Adjacent angles in a right triangle. What angles are called adjacent what is the sum of adjacent angles

How to find an adjacent angle?

Mathematics is the oldest exact science, which is mandatory studied in schools, colleges, institutes and universities. However, basic knowledge is always laid down at school. Sometimes, the child is given quite difficult tasks, and the parents are unable to help, because they simply forgot some things from mathematics. For example, how to find an adjacent angle by the value of the main angle, etc. The task is simple, but it can be difficult to solve due to not knowing which angles are called adjacent and how to find them.

Let's take a closer look at the definition and properties of adjacent corners, as well as how to calculate them from the data in the problem.

Definition and properties of adjacent corners

Two rays emanating from the same point form a figure called a "flat angle". In this case, this point is called the vertex of the angle, and the rays are its sides. If one of the rays is continued further than the starting point along a straight line, then another angle is formed, which is called adjacent. Each angle in this case has two adjacent angles, since the sides of the angle are equivalent. That is, there is always an adjacent angle of 180 degrees.

The main properties of adjacent angles include

Adjacent corners have a common vertex and one side;
The sum of adjacent angles is always 180 degrees, or pi if the calculation is in radians;
The sines of adjacent angles are always equal;
The cosines and tangents of adjacent angles are equal but have opposite signs.

How to find adjacent corners

Usually three variations of problems are given for finding the value of adjacent angles

The value of the main angle is given;
The ratio of the main and adjacent angle is given;
The value of the vertical angle is given.

Each version of the problem has its own solution. Let's consider them.

Given the value of the main angle

If the value of the main angle is indicated in the problem, then finding the adjacent angle is very simple. To do this, it is enough to subtract the value of the main angle from 180 degrees, and you will get the value of the adjacent angle. This solution comes from the property of an adjacent angle - the sum of adjacent angles is always 180 degrees.

If the value of the main angle is given in radians and in the problem it is required to find the adjacent angle in radians, then it is necessary to subtract the value of the main angle from the number Pi, since the value of the full angle of 180 degrees is equal to the number Pi.

Given the ratio of the main and adjacent angle

In the problem, the ratio of the main and adjacent angle can be given instead of degrees and radians of the magnitude of the main angle. In this case, the solution will look like an equation of proportion:

We denote the proportion of the proportion of the main angle as the variable "Y".
The proportion related to the adjacent corner is denoted as the variable "X".
The number of degrees that fall on each proportion, we denote, for example, "a".
The general formula will look like this - a*X+a*Y=180 or a*(X+Y)=180.
We find the common factor of the equation "a" by the formula a=180/(X+Y).
Then we multiply the obtained value of the common factor "a" by the fraction of the angle that needs to be determined.

This way we can find the value of the adjacent angle in degrees. However, if you need to find the value in radians, then you just need to convert degrees to radians. To do this, multiply the angle in degrees by pi and divide by 180 degrees. The resulting value will be in radians.

Given the value of the vertical angle

If the value of the main angle is not given in the problem, but the value of the vertical angle is given, then the adjacent angle can be calculated using the same formula as in the first paragraph, where the value of the main angle is given.

A vertical angle is an angle that comes from the same point as the main one, but at the same time it is directed in exactly the opposite direction. This results in a mirror image. This means that the vertical angle is equal in magnitude to the main one. In turn, the adjacent angle of the vertical angle is equal to the adjacent angle of the main angle. Thanks to this, it is possible to calculate the adjacent angle of the main angle. To do this, simply subtract the value of the vertical from 180 degrees and get the value of the adjacent angle of the main angle in degrees.

If the value is given in radians, then it is necessary to subtract the value of the vertical angle from the number Pi, since the value of the full angle of 180 degrees is equal to the number Pi.

You can also read our helpful articles and.

1. Adjacent corners.

If we continue the side of some angle beyond its vertex, we get two angles (Fig. 72): ∠ABC and ∠CBD, in which one side of BC is common, and the other two, AB and BD, form a straight line.

Two angles that have one side in common and the other two form a straight line are called adjacent angles.

Adjacent angles can also be obtained in this way: if we draw a ray from some point on a straight line (not lying on a given straight line), then we get adjacent angles.

For example, ∠ADF and ∠FDВ are adjacent angles (Fig. 73).

Adjacent corners can have a wide variety of positions (Fig. 74).

Adjacent angles add up to a straight angle, so the sum of two adjacent angles is 180°

Hence, a right angle can be defined as an angle equal to its adjacent angle.

Knowing the value of one of the adjacent angles, we can find the value of the other adjacent angle.

For example, if one of the adjacent angles is 54°, then the second angle will be:

180° - 54° = l26°.

2. Vertical angles.

If we extend the sides of an angle beyond its vertex, we get vertical angles. In Figure 75, the angles EOF and AOC are vertical; angles AOE and COF are also vertical.

