Angles in which one side is common, and the other sides lie on the same straight line (in the figure, angles 1 and 2 are adjacent). Rice. to Art. Adjacent corners... Great Soviet Encyclopedia
ADJACENT CORNERS- angles that have a common vertex and one common side, and their other two sides lie on the same straight line... Big Polytechnic Encyclopedia
See Angle... Big Encyclopedic Dictionary
ADJACENT ANGLES, two angles whose sum is 180°. Each of these angles complements the other to the full angle... Scientific and technical encyclopedic dictionary
See Angle. * * * ADJACENT CORNERS ADJACENT CORNERS, see Angle (see ANGLE) ... encyclopedic Dictionary
- (Angles adjacents) those that have a common vertex and a common side. Mostly this name refers to such C. angles, the other two sides of which lie in opposite directions of one straight line drawn through the vertex ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron
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Two straight lines intersect to create a pair of vertical angles. One pair consists of angles A and B, the other of C and D. In geometry, two angles are called vertical if they are created by the intersection of two ... Wikipedia
A pair of complementary angles that complement each other up to 90 degrees. Complementary angles are a pair of angles that complement each other up to 90 degrees. If two complementary angles are adjacent (i.e. they have a common vertex and are separated only... ... Wikipedia
A pair of complementary angles that complement each other up to 90 degrees Complementary angles are a pair of angles that complement each other up to 90 degrees. If two complementary angles are with... Wikipedia
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- About proof in geometry, A.I. Fetisov. This book will be produced in accordance with your order using Print-on-Demand technology. Once upon a time, at the very beginning school year, I had to hear a conversation between two girls. The eldest of them...
- A comprehensive notebook for knowledge control. Geometry. 7th grade. Federal State Educational Standard, Babenko Svetlana Pavlovna, Markova Irina Sergeevna. The manual presents control and measurement materials (CMM) in geometry for conducting current, thematic and final quality control of knowledge of 7th grade students. Contents of the manual...
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, angles AOB and BOC are adjacent.
The sum of adjacent angles is 180°
Theorem 1. The sum of adjacent angles is 180°.
Proof. Beam OB (see Fig. 1) passes between the sides of the unfolded angle. That's why ∠ AOB + ∠ BOS = 180°.
From Theorem 1 it follows that if two angles are equal, then their adjacent angles 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. Let's consider vertical angles AOB and COD (see Fig. 2). Angle BOD is adjacent to each of angles AOB and COD. By Theorem 1 ∠ AOB + ∠ BOD = 180°, ∠ COD + ∠ BOD = 180°.
From this 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 straight (angle 1 in Fig. 3), then the remaining angles are also right (angles 1 and 2, 1 and 4 are adjacent, angles 1 and 3 are vertical). In this case, they say that these lines intersect at right angles and are called perpendicular (or mutually perpendicular). The perpendicularity of lines AC and BD is denoted as follows: AC ⊥ BD.
A perpendicular bisector to a segment is a line perpendicular to this segment and passing through its midpoint.
AN - perpendicular to a line
Consider a straight line a and a point A not lying on it (Fig. 4). Let's connect point A with a segment to point H with straight line a. The segment AN is called a perpendicular drawn from point A to line a if lines AN and a are perpendicular. Point H is called the base of the perpendicular.
Drawing square
The following theorem is true.
Theorem 3. From any point not lying on a line, it is possible to draw a perpendicular to this line, and, moreover, only one.
To draw a perpendicular from a point to a straight line in a drawing, use a drawing square (Fig. 5).
Comment. The formulation 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 that the angles are vertical; conclusion - these angles are equal.
Any theorem can be expressed in detail in words so that its condition begins with the word “if” and its 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?
Solution.
Let us 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 = 136°. Therefore, the other angle is 136°.
Example 2. Let the angle COD in Figure 21 be 45°. What are the angles AOB and AOC?
Solution.
Angles COD and AOB are vertical, therefore, by Theorem 1.2 they are equal, i.e. ∠ AOB = 45°. Angle AOC is adjacent to angle COD, which means according to Theorem 1.
