Typically, angles are considered either with corresponding parallel sides or with corresponding perpendicular sides. Let's consider the first case first.
Let two angles ABC and DEF be given. Their sides are respectively parallel: AB || DE and BC || E.F. Such two angles will either be equal or their sum will be equal to 180°. In the figure below, in the first case ∠ABC = ∠DEF, and in the second ∠ABC + ∠DEF = 180°.
The proof that this is indeed the case comes down to the following.
Consider angles with correspondingly parallel sides, located as in the first figure. At the same time, we extend straight lines AB and EF until they intersect. Let us denote the intersection point by the letter G. In addition, for clarity of the subsequent proof, side BC is extended in the figure. 
Since lines BC and EF are parallel, if line AB intersects one of them, then it will certainly intersect the other. That is, line AB is a secant for two parallel lines. As is known, in this case, the crosswise angles at the secant are equal, one-sided angles add up to 180°, and the corresponding angles are equal.
That is, no matter what pair of angles we take at vertices B and G (one angle from one, the other from the second), we will always get either equal angles or giving a total of 180°.
However, lines AB and DE are also parallel. For them, the straight line EF is a secant. This means that any pairs of angles from vertices G and E will add up to either 180° or be equal to each other. It follows that pairs of angles from vertices B and E will obey this rule.
For example, consider the angles ∠ABC and ∠DEF. Angle ABC is equal to angle BGE, since these angles are corresponding to parallel lines BC and EF. In turn, angle BGE is equal to angle DEF, since these angles are corresponding when AB and DE are parallel. Thus it is proved, ∠ABC and ∠DEF.
Now consider the angles ∠ABC and ∠DEG. Angle ABC is equal to angle BGE. But ∠BGE and ∠DEG are one-sided angles with parallel lines (AB || DE) intersected by a transversal (EF). As you know, such angles add up to 180°. If we look at the second case in the first figure, we realize that it corresponds to the pair of angles ABC and DEG in the second figure.
Thus, two different angles whose sides are respectively parallel are either equal to each other or add up to 180°. The theorem has been proven.
A special case should be noted - when the corners are turned. In this case, they will obviously be equal to each other.
Now consider angles with correspondingly perpendicular sides. This case looks more complicated, since the relative positions of the angles are more varied. The figure below shows three examples of how corners can be positioned with correspondingly perpendicular sides. However, in either case, one side of the first angle (or its extension) is perpendicular to one side of the second angle, and the second side of the first angle is perpendicular to the second side of the second angle. 
Let's consider one of the cases. In this case, we draw a bisector in one corner and through an arbitrary point of it we draw perpendiculars to the sides of its angle. 
Here are angles ABC and DEF with respectively perpendicular sides: AB ⊥ DE and BC ⊥ EF. On the bisector of angle ABC, point G is taken, through which perpendiculars to the same angle are drawn: GH ⊥ AB and GI ⊥ BC.
Consider triangles BGH and BGI. They are rectangular because the angles H and I are right angles. In them, the angles at vertex B are equal, since BG is the bisector of angle ABC. Also, for the triangles under consideration, side BG is common and is the hypotenuse for each of them. As you know, right triangles are equal to each other if their hypotenuses and one of the acute angles are equal. Thus, ∆BGH = ∆BGI.
Since ∆BGH = ∆BGI, then ∠BGH = ∠BGI. Therefore, angle HGI can be represented not as the sum of these two angles, but as one of them multiplied by 2: ∠HGI = ∠BGH * 2.
Angle ABC can be represented as the sum of two angles: ∠ABC = ∠GBH + ∠GBI. Since the component angles are equal to each other (since they are formed by a bisector), angle ABC can be represented as the product of one of them and the number 2: ∠ABC = ∠GBH * 2.