Two angles are called vertical if the sides of one angle are extensions of the sides of the other angle.

Let ∠1 = \(\frac(7)(8)\) ⋅ 90° (Fig. 76). ∠2 adjacent to it will be equal to 180° - \(\frac(7)(8)\) ⋅ 90°, i.e. 1\(\frac(1)(8)\) ⋅ 90°.

In the same way, you can calculate what ∠3 and ∠4 are.

∠3 = 180° - 1\(\frac(1)(8)\) ⋅ 90° = \(\frac(7)(8)\) ⋅ 90°;

∠4 = 180° - \(\frac(7)(8)\) ⋅ 90° = 1\(\frac(1)(8)\) ⋅ 90° (Fig. 77).

We see that ∠1 = ∠3 and ∠2 = ∠4.

You can solve several more of the same problems, and each time you get the same result: the vertical angles are equal to each other.

However, to make sure that the vertical angles are always equal to each other, it is not enough to consider individual numerical examples, since conclusions drawn from particular examples can sometimes be erroneous.

It is necessary to verify the validity of the property of vertical angles by proof.

The proof can be carried out as follows (Fig. 78):

∠a +∠c= 180°;

∠b+∠c= 180°;

(since the sum of adjacent angles is 180°).

∠a +∠c = ∠b+∠c

(since the left side of this equality is 180°, and its right side is also 180°).

This equality includes the same angle with.

If we subtract equally from equal values, then it will remain equally. The result will be: ∠a = ∠b, i.e., the vertical angles are equal to each other.

3. The sum of angles that have a common vertex.

In drawing 79, ∠1, ∠2, ∠3 and ∠4 are located on the same side of the line and have a common vertex on this line. In sum, these angles make up a straight angle, i.e.

∠1 + ∠2 + ∠3 + ∠4 = 180°.

In drawing 80 ∠1, ∠2, ∠3, ∠4 and ∠5 have a common vertex. These angles add up to a full angle, i.e. ∠1 + ∠2 + ∠3 + ∠4 + ∠5 = 360°.

Other materials

Two angles are called adjacent if they have one side in common and the other sides of these angles are complementary rays. In figure 20, the angles AOB and BOC are adjacent.

The sum of adjacent angles is 180°

Theorem 1. The sum of adjacent angles is 180°.

Proof. The OB beam (see Fig. 1) passes between the sides of the developed angle. So ∠ AOB + ∠ BOC = 180°.

From Theorem 1 it follows that if two angles are equal, then the angles adjacent to them are equal.

Vertical angles are equal

Two angles are called vertical if the sides of one angle are complementary rays of the sides of the other. The angles AOB and COD, BOD and AOC, formed at the intersection of two straight lines, are vertical (Fig. 2).

Theorem 2. Vertical angles are equal.

Proof. Consider the vertical angles AOB and COD (see Fig. 2). Angle BOD is adjacent to each of the angles AOB and COD. By Theorem 1, ∠ AOB + ∠ BOD = 180°, ∠ COD + ∠ BOD = 180°.

Hence we conclude that ∠ AOB = ∠ COD.

Corollary 1. An angle adjacent to a right angle is a right angle.

Consider two intersecting straight lines AC and BD (Fig. 3). They form four corners. If one of them is right (angle 1 in Fig. 3), then the other angles are also right (angles 1 and 2, 1 and 4 are adjacent, angles 1 and 3 are vertical). In this case, these lines are said to intersect at right angles and are called perpendicular (or mutually perpendicular). The perpendicularity of lines AC and BD is denoted as follows: AC ⊥ BD.

The perpendicular bisector of a segment is a line perpendicular to this segment and passing through its midpoint.

AN - perpendicular to the line

Consider a line a and a point A not lying on it (Fig. 4). Connect the point A with a segment to the point H with a straight line a. A segment AH is called a perpendicular drawn from point A to line a if lines AN and a are perpendicular. The point H is called the base of the perpendicular.

Drawing square

The following theorem is true.

Theorem 3. From any point that does not lie on a line, one can draw a perpendicular to this line, and moreover, only one.

To draw a perpendicular from a point to a straight line in the drawing, a drawing square is used (Fig. 5).

Comment. The statement of the theorem usually consists of two parts. One part talks about what is given. This part is called the condition of the theorem. The other part talks about what needs to be proven. This part is called the conclusion of the theorem. For example, the condition of Theorem 2 is vertical angles; conclusion - these angles are equal.

Any theorem can be expressed in detail in words so that its condition will begin with the word “if”, and the conclusion with the word “then”. For example, Theorem 2 can be stated in detail as follows: "If two angles are vertical, then they are equal."

Example 1 One of the adjacent angles is 44°. What is the other equal to?