∠ AOC = 180° - ∠ COD = 180° - 45° = 135°.
Example 3. Find adjacent angles, if one of them is 3 times larger than the other.
Solution.
Let us denote the degree measure of the smaller angle by x. Then the degree measure of the larger angle will be 3x. Since the sum of adjacent angles is equal to 180° (Theorem 1), then x + 3x = 180°, whence x = 45°.
This means that adjacent angles are 45° and 135°.
Example 4. The sum of two vertical angles is 100°. Find the size of each of the four angles.
Solution.
Let Figure 2 meet the conditions 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 according to the condition is 100°). Angle BOD (also angle AOC) is adjacent to angle COD, and therefore, by Theorem 1
∠ BOD = ∠ AOC = 180° - 50° = 130°.
1. Adjacent angles.
If we extend the side of any angle beyond its vertex, we get two angles (Fig. 72): ∠ABC and ∠CBD, in which one side BC is common, and the other two, AB and BD, form a straight line.
Two angles in which one side is 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 line (not lying on a given line), we will obtain adjacent angles.
For example, ∠ADF and ∠FDB are adjacent angles (Fig. 73).
Adjacent angles 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 size of one of the adjacent angles, we can find the size of the other angle adjacent to it.
For example, if one of the adjacent angles is 54°, then the second angle will be equal to:
180° - 54° = l26°.
2. Vertical angles.
If we extend the sides of the 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 continuations 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 equal to.
∠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 will get the same result: the vertical angles are equal to each other.
However, to make sure that 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 properties 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 equal to 180°, and its right side is also equal to 180°).
This equality includes the same angle With.
If we subtract equal amounts from equal quantities, then equal amounts will remain. 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 one side of a 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 Figure 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 materialsGetting Started with Angles
Let us be given two arbitrary rays. Let's put them on top of each other. Then
Definition 1
We will call an angle two rays that have the same origin.
Definition 2
The point that is the beginning of the rays within the framework of Definition 3 is called the vertex of this angle.
We will denote the angle by its following three points: the vertex, a point on one of the rays and a point on the other ray, and the vertex of the angle is written in the middle of its designation (Fig. 1).
Let us now determine what the magnitude of the angle is.
To do this, we need to select some kind of “reference” angle, which we will take as a unit. Most often, this angle is the angle that is equal to the $\frac(1)(180)$ part of the unfolded angle. This quantity is called a degree. After choosing such an angle, we compare the angles with it, the value of which needs to be found.
There are 4 types of angles:
Definition 3
An angle is called acute if it is less than $90^0$.
Definition 4
An angle is called obtuse if it is greater than $90^0$.
Definition 5
An angle is called developed if it is equal to $180^0$.
Definition 6
An angle is called right if it is equal to $90^0$.
In addition to the types of angles described above, we can distinguish types of angles in relation to each other, namely vertical and adjacent angles.
Adjacent angles
Consider the reversed angle $COB$. From its vertex we draw a ray $OA$. This ray will split the original one into two angles. Then
Definition 7
We will call two angles adjacent if one pair of their sides is a developed angle, and the other pair coincides (Fig. 2).
IN in this case angles $COA$ and $BOA$ are adjacent.
Theorem 1
The sum of adjacent angles is $180^0$.
Proof.
Let's look at Figure 2.
By definition 7, the angle $COB$ in it will be equal to $180^0$. Since the second pair of sides of adjacent angles coincides, the ray $OA$ will divide the unfolded angle by 2, therefore
$∠COA+∠BOA=180^0$
The theorem has been proven.
Let's consider solving the problem using this concept.
Example 1
Find angle $C$ from the figure below
By Definition 7 we find that the angles $BDA$ and $ADC$ are adjacent. Therefore, by Theorem 1, we get
$∠BDA+∠ADC=180^0$
$∠ADC=180^0-∠BDA=180〗0-59^0=121^0$
By the theorem on the sum of angles in a triangle, we have
$∠A+∠ADC+∠C=180^0$
$∠C=180^0-∠A-∠ADC=180^0-19^0-121^0=40^0$
Answer: $40^0$.
