Two triangles are said to be congruent if they can be overlapped. Figure 1 shows equal triangles ABC and A 1 B 1 C 1. Each of these triangles can be superimposed on another so that they are completely compatible, that is, their vertices and sides are paired together. It is clear that in this case the angles of these triangles will be combined in pairs.

Thus, if two triangles are equal, then the elements (i.e., sides and angles) of one triangle are respectively equal to the elements of the other triangle. Note that in equal triangles against respectively equal sides(i.e. overlapping when superimposed) lie equal angles and back: opposite correspondingly equal angles lie equal sides.

So, for example, in equal triangles ABC and A 1 B 1 C 1, shown in Figure 1, equal angles C and C 1 lie against respectively equal sides AB and A 1 B 1. The equality of triangles ABC and A 1 B 1 C 1 will be denoted as follows: Δ ABC = Δ A 1 B 1 C 1. It turns out that the equality of two triangles can be established by comparing some of their elements.

Theorem 1. The first sign of equality of triangles. 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 equal (Fig. 2).

Proof. Consider triangles ABC and A 1 B 1 C 1, in which AB \u003d A 1 B 1, AC \u003d A 1 C 1 ∠ A \u003d ∠ A 1 (see Fig. 2). Let us prove that Δ ABC = Δ A 1 B 1 C 1 .

Since ∠ A \u003d ∠ A 1, then the triangle ABC can be superimposed on the triangle A 1 B 1 C 1 so that the vertex A is aligned with the vertex A 1, and the sides AB and AC overlap, respectively, on the rays A 1 B 1 and A 1 C one . Since AB \u003d A 1 B 1, AC \u003d A 1 C 1, then side AB will be combined with side A 1 B 1 and side AC - with side A 1 C 1; in particular, points B and B 1 , C and C 1 will coincide. Therefore, the sides BC and B 1 C 1 will be aligned. So, triangles ABC and A 1 B 1 C 1 are completely compatible, which means they are equal.

Theorem 2 is proved similarly by the superposition method.

Theorem 2. The second sign of the equality of triangles. If the side 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 equal (Fig. 34).

Comment. Based on Theorem 2, Theorem 3 is established.

Theorem 3. The sum of any two interior angles of a triangle is less than 180°.

Theorem 4 follows from the last theorem.

Theorem 4. An external angle of a triangle is greater than any internal angle not adjacent to it.

Theorem 5. The third sign of the equality of triangles. If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are equal ().

Example 1 In triangles ABC and DEF (Fig. 4)

∠ A = ∠ E, AB = 20 cm, AC = 18 cm, DE = 18 cm, EF = 20 cm. Compare triangles ABC and DEF. What angle in triangle DEF is equal to angle B?

Decision. These triangles are equal in the first sign. Angle F of triangle DEF is equal to angle B of triangle ABC, since these angles lie opposite the corresponding equal sides DE and AC.

Example 2 Segments AB and CD (Fig. 5) intersect at point O, which is the midpoint of each of them. What is segment BD equal to if segment AC is 6 m?

Decision. Triangles AOC and BOD are equal (by the first criterion): ∠ AOC = ∠ BOD (vertical), AO = OB, CO = OD (by condition).
From the equality of these triangles follows the equality of their sides, i.e. AC = BD. But since, according to the condition, AC = 6 m, then BD = 6 m.

Proof: We impose ABC on A 1 B 1 C 1 so that point A 1 coincides with A. Since AC \u003d A 1 C 1, then, according to the axiom of postponing segments, point C 1 will coincide with C. Since A \u003d A 1 , then, according to the axiom of laying off angles, the beam A 1 B 1 will coincide with the beam AB. Since AB \u003d A 1 B 1, then, according to the axiom of postponing segments, point B 1 will coincide with point B. Triangles A 1 B 1 C 1 and ABC coincided, which means ABC \u003d A 1 B 1 C 1 Ch.T.D .

