smaller right angle (AOB) is called sharp.

Every corner ( AOD) the larger direct one is called stupid.

An angle is a geometric figure that consists of two different rays emanating from one point. IN in this case, these rays are called sides of the angle. The point that is the beginning of the rays is called the vertex of the angle. In the picture you can see the angle with the vertex at the point ABOUT, and the parties k And m.

Points A and C are marked on the sides of the angle. This angle can be designated as angle AOC. In the middle there must be the name of the point at which the vertex of the angle is located. There are also other designations, angle O or angle km. In geometry, instead of the word angle, a special symbol is often written.

Developed and non-expanded angle

If both sides of an angle lie on the same straight line, then such an angle is called expanded angle. That is, one side of the angle is a continuation of the other side of the angle. The figure below shows the expanded angle O.

It should be noted that any angle divides the plane into two parts. If the angle is not unfolded, then one of the parts is called the internal region of the angle, and the other is called the external region of this angle. The figure below shows an undeveloped angle and marks the outer and inner regions of this angle.

In the case of a developed angle, either of the two parts into which it divides the plane can be considered the outer region of the angle. We can talk about the position of a point relative to an angle. A point can lie outside the corner (in the outer region), can be located on one of its sides, or can lie inside the corner (in the inner region).

In the figure below, point A lies outside angle O, point B lies on one side of the angle, and point C lies inside the angle.

Measuring angles

To measure angles there is a device called a protractor. The unit of angle is degree. It should be noted that each angle has a certain degree measure, which is greater than zero.

Depending on the degree measure, angles are divided into several groups.