CHAPTER I.
BASIC CONCEPTS.
§eleven. ADJACENT AND VERTICAL CORNERS.
1. Adjacent angles.
If we extend the side of any angle beyond its vertex, we get two angles (Fig. 72): / And the sun and / SVD, in which one side BC is common, and the other two A 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 /
FDВ - 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 umma of two adjacent angles is equal 2d.
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 3/5 d, then the second angle will be equal to:
2d- 3 / 5 d= l 2 / 5 d.
2. Vertical angles.
If we extend the sides of the angle beyond its vertex, we get vertical angles. In drawing 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 = 7 / 8 d(Figure 76). Adjacent to it / 2 will be equal to 2 d- 7 / 8 d, i.e. 1 1/8 d.
In the same way you can calculate what they are equal to /
3 and /
4.
/
3 = 2d - 1 1 / 8 d = 7 / 8 d; /
4 = 2d - 7 / 8 d = 1 1 / 8 d(Figure 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 reasoning, by proof.
The proof can be carried out as follows (Fig. 78):
/
a +/
c = 2d;
/
b+/
c = 2d;
(since the sum of adjacent angles is 2 d).
/ a +/ c = / b+/ c
(since the left side of this equality is also equal to 2 d, and its right side is also equal to 2 d).
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.
When considering the issue of vertical angles, we first explained which angles are called vertical, i.e. definition vertical angles.
Then we made a judgment (statement) about the equality of the vertical angles and were convinced of the validity of this judgment through proof. Such judgments, the validity of which must be proven, are called theorems. Thus, in this section we gave a definition of vertical angles, and also stated and proved a theorem about their properties.
In the future, when studying geometry, we will constantly have to encounter definitions and proofs of theorems.
3. The sum of angles that have a common vertex.
On the 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 = 2d.
On the drawing 80 / 1, / 2, / 3, / 4 and / 5 have a common vertex. In sum, these angles make up a full angle, i.e. / 1 + / 2 + / 3 + / 4 + / 5 = 4d.
Exercises.
1. One of the adjacent angles is 0.72 d. Calculate the angle formed by the bisectors of these adjacent angles.
2. Prove that the bisectors of two adjacent angles form a right angle.
3. Prove that if two angles are equal, then their adjacent angles are also equal.
4. How many pairs of adjacent angles are there in the drawing 81?
5. Can a pair of adjacent angles consist of two acute angles? from two obtuse angles? from right and obtuse angles? from a right and acute angle?
6. If one of the adjacent angles is right, then what can be said about the size of the angle adjacent to it?
7. If at the intersection of two straight lines one angle is right, then what can be said about the size of the other three angles?
Geometry is a very multifaceted science. It develops logic, imagination and intelligence. Of course, due to its complexity and the huge number of theorems and axioms, schoolchildren do not always like it. In addition, there is a need to constantly prove your conclusions using generally accepted standards and rules.
Adjacent and vertical angles are an integral part of geometry. Surely many schoolchildren simply adore them for the reason that their properties are clear and easy to prove.
Formation of corners
Any angle is formed by intersecting two straight lines or drawing two rays from one point. They can be called either one letter or three, which sequentially designate the points at which the angle is constructed.
Angles are measured in degrees and can (depending on their value) be called differently. So, there is a right angle, acute, obtuse and unfolded. Each of the names corresponds to a certain degree measure or its interval.
An acute angle is an angle whose measure does not exceed 90 degrees.
An obtuse angle is an angle greater than 90 degrees.
An angle is called right when its degree measure is 90.
In the case when it is formed by one continuous straight line and its degree measure is 180, it is called expanded.
Angles that have a common side, the second side of which continues each other, are called adjacent. They can be either sharp or blunt. The intersection of the line forms adjacent angles. Their properties are as follows:
- The sum of such angles will be equal to 180 degrees (there is a theorem that proves this). Therefore, one can easily calculate one of them if the other is known.
