Introduction to corners
Let us be given two arbitrary rays. Let's put them on top of each other. Then
Definition 1
An angle is a name given to two rays that have the same origin.
Definition 2
The point, which is the beginning of the rays within the framework of Definition 3, is called the vertex of this angle.
An angle will be denoted by its following three points: a 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).
Now let's define what the value of the angle is.
To do this, you need to choose some kind of "reference" angle, which we will take as a unit. Most often, such an angle is an angle that is equal to $\frac(1)(180)$ of a part of a straight angle. This value is called a degree. After choosing such an angle, we compare the angles with it, the value of which must be found.
There are 4 types of corners:
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 straight if it is equal to $180^0$.
Definition 6
An angle is called a right angle if it is equal to $90^0$.
In addition to such types of angles, which are described above, it is possible to distinguish types of angles in relation to each other, namely vertical and adjacent angles.
Adjacent corners
Consider a straight angle $COB$. Draw a ray $OA$ from its vertex. This ray will divide the original one into two angles. Then
Definition 7
Two angles will be called adjacent if one pair of their sides is a straight angle, and the other pair coincides (Fig. 2).
AT this case the angles $COA$ and $BOA$ are adjacent.
Theorem 1
Sum adjacent corners equals $180^0$.
Proof.
Consider 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 coincide, then the ray $OA$ will divide the straight angle by 2, therefore
$∠COA+∠BOA=180^0$
The theorem has been proven.
Consider the solution of the problem using this concept.
Example 1
Find the angle $C$ from the figure below
By Definition 7, we get that the angles $BDA$ and $ADC$ are adjacent. Therefore, by Theorem 1, we obtain
$∠BDA+∠ADC=180^0$
$∠ADC=180^0-∠BDA=180〗0-59^0=121^0$
By the theorem on the sum of angles in a triangle, we will 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 developed angles $AOB$ and $MOC$. Let's match their vertices with each other (that is, put the point $O"$ on the point $O$) so that none of the sides of these angles coincide. Then
Definition 8
Two angles will be called vertical if the pairs of their sides are straight angles and their values are the same (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.
Consider Figure 3. Let's prove, for example, that the angle $MOA$ is equal to the angle $BOC$.
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 two other sides of them lie on the same straight line ... Great Polytechnic Encyclopedia
See Angle... Big Encyclopedic Dictionary
ADJACENT ANGLES, two angles whose sum is 180°. Each of these corners complements the other to a full angle... Scientific and technical encyclopedic dictionary
See Angle. * * * ADJACENT CORNERS ADJACENT CORNERS, see Corner (see CORNER) … encyclopedic Dictionary
- (Angles adjacent) those that have a common vertex and a common side. Mostly, this name means such S. angles, of which the other two sides lie in opposite directions of one straight line drawn through the vertex ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron
See Angle... Natural science. encyclopedic Dictionary
The two lines intersect, creating 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 A complementary angle is a pair of angles that complement each other up to 90 degrees. If two complementary angles are adjacent (that is, 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 additional angles are c ... Wikipedia
Books
- About Proof in Geometry, Fetisov A.I. This book will be produced in accordance with your order using Print-on-Demand technology. Once, at the very beginning of the school year, I happened to overhear a conversation between two girls. The oldest one…
- A comprehensive notebook for knowledge control. Geometry. 7th grade. Federal State Educational Standard, Babenko Svetlana Pavlovna, Markova Irina Sergeevna. The manual presents control and measuring materials (KMI) in geometry for conducting current, thematic and final quality control of knowledge of students in grade 7. The contents of the guide…
2) How many common points can 2 lines have?
3) Explain what a segment is?
4) Explain what a ray is. How are rays designated?
5) What figure is called an angle? Explain what a vertex and sides of an angle are?
6) What angle is called deployed?
7) What figures are called equal?
8) Explain how to compare 2 segments
9) What point is called the midpoint of the segment?
10) Explain how to compare 2 angles.
11) Which ray is called the angle bisector?
12) Point C divides segment AB into 2 segments. How to find the length of segment AB if the lengths of segments AC and CB are known?
13) What tools are used to measure distances?
14) What is the degree measure of an angle?
15) Ray OS divides the angle AOB into 2 angles. How to find the degree measure of the angle AOB if the degree measures of the angles AOC and COB are known?
16) What angle is called acute? Right? Obtuse?
17) What angles are called adjacent? What is the sum of adjacent angles?
18) What angles are called vertical? What property do vertical angles have?
19) What lines are called perpendicular?
20) Explain why 2 lines perpendicular to the 3rd do not intersect?
21) What instruments are used to construct right angles on the ground?
How many common points can two lines have?