Angles BGH and GBH are acute angles of a right triangle and therefore add up to 90°. Let's look at the resulting equalities:
∠BGH + ∠GBH = 90°
∠HGI = ∠BGH * 2
∠ABC = ∠GBH * 2
Let's add the last two:
∠HGI + ∠ABC = ∠BGH * 2 + ∠GBH * 2
Let's take the common factor out of brackets:
∠HGI + ∠ABC = 2(∠BGH + ∠GBH)
Since the sum of the angles in brackets is 90°, it turns out that angles HGI and ABC add up to 180°:
∠ABC + ∠HGI = 2 * 90° = 180°
So, we have proven that the sum of the angles HGI and ABC is 180°. Now let's look at the drawing again and return our attention to the angle with which angle ABC has correspondingly perpendicular sides. This is the DEF angle.
Lines GI and EF are parallel to each other because they are both perpendicular to the same line BC. And as you know, lines that are perpendicular to the same line are parallel to each other. For the same reason DE || GH.
As has been previously proven, angles with correspondingly parallel sides either add up to 180° or are equal to each other. This means that either ∠DEF = ∠HGI, or ∠DEF + ∠HGI = 180°.
However, ∠ABC + ∠HGI = 180°. From this it is concluded that in the case of correspondingly perpendicular sides, the angles are either equal or add up to 180°.
Although in this case we limited ourselves to proving only the amount. But if we mentally extend side EF in the opposite direction, we will see an angle that is equal to angle ABC, and at the same time its sides are also perpendicular to angle ABC. The equality of such angles can be proven by considering angles with correspondingly parallel sides: ∠DEF and ∠HGI.
53.Angles (internal angles) of a triangle three angles are called, each of which is formed by three rays emerging from the vertices of the triangle and passing through the other two vertices.
54. Triangle Angle Sum Theorem. The sum of the angles of a triangle is 180°.
55. External corner of a triangle is an angle adjacent to some angle of this triangle.
56. External corner of a triangle is equal to the sum of two angles of a triangle that are not adjacent to it.
57. If all three corners triangle spicy, then the triangle is called acute-angled. 
58. If one of the corners triangle blunt, then the triangle is called obtuse-angled. 
59. If one of the corners triangle direct, then the triangle is called rectangular.
60. The side of a right triangle lying opposite the right angle is called hypotenuse(Greek word gyipotenusa - “contracting”), and two sides forming a right angle - legs(Latin word katetos - “plumb”) .
61. Theorem on the relationships between the sides and angles of a triangle. In a triangle the larger angle is opposite the larger side, and back, The larger side lies opposite the larger angle.
62. In a right triangle The hypotenuse is longer than the leg.
because The larger side always lies opposite the larger angle.
Signs of an isosceles triangle.
If in a triangle two angles are equal, then it is isosceles;
If in a triangle the bisector is the median or height,
then this triangle is isosceles;
If in a triangle the median is the bisector or height, That
this triangle is isosceles;
If in a triangle height is median or bisector,
then this triangle is isosceles.
64. Theorem. Triangle inequality. The length of each side of a triangle is greater than the difference and less than the sum of the lengths of the other two sides:
Properties of the angles of a right triangle.
The sum of two acute angles of a right triangle is 90°.
A + B = 90°
66. Right Triangle Property.
A leg of a right triangle lying opposite an angle of 30° is equal to half the hypotenuse.
If/
A = 30°, then BC = ½ AB
67. Properties of a right triangle.
a) If a leg of a right triangle is equal to half the hypotenuse, then the angle opposite this leg is 30°.
If BC = ½ AB, then /
B = 30°
B) The median drawn to the hypotenuse is equal to half the hypotenuse.
median CF = ½ AB
Sign of equality of right triangles on two sides.
If the legs of one right triangle are correspondingly equal to the legs of another, then such triangles are congruent.
The theorem on the property of angles with correspondingly parallel sides should be considered for cases where the given angles are either both acute, or both obtuse, or one of them is acute and the other is obtuse.
The theorem is widely used in studying the properties of various figures and, in particular, the quadrilateral.
The indication that the sides of angles with correspondingly parallel sides can have either the same or opposite direction, which is sometimes found in the formulation of theorems, is considered unnecessary. If we use the term “direction,” then it would be necessary to clarify what should be understood by this word. It is enough to draw students' attention to the fact that angles with correspondingly parallel sides are equal if they are both acute or both obtuse, but if one of the angles is obtuse and the other acute, then they add up to 2d.