Proof: We impose ABC on A 1 B 1 C 1 so that point A 1 coincides with A. Since AC \u003d A 1 C 1, then, according to the axiom of postponing segments, point C 1 will coincide with C. Since A \u003d A 1 , then, according to the axiom of laying off angles, the beam A 1 B 1 will coincide with the beam AB. Since C \u003d C 1, then, according to the axiom of laying off angles, the ray C 1 IN 1 will coincide with the ray CB. Point B 1 will coincide with point B. Triangles A 1 B 1 C 1 and ABC coincided, which means ABC \u003d A 1 B 1 C 1 FTD

Mediana A segment of the bisector of the angle of a triangle connecting the vertex of the triangle with a point on the opposite side is called the bisector of the triangle. medianabisector 1 HEIGHT The perpendicular drawn from the vertex of the triangle to the line containing the opposite side is called the height of the triangle. The line segment connecting the vertex of a triangle with the midpoint of the opposite side is called the median of the triangle. height

A B C K M O T The heights of a right triangle intersect at vertex C. The heights of an acute triangle intersect at point O, which lies in the interior of the triangle. O A B C The point of intersection of the heights is called the orthocenter.

The segment of the bisector of the angle of a triangle connecting the vertex of the triangle with a point on the opposite side is called the bisector of the triangle. This point is also remarkable - the point of intersection of the bisectors is the center of the inscribed circle. O b i s s e c t r i c a

1 The perpendicular drawn from the vertex of a triangle to the line containing the opposite side is called the height of the triangle. HEIGHT A height in a right triangle, drawn from the vertex of an acute angle, coincides with the leg. The height in an obtuse triangle, drawn from the vertex of an acute angle, passes in the outer region of the triangle. HEIGHT 11

Conclusion 1. In an isosceles triangle, the height drawn to the base is the median and the bisector. 2. In an isosceles triangle, the median drawn to the base is the height and the bisector. 3. In an isosceles triangle, the bisector drawn to the base is the median and height.

Which of these statements are correct? Write down their numbers.
1) If two sides of one triangle are respectively equal to two sides of another triangle, then such triangles are congruent.
2) If the diagonals in a quadrilateral are perpendicular, then this quadrilateral is a rhombus.
3) The area of ​​a circle is less than the square of the length of its diameter.

The solution of the problem:

Let's consider each statement.
1) "If two sides of one triangle are respectively equal to two sides of another triangle, then such triangles are congruent", this statement is false, because does not correspond to any of the criteria for the equality of triangles.
2) "If the diagonals in a quadrilateral are perpendicular, then this quadrilateral is a rhombus", this statement is false, because does not fully correspond to any property of the rhombus. For example, the quadrilateral shown in the figure, its diagonals are perpendicular, but it is obvious that this is not a rhombus.
3) "The area of ​​a circle is less than the square of the length of its diameter." The area of ​​the circle is ΠR 2 , or ΠD 2 /4. The number Π (Pi) is approximately 3.14. Then S circle \u003d 0.785D 2. And this, of course, is less than D 2 . The statement is true

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Task #03A3EF

The area of ​​a right triangle is 722 √ 3 . One of the acute angles is 30°. Find the length of the leg opposite this angle.

Problem #9FCAB9

In triangle ABC, the bisector BE and median AD are perpendicular and have the same length equal to 96. Find the sides of triangle ABC.

Signs of equality of triangles

Equal triangles are those whose corresponding sides are equal.

Theorem (the first criterion for the equality of triangles).
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.

Theorem (the second criterion for the equality of triangles).
If a side and two adjacent angles of one triangle are respectively equal to a side and two adjacent angles of another triangle, then such triangles are congruent.

Theorem (the third criterion for the equality of triangles).
If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are congruent.

Signs of similarity of triangles

Triangles are called similar if the angles are equal and the similar sides are proportional: , where is the similarity coefficient.

I sign of similarity of triangles. If two angles of one triangle are respectively equal to two angles of another, then these triangles are similar.

II sign of similarity of triangles. If three sides of one triangle are proportional to three sides of another triangle, then such triangles are similar.

III sign of similarity of triangles. If two sides of one triangle are proportional to two sides of another triangle, and the angles included between these sides are equal, then such triangles are similar.

Theorem 1.1. If a line that does not pass through any of the vertices of a triangle intersects one of its sides, then it intersects only one of the other two sides.

Theorem 2.1. The sum of adjacent angles is 180 about .
Consequences:
If two angles are equal, then the angles adjacent to them are equal.
If the angle is not developed, then its degree measure is less than 180 about .
An angle adjacent to a right angle is a right angle.

Theorem 2.2. Vertical angles are equal.

Theorem 2.3. Through each point of a line, one can draw a line perpendicular to it, and only one.

Theorem 3.1 (The first criterion for the equality of triangles). If two sides and the angle between them of one triangle are equal, respectively, to two sides and the angle between them of another triangle, then such triangles are congruent.