Decision. Denote the degree measure of another angle by x, then according to Theorem 1.
44° + x = 180°.
Solving the resulting equation, we find that x \u003d 136 °. Therefore, the other angle is 136°.

Example 2 Let the COD angle in Figure 21 be 45°. What are angles AOB and AOC?

Decision. The angles COD and AOB are vertical, therefore, by Theorem 1.2 they are equal, i.e., ∠ AOB = 45°. The angle AOC is adjacent to the angle COD, hence, by Theorem 1.
∠ AOC = 180° - ∠ COD = 180° - 45° = 135°.

Example 3 Find adjacent angles if one of them is 3 times the other.

Decision. Denote the degree measure of the smaller angle by x. Then the degree measure of the larger angle will be Zx. Since the sum of adjacent angles is 180° (Theorem 1), then x + 3x = 180°, whence x = 45°.
So the adjacent angles are 45° and 135°.

Example 4 The sum of two vertical angles is 100°. Find the value of each of the four angles.

Decision. Let figure 2 correspond to the condition of the problem. The vertical angles COD to AOB are equal (Theorem 2), which means that their degree measures are also equal. Therefore, ∠ COD = ∠ AOB = 50° (their sum is 100° by condition). The angle BOD (also the angle AOC) is adjacent to the angle COD, and, therefore, by Theorem 1
∠ BOD = ∠ AOC = 180° - 50° = 130°.

Question 1. What angles are called adjacent?
Answer. Two angles are called adjacent if they have one side in common and the other sides of these angles are complementary half-lines.
In figure 31, the corners (a 1 b) and (a 2 b) are adjacent. They have a common side b, and sides a 1 and a 2 are additional half-lines.

Question 2. Prove that the sum of adjacent angles is 180°.
Answer. Theorem 2.1. The sum of adjacent angles is 180°.
Proof. Let the angle (a 1 b) and the angle (a 2 b) be given adjacent angles (see Fig. 31). The beam b passes between the sides a 1 and a 2 of the developed angle. Therefore, the sum of the angles (a 1 b) and (a 2 b) is equal to the developed angle, i.e. 180 °. Q.E.D.

Question 3. Prove that if two angles are equal, then the angles adjacent to them are also equal.
Answer.

From the theorem 2.1 It follows that if two angles are equal, then the angles adjacent to them are equal.
Let's say the angles (a 1 b) and (c 1 d) are equal. We need to prove that the angles (a 2 b) and (c 2 d) are also equal.
The sum of adjacent angles is 180°. It follows from this that a 1 b + a 2 b = 180° and c 1 d + c 2 d = 180°. Hence, a 2 b \u003d 180 ° - a 1 b and c 2 d \u003d 180 ° - c 1 d. Since the angles (a 1 b) and (c 1 d) are equal, we get that a 2 b \u003d 180 ° - a 1 b \u003d c 2 d. By the property of transitivity of the equal sign, it follows that a 2 b = c 2 d. Q.E.D.

Question 4. What angle is called right (acute, obtuse)?
Answer. An angle equal to 90° is called a right angle.
An angle less than 90° is called an acute angle.
An angle greater than 90° and less than 180° is called an obtuse angle.

Question 5. Prove that an angle adjacent to a right angle is a right angle.
Answer. From the theorem on the sum of adjacent angles it follows that the angle adjacent to a right angle is a right angle: x + 90° = 180°, x= 180° - 90°, x = 90°.

Question 6. What are the vertical angles?
Answer. Two angles are called vertical if the sides of one angle are the complementary half-lines of the sides of the other.

Question 7. Prove that the vertical angles are equal.
Answer. Theorem 2.2. Vertical angles are equal.
Proof. Let (a 1 b 1) and (a 2 b 2) be given vertical angles (Fig. 34). The corner (a 1 b 2) is adjacent to the corner (a 1 b 1) and to the corner (a 2 b 2). From here, by the theorem on the sum of adjacent angles, we conclude that each of the angles (a 1 b 1) and (a 2 b 2) complements the angle (a 1 b 2) up to 180 °, i.e. the angles (a 1 b 1) and (a 2 b 2) are equal. Q.E.D.

Question 8. Prove that if at the intersection of two lines one of the angles is a right angle, then the other three angles are also right.
Answer. Assume that lines AB and CD intersect each other at point O. Assume that angle AOD is 90°. Since the sum of adjacent angles is 180°, we get that AOC = 180°-AOD = 180°- 90°=90°. The COB angle is vertical to the AOD angle, so they are equal. That is, the angle COB = 90°. COA is vertical to BOD, so they are equal. That is, the angle BOD = 90°. Thus, all angles are equal to 90 °, that is, they are all right. Q.E.D.