3 Explain what a segment is
4explain what a ray is. How are rays designated?
What figure is called an angle? Explain what a vertex and sides of an angle are
6what angle is called unfolded
7 what figures are called equal
8explain how to compare two segments
What point is called the midpoint of a segment
10explain how to compare two angles
11 which ray is called the angle bisector
12point c divides the segment ab into two segments. How to find the length of the segment ab if the lengths of the segments ac and sb are known
13what tools are used to measure distances
14 what is the degree measure of an angle
The ray os divides the angle aob into two angles. How to find the degree measure of the angle aob if the measures of the angles aos
What angle is called acute?, right?, obtuse?.
17What angles are called adjacent? What is the sum of adjacent angles?
18what kind of angles are called vertical? what property do vertical angles have
19 which lines are called perpendicular
20explain why two lines perpendicular to a third do not intersect
21What instruments are used to construct right angles on the ground?
vertical what property do vertical angles have 5)
Help plz!! plzz=**7. Prove that if two parallel lines are intersected by a third line, then the interior cross-lying angles are equal, and the sum of interior one-sided angles is 180 degrees.
8. Prove that two lines perpendicular to the third are parallel. If a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.
9. Prove that the sum of the angles of a triangle is 180 degrees.
10. Prove that any triangle has at least two acute angles.
11. What is outer corner triangle?
12. Prove that the exterior angle of a triangle is equal to the sum of two interior angles not adjacent to it.
13. Prove that the external angle of a triangle is greater than any inner corner, not adjacent to it.
14. What triangle is called a right triangle?
15. What is the sum of the acute angles of a right triangle?
16. Which side of a right triangle is called the hypotenuse? What sides are called legs?
17. Formulate a sign of equality of right triangles along the hypotenuse and leg.
18. Prove that from any point not lying on a given line, one can drop a perpendicular to this line, and only one.
19. What is called the distance from a point to a line?
20. Explain what is the distance between parallel lines.
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. 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 axioms and proved 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.
In the process of studying the geometry course, the concepts of “angle”, “vertical angles”, “adjacent angles” are encountered quite often. Understanding each of the terms will help to understand the task and solve it correctly. What are adjacent angles and how to determine them?
Adjacent corners - definition of the concept
The term "adjacent angles" characterizes two angles formed by a common ray and two additional half-lines lying on the same line. All three beams come from the same point. The common half-line is at the same time the side of both one and the second angle.
Adjacent corners - basic properties
1. Based on the formulation of adjacent angles, it is easy to see that the sum of such angles always forms a straight angle, the degree measure of which is 180 °:
- If μ and η are adjacent angles, then μ + η = 180°.
- Knowing the value of one of the adjacent angles (for example, μ), one can easily calculate the degree measure of the second angle (η) using the expression η = 180° - μ.
2. This property angles allows us to draw the following conclusion: an angle that is adjacent right angle, will also be straight.
3. Considering trigonometric functions(sin, cos, tg, ctg), based on the reduction formulas for adjacent angles μ and η, the following is true:
- sinη = sin(180° - μ) = sinμ,
- cosη = cos(180° - μ) = -cosμ,
- tgη = tg(180° - μ) = -tgμ,
- ctgη = ctg(180° - μ) = -ctgμ.
Adjacent corners - examples
Example 1
Given a triangle with vertices M, P, Q – ΔMPQ. Find the angles adjacent to the angles ∠QMP, ∠MPQ, ∠PQM.
- Let us extend each side of the triangle as a straight line.
- Knowing that adjacent angles complement each other to a straight angle, we find out that:
adjacent to the angle ∠QMP is ∠LMP,
adjacent to the angle ∠MPQ is ∠SPQ,
the adjacent angle for ∠PQM is ∠HQP.
Example 2
The value of one adjacent angle is 35°. What is the degree measure of the second adjacent angle?
- Two adjacent angles add up to 180°.
- If ∠μ = 35°, then adjacent ∠η = 180° – 35° = 145°.
Example 3
Determine the values of adjacent angles, if it is known that the degree measure of one of the bottom is three times greater than the degree measure of the other angle.
- Let us denote the value of one (smaller) angle through – ∠μ = λ.
- Then, according to the condition of the problem, the value of the second angle will be equal to ∠η = 3λ.
- Based on the basic property of adjacent angles, μ + η = 180° follows
λ + 3λ = μ + η = 180°,
λ = 180°/4 = 45°.
So the first one angle is ∠μ = λ = 45°, and the second angle is ∠η = 3λ = 135°.
The ability to appeal to terminology, as well as knowledge of the basic properties of adjacent angles, will help to cope with the solution of many geometric problems.