Geometry is a very multifaceted science. It develops logic, imagination and intelligence. Of course, due to its complexity and huge amount theorems and axioms, schoolchildren do not always like it. In addition, there is a need to constantly prove their 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 the intersection of two lines or by drawing two rays from one point. They can be called either one letter or three, which successively designate the points of construction of the corner.

Angles are measured in degrees and can (depending on their value) be called differently. So, there is a right angle, acute, obtuse and deployed. 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 measure is 90.

In the case when it is formed by one continuous straight line, and its degree measure is 180, it is called deployed.

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:

1. The sum of such angles will be equal to 180 degrees (there is a theorem proving this). Therefore, one of them can be easily calculated if the other is known.
2. It follows from the first point that adjacent angles cannot be formed by two obtuse or two acute angles.

Thanks to these properties, one can always 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 lines intersect. Together with them, adjacent corners are always present. An angle can be both adjacent for one and vertical for the other.

When crossing an arbitrary line, several more types of angles are also considered. Such a line is called a secant, and it forms the 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 corners seems to be quite simple and understandable. All their properties are easy to remember and prove. Solving problems is not difficult as long as the angles correspond to a numerical value. Already further, 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 in which you need to find adjacent corners.

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. That's why ∠ 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?

Solution. 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?

Solution. 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.

Solution. 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.

Solution. 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°.

Adjacent corners- two angles that have one side in common, and the other two are continuations of one another.

The sum of adjacent angles is 180°

Vertical angles are two angles in which the sides of one angle are the continuation of the sides of the other.

Vertical angles are equal.

2. Signs of equality of triangles:

I sign: If two sides and the angle between them of one triangle are respectively equal to two sides and the angle between them of another triangle, then such triangles are congruent.

II sign: If the sides and two angles adjacent to it of one triangle are respectively equal to the side and two angles adjacent to it of another triangle, then such triangles are congruent.

III sign: If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are congruent

3. Signs of parallelism of two lines: one-sided angles, lying crosswise and corresponding:

Two lines in a plane are called parallel if they do not intersect.

Crosswise lying angles: 3 and 5, 4 and 6;

Unilateral corners: 4 and 5, 3 and 6; rice. Page55

Corresponding angles: 1 and 5, 4 and 8, 2 and 6, 3 and 7;

Theorem: If at the intersection of two lines of a transversal, the lying angles are equal, then the lines are parallel.

Theorem: If at the intersection of two lines of a secant, the corresponding angles are equal, then the lines are parallel.

Theorem: If at the intersection of two lines of a secant the sum of one-sided angles is equal to 180 °, then the lines are parallel.

Theorem: if two parallel lines are intersected by a secant, then the crosswise lying angles are equal

Theorem: if two parallel lines are intersected by a secant, then the corresponding angles are equal

Theorem: if two parallel lines are intersected by a secant, then the sum of one-sided angles is 180°

4. The sum of the angles of a triangle:

The sum of the angles of a triangle is 180°

5. Properties of an isosceles triangle:

Theorem: In an isosceles triangle, the angles at the base are equal.

Theorem: In an isosceles triangle, the bisector drawn to the base is the median and the height (the median is vice versa), (the bisector bisects the angle, the median bisects the side, the height forms an angle of 90 °)

Sign: If two angles of a triangle are equal, then the triangle is isosceles.

6. Right Triangle:

Right triangle is a triangle in which one angle is a right angle (that is, it is 90 degrees)

In a right triangle, the hypotenuse is longer than the leg

1. The sum of two acute angles of a right triangle is 90°

2. The leg of a right triangle, lying opposite an angle of 30 °, is equal to half the hypotenuse

3. If the leg of a right triangle is equal to half of the hypotenuse, then the angle opposite this leg is 30 °

7. Equilateral Triangle:

EQUILATERAL TRIANGLE, a flat figure having three sides of equal length; the three internal angles formed by the sides are also equal and equal to 60 °C.

8. Sin, cos, tg, ctg:

Sin= , Cos= , tg= , ctg= , tg= ,ctg=

9. Signs of a quadrilateral^

The sum of the angles of the quadrilateral is 2 π = 360°.

A quadrilateral can be inscribed in a circle if and only if the sum of the opposite angles is 180°

10. Signs of similarity of triangles:

I sign: if two angles of one triangle are respectively equal to two angles of another, then such triangles are similar

II sign: if two sides of one triangle are proportional to two sides of another triangle and the angles enclosed between these sides are equal, then such triangles are similar.

III sign: if three sides of one triangle are proportional to three sides of another, then such triangles are similar

11. Formulas:

· Pythagorean theorem: a 2 +b 2 =c 2

· The sin theorem:

· cos theorem:

· 3 triangle area formulas:

· Area of ​​a right triangle: S= S=

· Area of ​​an equilateral triangle:

· Parallelogram area: S=ah

· Square area: S = a2

· Trapezium area:

· Rhombus area:

· Rectangle area: S=ab

· Equilateral triangle. Height: h=

· Trigonometric unit: sin 2 a+cos 2 a=1

· Middle line of the triangle: S=

· Median line of the trapezoid:MK=

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CHAPTER I.

BASIC CONCEPTS.

§eleven. ADJACENT AND VERTICAL ANGLES.

1. Adjacent corners.

If we continue the side of some corner beyond its vertex, we will get two corners (Fig. 72): / A sun and / SVD, in which one side 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 - adjacent corners (Fig. 73).

Adjacent corners 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 2d.

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 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 an 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 extensions of the sides of the other angle.

Let / 1 = 7 / 8 d(Fig. 76). Adjacent to it / 2 will equal 2 d- 7 / 8 d, i.e. 1 1/8 d.

In the same way, you can calculate what are equal to / 3 and / 4.
/ 3 = 2d - 1 1 / 8 d = 7 / 8 d; / 4 = 2d - 7 / 8 d = 1 1 / 8 d(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 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 equal to 2 d, and its right side is also equal to 2 d).

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.

When considering the question of vertical angles, we first explained which angles are called vertical, i.e., we gave definition vertical corners.

Then we made a judgment (statement) about the equality of vertical angles and we were convinced of the validity of this judgment by proof. Such judgments, the validity of which must be proved, are called theorems. Thus, in this section we have given the definition of vertical angles, and also stated and proved a theorem about their property.

In the future, when studying geometry, we will constantly have to meet with 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 the same side of a straight line and have a common vertex on this straight 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 top. 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 corners are in drawing 81?

5. Can a pair of adjacent angles consist of two acute angles? from two obtuse corners? 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 value of the angle adjacent to it?

7. If at the intersection of two straight lines there is one right angle, then what can be said about the size of the remaining three angles?