Question 9. Which lines are called perpendicular? What sign is used to indicate perpendicularity of lines?
Answer. Two lines are called perpendicular if they intersect at a right angle.
The perpendicularity of lines is denoted by \(\perp\). The entry \(a\perp b\) reads: "Line a is perpendicular to line b".

Question 10. Prove that through any point of a line one can draw a line perpendicular to it, and only one.
Answer. Theorem 2.3. Through each line, you can draw a line perpendicular to it, and only one.
Proof. Let a be a given line and A be a given point on it. Denote by a 1 one of the half-lines by the straight line a with the starting point A (Fig. 38). Set aside from the half-line a 1 the angle (a 1 b 1) equal to 90 °. Then the line containing the ray b 1 will be perpendicular to the line a.

Assume that there is another line that also passes through the point A and is perpendicular to the line a. Denote by c 1 the half-line of this line lying in the same half-plane with the ray b 1 .
Angles (a 1 b 1) and (a 1 c 1), equal to 90° each, are laid out in one half-plane from the half-line a 1 . But from the half-line a 1, only one angle equal to 90 ° can be set aside in this half-plane. Therefore, there cannot be another line passing through the point A and perpendicular to the line a. The theorem has been proven.

Question 11. What is a perpendicular to a line?
Answer. A perpendicular to a given line is a line segment perpendicular to the given one, which has one of its ends at their intersection point. This end of the segment is called basis perpendicular.

Question 12. Explain what proof by contradiction is.
Answer. The method of proof that we used in Theorem 2.3 is called proof by contradiction. This way of proof consists in that we first make an assumption opposite to what is stated by the theorem. Then, by reasoning, relying on the axioms and proven theorems, we come to a conclusion that contradicts either the condition of the theorem, or one of the axioms, or the previously proven theorem. On this basis, we conclude that our assumption was wrong, which means that the assertion of the theorem is true.

Question 13. What is an angle bisector?
Answer. The bisector of an angle is a ray that comes from the vertex of the angle, passes between its sides and divides the angle in half.

The known value of the main angle α₁ = α₂ = 180°-α.

From this there are . If two angles are both adjacent and equal at the same time, then they are right angles. If one of the adjacent angles is right, that is, it is 90 degrees, then the other angle is also right. If one of the adjacent angles is acute, then the other will be obtuse. Similarly, if one of the angles is obtuse, then the second, respectively, will be acute.

An acute angle is one whose measure is less than 90 degrees but greater than 0. An obtuse angle has a measure greater than 90 degrees but less than 180.

Another property of adjacent angles is formulated as follows: if two angles are equal, then the angles adjacent to them are also equal. This is that if there are two angles for which the degree measure is the same (for example, it is 50 degrees) and at the same time one of them has an adjacent angle, then the values \u200b\u200bof these adjacent angles also coincide (in the example, their degree measure will be 130 degrees).

Sources:

Big Encyclopedic Dictionary - Adjacent corners
180 degree angle

The word "" has various interpretations. In geometry, an angle is a part of a plane bounded by two rays coming out of one point - a vertex. When it comes to straight, sharp, developed angles, it is geometric angles that are meant.

Like any shape in geometry, angles can be compared. The equality of angles is determined by movement. An angle is easy to divide into two equal parts. Dividing into three parts is a little more difficult, but it can still be done with a ruler and compass. By the way, this task seemed quite difficult. It is geometrically easy to describe that one angle is greater or less than another.

The unit of measurement for angles is 1/180 of the expanded angle. The angle value is a number showing how many times the angle chosen as a unit of measurement fits into the figure in question.

Each angle has a degree measure greater than zero. The straight angle is 180 degrees. The degree measure of an angle is considered to be equal to the sum of the degree measures of the angles into which it is divided by any ray on the plane bounded by its sides.

From any ray to a given plane, you can set aside an angle with a certain degree measure not exceeding 180 . Moreover, there will be only one such angle. The measure of a flat angle, which is part of a half-plane, is the degree measure of an angle with similar sides. The measure of the plane of the angle containing the half-plane is the value 360 – α, where α is the degree measure of the complementary flat angle.

The degree measure of an angle makes it possible to move from their geometric description to a numerical one. So, a right angle is understood as an angle equal to 90 degrees, an obtuse angle is an angle less than 180 degrees, but more than 90, an acute angle does not exceed 90 degrees.

In addition to degrees, there is a radian measure of an angle. In planimetry, the length is L, the radius is r, and the corresponding central angle is α. Moreover, these parameters are related by the relation α = L/r. This is the basis of the radian measure of angles. If L=r, then the angle α will be equal to one radian. So, the radian measure of an angle is the ratio of the length of an arc drawn by an arbitrary radius and enclosed between the sides of this angle to the radius of the arc. A complete rotation in degrees (360 degrees) corresponds to 2π in radians. One is 57.2958 degrees.