Vertical angles
Consider the unfolded angles $AOB$ and $MOC$. Let's align their vertices with each other (that is, put the point $O"$ on the point $O$) so that no sides of these angles coincide. Then
Definition 8
We will call two angles vertical if the pairs of their sides are unfolded angles and their values coincide (Fig. 3).
In this case, the angles $MOA$ and $BOC$ are vertical and the angles $MOB$ and $AOC$ are also vertical.
Theorem 2
Vertical angles are equal to each other.
Proof.
Let's look at Figure 3. Let's prove, for example, that the angle $MOA$ is equal to the angle $BOC$.
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 angles (a 1 b) and (a 2 b) are adjacent. They have side b in common, 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 angle (a 1 b) and angle (a 2 b) be given adjacent angles (see Fig. 31). Ray b passes between sides a 1 and a 2 of a straight angle. Therefore, the sum of the angles (a 1 b) and (a 2 b) is equal to the unfolded angle, i.e. 180°. Q.E.D.
Question 3. Prove that if two angles are equal, then their adjacent angles are also equal.
Answer.
From the theorem 2.1
It follows that if two angles are equal, then their adjacent angles 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 = 180° - a 1 b and c 2 d = 180° - c 1 d. Since the angles (a 1 b) and (c 1 d) are equal, we get that a 2 b = 180° - a 1 b = 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 obtuse.
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 an angle adjacent to a right angle is a right angle: x + 90° = 180°, x = 180° - 90°, x = 90°.
Question 6. What angles are called vertical?
Answer. Two angles are called vertical if the sides of one angle are 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 the given vertical angles (Fig. 34). Angle (a 1 b 2) is adjacent to angle (a 1 b 1) and to angle (a 2 b 2). From here, using 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) to 180°, i.e. angles (a 1 b 1) and (a 2 b 2) are equal. Q.E.D.
Question 8. Prove that if, when two lines intersect, one of the angles is right, then the other three angles are also right.
Answer. Suppose lines AB and CD intersect each other at point O. Suppose angle AOD is 90°. Since the sum of adjacent angles is 180°, we get that AOC = 180° - AOD = 180° - 90° = 90°. Angle COB is vertical to angle AOD, so they are equal. That is, angle COB = 90°. Angle COA is vertical to angle BOD, so they are equal. That is, angle BOD = 90°. Thus, all angles are equal to 90°, that is, they are all right angles. 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 right angles.
The perpendicularity of lines is indicated by the sign \(\perp\). The entry \(a\perp b\) reads: “Line a is perpendicular to line b.”
Question 10. Prove that through any point on a line you 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 a given point on it. Let us denote by a 1 one of the half-lines of the straight line a with the starting point A (Fig. 38). Let us subtract an angle (a 1 b 1) equal to 90° from the half-line a 1. Then the straight line containing the ray b 1 will be perpendicular to the straight line a.
Let us assume that there is another line, also passing through point A and perpendicular to line a. Let us 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), each equal to 90°, 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 put into a given half-plane. Therefore, there cannot be another line passing through point A and perpendicular to line a. The theorem has been proven.
Question 11. What is perpendicular to a line?
Answer. A perpendicular to a given line is a segment of a line perpendicular to a given line, 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 consists of.
Answer. The proof method we used in Theorem 2.3 is called proof by contradiction. This method of proof consists of first making an assumption opposite to what the theorem states. Then, by reasoning, relying on axioms and proven theorems, we come to a conclusion that contradicts either the conditions of the theorem, or one of the axioms, or a previously proven theorem. On this basis, we conclude that our assumption was incorrect, and therefore the statement of the theorem is true.
Question 13. What is the bisector of an angle?
Answer. The bisector of an angle is a ray that emanates from the vertex of the angle, passes between its sides and divides the angle in half.
