Signs of equality of right triangles. Summary of theoretical issues

Basic properties

1. If the angle of one triangle is equal to the angle of another triangle, then the areas of these triangles are related as the product of the sides enclosing equal angles.

2. The ratio of the areas of triangles that have common heights is equal to the ratio of the bases corresponding to these heights.

3. The ratio of the areas of triangles that have common bases is equal to the ratio of the heights corresponding to these sides of the triangle.

4. In similar triangles, similar elements are proportional, the radii of inscribed and circumscribed circles, the perimeters of triangles, square roots from squares.

5.The radius of the inscribed circle can be found using the formula:

6. It is convenient to find the radius of the circumscribed circle using the theorem of sines and cosines:

7.Each median divides the triangle into 2 equal triangles.

8. Three medians divide a triangle into 6 equal triangles.

9. The point of intersection of the bisectors divides the bisector in the ratio:

the sum of the sides forming the angle from which the bisector is drawn to the third side.

10.The medians of a triangle and sides are related by the formula:

11. A straight line parallel to a side of a triangle and intersecting two others cuts off a triangle similar to this one from it.

12. If the bisectors of the anglesB and C triangle ABCintersect at point M, then .

13. The angle between the bisectors of adjacent angles is 90.

14. If M is the point of tangency with side AC of a circle inscribed in triangle ABC, then Where - semi-perimeter of a triangle.

15. The circle touches side BC of triangle ABC and the extensions of sides AB and AC. Then the distance from vertex A to the point of contact of the circle with line AB is equal to the semi-perimeter of triangle ABC.

16. A circle inscribed in triangle ABC touches sides AB, BC and AC respectively at pointsK, L And M. If, then.

17. Menelaus's theorem. Given triangle ABC. A certain straight line intersects its sides AB, BC and the continuation of side AC at points C 1, A 1, B 1 respectively. Then

18. Ceva's theorem. Let points A 1, B 1 and C 1 belong respectively to sides BC, AC and AB of triangle ABC. AA segments 1, BB 1 and SS 1 intersect at one point if and only if

19. Steiner-Lemus theorem. If two bisectors of a triangle are equal, then it is isosceles.

20. Stewart's theorem. Dot Dis located on side BC of triangle ABC, then .

21. An excircle is a circle tangent to one of its sides and the extensions of the other two.

22.For each triangle, there are three excircles that are located outside the triangle.

23.The center of an excircle is the point of intersection of the bisectors external corners a triangle and an internal bisector that is not adjacent to these two external ones.

24.If the circle touches side BC of triangle ABC and the extensions of sides AB and AC. Then the distance from vertex A to the point of contact of the circle with line AB is equal to the semi-perimeter of triangle ABC.

On straight line a (Fig. 7, o) points A, B and C are taken. Point B lies between points A and C. We can also say that points A and C lie on opposite sides of point B. Points A and B lie along one side from point C, they are not separated by point C. Points B and C lie on the same side of point A.

A segment is a part of a line that consists of all points of this line lying between two given points. These points are called the ends of the segment. A segment is indicated by indicating its ends.

In Figure 7, b, segment AB is part of straight line a. Point M lies between points A and B, and therefore belongs to the segment AB; Point K does not lie between points A and B, and therefore does not belong to segment AB.

The axiom (main property) of the location of points on a straight line is formulated as follows:

Of the three points on a line, one and only one lies between the other two.

The following axiom expresses the basic property of measuring segments.

Each segment has a certain length greater than zero. The length of a segment is equal to the sum of the lengths of the parts into which it is divided by any of its points.

This means that if we take any point C on the segment MK, then the length of the segment MK is equal to the sum of the lengths of the segments MS and SK (Fig. 7, c).

The length of the segment MK is also called the distance between points M and K.

Example 1. Three points O, P and M are given on a line. It is known that . Does point P lie between O and M? Can point B belong to the segment PM if ? Explain the answer.