Theorem 3.2 (The second criterion for the equality of triangles). If a side and the angles adjacent to it of one triangle are equal, respectively, to the side and angles adjacent to it of another triangle, then such triangles are congruent.

Theorem 3.3 (Property of the angles of an isosceles triangle). In an isosceles triangle, the angles at the base are equal.

Theorem 3.4 (Sign of an isosceles triangle). If two angles are equal in a triangle, then it is isosceles.

Theorem 3.5 (Property of the median of an isosceles triangle). In an isosceles triangle, the median drawn to the base is the bisector and height.

Theorem 3.6 (The third criterion for the equality of triangles). If three sides of one triangle are equal, respectively, to three sides of another triangle, then such triangles are congruent.

Theorem 4.1. Two lines parallel to a third are parallel.

Theorem 4.2 (A criterion for parallel lines). If interior cross-lying angles are equal or the sum of interior one-sided angles is 180 about , then the lines are parallel.

Theorem 4.3 (Converse to Theorem 4.2). 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 about .

Theorem 4.4. The sum of the angles of a triangle is 180 about .
Consequence: Every triangle has at least two acute angles.

Theorem 4.5. An exterior angle of a triangle is equal to the sum of two interior angles that are not adjacent to it.
Consequence: An exterior angle of a triangle is greater than any interior angle not adjacent to it.

Theorem 4.6. From any point not lying on a given line, one can drop a perpendicular to this line, and only one.

Theorem 5.1. The center of a circle circumscribed about a triangle is the point of intersection of the perpendiculars to the sides of the triangle, drawn through the midpoints of these sides.

Theorem 5.2. The center of a circle inscribed in a triangle is the intersection point of its bisectors.

Theorem 5.3. The locus of points equidistant from two given points is a line perpendicular to the line segment connecting these points and passing through its midpoint.

Theorem 6.1. If the diagonals of a quadrilateral intersect and the intersection point is bisected, then the quadrilateral is a parallelogram.

Theorem 6.2 (Converse to Theorem 6.1). The diagonals of a parallelogram intersect and the intersection point is bisected.

Theorem 6.3. A parallelogram has opposite sides equal and opposite angles equal.

Theorem 6.4. The diagonals of a rectangle are equal.

Theorem 6.5. The diagonals of the rhombus intersect at right angles. The diagonals of a rhombus are the bisectors of its angles.

Theorem 6.6 (Thales' theorem). If parallel lines intersecting the sides of an angle cut off equal segments on one of its sides, then they cut off equal segments on its other side.

Theorem 6.7. The midline of a triangle connecting the midpoints of two given sides is parallel to the third side and equal to half of it.

Theorem 6.8. The median line of the trapezoid is parallel to the bases and equal to half their sum.

Theorem 6.9. Parallel lines intersecting the sides of the angle cut off proportional segments from the sides of the angle.

Theorem 7.1. The cosine of an angle depends only on the degree measure of the angle and does not depend on the location and size of the triangle.

Theorem 7.2 (Pythagorean theorem). In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
Consequences:
-In a right triangle, either leg is less than the hypotenuse.
- cosA
-If a perpendicular and oblique are drawn to a straight line from one point, then any oblique is greater than the perpendicular, equal obliques have equal projections, of two obliques, the one with the largest projection is greater.

Theorem 7.3 (Triangle Inequality). Whatever the three points are, the distance between any two of these points is not greater than the sum of their distances to the third point.
Consequence: In any triangle, each side is less than the sum of the other two.

Theorem 7.4. For any acute angle A.
sin(90 o -A) = cosA, cos(90 o -A) = sinA.

Theorem 7.5. As the acute angle increasessinAandtgAare increasing, andcosAdecreases.

Theorem 9.1. Points lying on a straight line, when moving, pass into points lying on a straight line, and the order of their mutual arrangement is preserved.
Consequence: When moving, straight lines turn into straight lines, half-lines into half-lines, segments into segments.

Theorem 9.2. A symmetry transformation about a point is a movement.

Theorem 9.3. A symmetry transformation about a line is a movement.

Theorem 9.4. Whatever the two pointsBUT andBUT ', there is one and only one parallel translation in which the pointBUT goes to the pointBUT ’.