The theorem on angles with correspondingly perpendicular sides can be given immediately after the theorem on the property of angles with correspondingly parallel sides. Students are given examples of using the properties of angles with parallel and perpendicular sides, respectively, in devices and machine parts.
Sum of triangle angles
When deriving the theorem on the sum of the angles of a triangle, you can use visual aids. Triangle ABC is cut out, its corners are numbered, then they are cut off and applied to each other. It turns out l+2+3=2d. Draw height CD from vertex C of triangle ABC and bend the triangle so that the height is divided in half, i.e. vertex C fell to point D - the base of the height. The inflection line MN is the midline of triangle ABC. Then the isosceles triangles AMD and DNB are bent along their heights, with vertices A and B coinciding with point D and l+2+3=2d.
It should be remembered that the use of visual aids in a systematic course of geometry is not intended to replace the logical proof of a proposition with its experimental verification. Visual aids should only facilitate students’ understanding of this or that geometric fact, the properties of this or that geometric figure and the relative position of its individual elements. When determining the size of the angle of a triangle, students should be reminded of the previously discussed theorem on the external angle of a triangle and indicate that the theorem on the sum of the angles of a triangle allows, by construction and calculation, to establish a numerical relationship between external and internal angles that are not adjacent to them.
As a consequence of the theorem on the sum of the angles of a triangle, it is proved that in a right triangle the leg opposite the angle of 30 degrees is equal to half the hypotenuse.
As the material is presented, students should be asked questions and simple tasks to facilitate better understanding of the new material. For example, Which lines are called parallel?
At what position of the transversal are all angles formed by two parallel lines and this transversal equal?
A straight line drawn in a triangle parallel to the base cuts off a small triangle from it. Prove that the triangle being cut off and the given triangle are congruent.
Calculate all the angles subtended by two parallels and a transversal if it is known that one of the angles is 72 degrees.
The internal one-sided angles are respectively equal to 540 and 1230. By how many degrees should one of the lines be rotated around the point of its intersection with the transversal so that the lines are parallel?
Prove that the bisectors of: a) two equal but not opposite angles formed by two parallel lines and a transversal are parallel, b) two unequal angles with the same lines and a transversal are perpendicular.
Given two parallel lines AB and CD and a secant EF intersecting these lines at points K and L. The drawn bisectors KM and KN of the angles AKL and BKL cut off the segment MN on the straight line CD. Find the length MN if it is known that the secant segment KL enclosed between the parallel ones is equal to a.
What is the type of triangle in which: a) the sum of any two angles is greater than d, b) the sum of two angles is equal to d, c) the sum of two angles is less than d? Answer: a) acute-angled, b) rectangular, c) obtuse-angled. How many times is the sum of the exterior angles of a triangle greater than the sum of its interior angles? Answer: 2 times.
Can all the external angles of a triangle be: a) acute, b) obtuse, c) straight? Answer: a) no, b) yes, c) no.
Which triangle has each exterior angle twice the size of each interior angle? Answer: equilateral.
When studying the technique of parallel lines, it is necessary to use historical, theoretical and methodological literature to fully formulate the concept of parallel lines.
An angle is a part of a plane bounded by two rays emanating from one point. The rays that limit the angle are called the sides of the angle. The point from which the rays emerge is called the vertex of the angle.
Corner designation scheme Let's look at the example of the angle shown in Figure 1.
The angle shown in Figure 1 can be designated in three ways:
Angles are called equal angles if they can be combined.
If the intersection of two lines produces four equal angles , then such angles are called right angles (Fig. 2). Intersecting straight lines forming right angles are called perpendicular lines.
If through a point A, not lying on a line l, a line is drawn perpendicular to line l and intersecting line l to point B, then they say that from the point B perpendicular AB is dropped onto line l(Fig. 3). Point B is called base of perpendicular AB.
Note. The length of segment AB is called distance from point A to straight line l.
Angle of 1° (one degree) called the angle that makes up one ninetieth part right angle.
An angle k times larger than an angle of 1° is called an angle of k° (k degrees).
Angles are also measured in radians. You can read about radians in the section of our reference book “Measuring angles. Degrees and radians".