- From the first point it follows that adjacent angles cannot be formed by two obtuse or two acute angles.
Thanks to these properties, it is always possible to calculate the degree measure of an angle given the value of another angle, or at least the ratio between them.
Vertical angles
Angles whose sides are continuations of each other are called vertical. Any of their varieties can act as such a pair. Vertical angles are always equal to each other.
They are formed when straight lines intersect. Along with them, adjacent angles are always present. An angle can be simultaneously adjacent for one and vertical for another.
When crossing an arbitrary line, several other types of angles are also considered. Such a line is called a secant line, and it forms corresponding, one-sided and cross-lying angles. They are equal to each other. They can be viewed in light of the properties that vertical and adjacent angles have.
Thus, the topic of angles seems quite simple and understandable. All their properties are easy to remember and prove. Solving problems is not difficult as long as the angles have a numerical value. Later, when the study of sin and cos begins, you will have to memorize many complex formulas, their conclusions and consequences. Until then, you can just enjoy easy puzzles where you need to find adjacent angles.
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 the 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
Let's consider a straight line a and a point A that does not lie 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°.
Indicate the numbers of the correct statements.
1) Any three lines have at most one common point.
2) If an angle is 120°, then the adjacent one is 120°.
3) If the distance from a point to a straight line is greater than 3, then the length of any inclined line drawn from a given point to a straight line is greater than 3.
If there are several statements, write down their numbers in ascending order.
Solution.
We verify each of the statements.
1) “Any three lines have at most one common point” - right. If straight lines have two or more common points, then they coincide. (See com-men-ta-rii to za-da-che.)
2) “If an angle is 120°, then the adjacent one is 120°” - wrong. The sum of adjacent angles is 180°.
3) “If the distance from a point to a straight line is greater than 3, then the length of any inclined line drawn from a given point to a straight line is greater than 3.” - right. Because the distance is the shortest length from the cut to the straight line, and all oblique ones are longer.
Answer: 13.
Answer: 13
· Task prototype ·
Guest 19.02.2015 12:42
In the school textbook by Atanasyan L.S. et al. “Geometry 7--9”, “Enlightenment”, 2014, chapter 1, paragraph 1, the following is stated.
1) Axiom of planimetry: through any two points you can draw a straight line and, moreover, only one.
2) The position adopted in the school course: when we say “two points”, “three points”, “two lines”, etc., we will assume that these points and lines are different.
The conclusion that the student must learn is that two lines either have only one common point or have no common points.
Therefore, the answer to question 1 should be “true”. If all three lines coincide, then it is one line, not three.
Petr Murzin
It would be correct to write in the condition "any three various straight lines have at most one common point", but this is not true.
Guest 10.04.2015 16:38
Dear editor!
I agree with the Guest’s remark dated 02/19/2015 on the merits of the statement of paragraph 1 of this problem: in the mentioned Textbook “Geometry 7-9” (clause 1 of paragraph 1, note 1) it is said: “hereinafter, saying “two points”, “three points”, “two lines”, etc., we will assume that these points and lines are different.”
Taking into account the above, the reasoning given on the site in solving this problem (in part of point 1) is erroneous, since the formulation of the “three lines” problem implies that these three lines are different (i.e. they cannot coincide!). Three lines (different, which is the default!): either have one common point (which belongs to each of these three lines) - in the case when three lines intersect at one point; or do not have common points.
This conclusion is confirmed by the conclusion of paragraph 1 of paragraph 1 of the mentioned textbook: “two straight lines either have only one common point or have no common points.” Proof by contradiction: suppose that three lines have more than one common point; therefore, two of these lines have at least more than one common point (since for these two lines the common points will be those that are common to all three lines); but this contradicts the textbook conclusion mentioned that two lines either have only one common point or have no common points.
Best regards, guest.
Support
Each angle, depending on its size, has its own name:
| Angle type | Size in degrees | Example |
|---|---|---|
| Spicy | Less than 90° | |
| Straight | 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.
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