Theorem 10.1. Whatever the pointsBUT , AT , With , the vector equality

Theorem 10.2. The absolute value of the vector is equal to . vector direction at coincides with the direction of the vector , ifl > 0, and opposite to the direction of the vector , ifl

Theorem 10.3. The scalar product of vectors is equal to the product of their absolute values ​​and the cosine of the angle between them.
Consequences:
If the vectors are perpendicular, then their dot product is 0.
If the dot product of non-0 vectors is 0, then the vectors are perpendicular.

Theorem 11.1. Homothety is a similarity transformation.

Theorem 11.2 (A test for the similarity of triangles in two angles). If two angles of one triangle are equal to two angles of another triangle, then such triangles are similar.

Theorem 11.3 (A test for the similarity of triangles on two sides and the angle between them). If two sides of one triangle are proportional to two sides of another triangle and the angles formed by these sides are equal, then the triangles are similar.

Theorem 11.4 (Triangles similarity criterion on three sides). If the sides of one triangle are proportional to the sides of another triangle, then the triangles are similar.

Theorem 11.5. An inscribed angle in a circle is half the corresponding central angle.
Consequences:
-Inscribed angles whose sides pass through points A and B of the circle, and whose vertices lie on the same side of line AB, are equal.
-The inscribed angles based on the diameter are straight.

Theorem 12.1 (Cosine theorem). The square of any side of a triangle is equal to the sum of the squares of the other two sides without doubling the product of those sides times the cosine of the angle between them.

Theorem 12.2 (Sine theorem). The sides of a triangle are proportional to the sines of the opposite angles.

Theorem 13.1. The length of the polyline is not less than the length of the segment connecting its ends.

Theorem 13.2. The sum of the angles of a convexn-gon is 180 0 (n – 2).

Theorem 13.3. A regular convex polygon is inscribed in a circle and circumscribed about the circle.

Theorem 13.4. Regular convexn-gons are similar. In particular, if their sides are the same, then they are equal.

Theorem 13.5. The ratio of the circumference of a circle to its diameter does not depend on the circle, i.e. the same for any two circles.

Theorem 15.1.

Theorem 15.2.
Consequence:

Theorem 15.3.

Theorem 15.4. XandYXYXandYXYcrosses the plane.

Theorem 16.1.

Theorem 16.2.

Theorem 16.5.

Theorem 17.3.

Theorem 17.4.

Theorem 17.6.

Theorem 15.1. Through a line and a point not lying on it, one can draw a plane, and moreover, only one.

Theorem 15.2. If two points of a line belong to a plane, then the whole line belongs to that plane.
Consequence: A plane and a line not lying on it either do not intersect or intersect at one point.

Theorem 15.3. Through three points that do not lie on the same straight line, it is possible to draw a plane, and moreover, only one.

Theorem 15.4. The plane divides the space into two half-spaces. If the pointsXandYbelong to the same half-space, then the segmentXYdoes not cross the plane. If the pointsXandYbelong to different half-spaces, then the segmentXYcrosses the plane.

Theorem 16.1. Through a point outside a given line, one can draw a line parallel to this line, and moreover, only one.

Theorem 16.2. Two lines parallel to a third line are parallel.

Theorem 16.3. If a line not belonging to a plane is parallel to any line in that plane, then it is also parallel to the plane itself.

Theorem 16.4. If two intersecting lines of one plane are respectively parallel to two lines of another plane, then these planes are parallel.

Theorem 16.5. Through a point outside a given plane, one can draw a plane parallel to the given one, and moreover, only one.

Theorem 17.1. If two intersecting lines are parallel, respectively, to two perpendicular lines, then they are also perpendicular.

Theorem 17.2. If a line is perpendicular to two intersecting lines lying in a plane, then it is perpendicular to the given plane.

Theorem 17.3. If a plane is perpendicular to one of two parallel lines, then it is also perpendicular to the other.

Theorem 17.4. Two lines perpendicular to the same plane are parallel.

Theorem 17.5. If a straight line drawn in a plane through the base of an oblique line is perpendicular to its projection, then it is perpendicular to the oblique line. And back: if a straight line in a plane is perpendicular to an oblique one, then it is also perpendicular to the projection of the oblique one.

Theorem 17.6. If a plane passes through a line perpendicular to another plane, then these planes are perpendicular.

Theorem 18.1. The area of ​​an orthogonal projection of a polygon onto a plane is equal to the product of its area and the cosine of the angle between the plane of the polygon and the projection plane.