Table 1 - Types of angles depending on the value in degrees
| Drawing | Types of angles | Properties of corners |
 | Right angle | A right angle is 90° |
 | Acute angle | Acute angle less than 90° |
 | Obtuse angle | Obtuse angle greater than 90° but less than 180° |
 | Straight angle | The rotated angle is 180° |
 | This angle is greater than 180° but less than 360° | |
 | Full Angle | Full angle is 360° |
 | Angle equal to zero | This angle is 0° |
| Right angle |
Property: A right angle is 90° |
| Acute angle |
Property: Acute angle less than 90° |
| Obtuse angle |
Property: Obtuse angle greater than 90° but less than 180° |
| Straight angle |
Property: The rotated angle is 180° |
| Angle greater than straight |
Property: This angle is greater than 180° but less than 360° |
| Full Angle |
Property: Full angle is 360° |
| Angle equal to zero |
Property: This angle is 0° |
Table 2 - Types of angles depending on the location of the sides
| Drawing | Types of angles | Properties of corners |
 | Vertical angles | Vertical angles are equal |
 | Adjacent angles | The sum of adjacent angles is 180° |
 | Angles with respectively parallel sides are equal if both are acute or both are obtuse | |
 | The sum of angles with correspondingly parallel sides is equal to 180°, if one of them is acute and the other is obtuse | |
 | Angles with respectively perpendicular sides are equal if both are acute or both are obtuse | |
 | The sum of angles with correspondingly perpendicular sides is equal to 180°, if one of them is acute and the other is obtuse |
| Vertical angles |
Property of vertical angles: Vertical angles are equal |
| Adjacent angles |
Property of adjacent angles: The sum of adjacent angles is 180° |
| Angles with correspondingly parallel sides |
Angles with respectively parallel sides are equal if both are acute or both are obtuse |
Property of angles with correspondingly parallel sides: The sum of angles with correspondingly parallel sides is equal to 180°, if one of them is acute and the other is obtuse |
| Angles with correspondingly perpendicular sides |
Angles with respectively perpendicular sides are equal if both are acute or both are obtuse |
Property of angles with correspondingly perpendicular sides: The sum of angles with correspondingly perpendicular sides is equal to 180°, if one of them is acute and the other is obtuse |
Definition . The bisector of an angle is the ray that bisects the angle.
Task . Prove that the bisectors of adjacent angles are perpendicular.
Solution . Consider Figure 4.
In this figure, angles AOB and BOC are adjacent, and rays OE and OD are bisectors of these angles. Since
2α + 2β = 180°.
Q.E.D.
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For angles with correspondingly parallel sides, the following propositions are valid:
1. If sides a and b of one angle are respectively parallel to sides a and b of another angle and have the same directions as them, then the angles are equal.
2. If, under the same condition of parallelism, sides a and b are adjusted opposite to sides a and b, then the angles are also equal.
3. If, finally, the sides a and are parallel and in the same direction, and the sides are parallel and opposite in direction, then the angles complement each other until they are reversed.
Proof. Let us prove the first of these propositions. Let the sides of the angles be parallel and equally directed (Fig. 191). Let's connect the vertices of the corners with a straight line.
In this case, two cases are possible: the straight line passes inside the corners or outside these corners (Fig. 191, b). In both cases the proof is obvious: so, in the first case
but where do we get it from? In the second case we have
and the result again follows from the equalities
We leave the proofs of Propositions 2 and 3 to the reader. We can say that if the sides of the angles are respectively parallel, then the angles are either equal or add up to the opposite angle.
Obviously, they are equal if both are simultaneously acute or both are obtuse, and their sum is equal if one of them is acute and the other is obtuse.
Angles with correspondingly perpendicular sides are equal or complementary to each other up to a straight angle.
Proof. Let a be some angle (Fig. 192), and O be the vertex of the angle formed by straight lines; therefore, let there be any of the four angles formed by these two straight lines). Let us rotate the angle (i.e., both of its sides) around its vertex O at a right angle; we obtain an angle equal to it, but one whose sides are perpendicular to the sides of the rotated angle indicated in Fig. 192 through They are parallel to the straight lines forming a given angle a. Therefore, angles mean that angles are either equal or form a reverse angle in total.