Solution. Point P lies between points O and M, if Let's check whether this condition is met: . Conclusion: point P lies between points O and M.

Point B belongs to the segment RM if it lies between points P and M, i.e. Let's check: , and by condition . Conclusion: point B does not belong to the segment PM.

Example 2. Is it possible to arrange 6, 7 and 8 segments on a plane so that each of them intersects exactly three others?

Solution. 6 segments can be arranged this way (Fig. 8, o). 8 segments can also be arranged in this way (Fig. 8, b). 7 segments cannot be arranged like this.

Let us prove the last statement. Let us assume that such an arrangement of seven segments is possible. Let's number the segments and make such a table in a cell at the intersection of a row and a column, put “+” if the segment intersects with the j-th one, and “-” if it does not intersect. If then we also put Let’s count in two ways how many characters are in the table.

On the one hand, there are 3 of them in each line, so the total number of characters is . On the other hand, the table is filled symmetrically about the diagonal:

if in cell C: j) stands then in the cell too. This means that the total number of characters must be even. We got a contradiction.

Here we used the proof by contradiction.

5. Beam.

A half-line or ray is a part of a line that consists of all points of this line lying on one side of a given point. This point is called the starting point of the half-line or the beginning of the ray. Different half-lines of the same line with a common starting point are called complementary.

Half-lines are indicated by lowercase in Latin letters. You can denote a half-line by two letters: an initial letter and some other letter corresponding to a point belonging to the half-line. In this case, the starting point is placed in the first place. For example, in Figure 9, a, rays AB and AC are shown, which are additional; in Figure 9, b, rays MA, MB and ray c are shown.

The following axiom reflects the main property of laying down segments.

On any half-line from its starting point, you can plot a segment of a given length, and only one.

Example. Given two points A and B. How many lines can be drawn through points A and B? How many rays are there on straight line AB starting at point A and point B? Mark two points on line A B that are different from A and B. Do they belong to segment AB?

Solution. 1) According to the axiom, it is always possible to draw a straight line through points A and B, and only one.

2) On the line AB with the beginning at point A, there are two rays, which are called additional. Likewise for point B.

3) The answer depends on the location of the marked points. Let's consider possible cases(Fig. 10). It is clear that in case a) the points belong to the segment AB; in cases b), c) one point

belongs to the segment, but the other does not; in cases d) and e) points M and N do not belong to the segment AB.

6. Circle. Circle.

A circle is a figure that consists of all points of the plane located at a given distance from a given point. This point is called the center of the circle.

The distance from the points of a circle to its center is called the radius of the circle. A radius is also called any segment connecting a point on a circle to its center.

A segment connecting two points on a circle is called a chord. The chord passing through the center is called the diameter.

Figure 11, a shows a circle with a center at point O. Segment OA is the radius of this circle, BD is the chord of the circle, CM is the diameter of the circle.

A circle is a figure that consists of all points of the plane located at a distance not greater than a given one from a given point. This point is called the center of the circle, and this distance is called the radius of the circle. The boundary of a circle is a circle with the same center and radius (Fig. 11, b).

Example. For what greatest number various parts, having no common points except their boundaries, can divide a plane: a) a straight line and a circle; b) two circles; c) three circles?

Solution. Let us depict in the figure the cases of mutual arrangement of figures corresponding to the condition. Let's write down the answer: a) four parts (Fig. 12, o); b) four parts (Fig. 12, b); c) eight parts (Fig. 12, c).

7. Half-plane.

Let us formulate another axiom of geometry.

A straight line divides a plane into two half-planes.

In Figure 13, line a divides the plane into two half-planes so that every point of the plane that does not belong to line o lies in one of them. This partition has the following property: if the ends of a segment belong to the same half-plane, then the segment does not intersect with a line; if the ends of a segment belong to different half-planes, then the segment intersects the line. In Figure 13, the points lie in one of the half-planes into which Line a divides the plane. Therefore, segment AB does not intersect line a. Points C and D lie in different half-planes. Therefore, segment CD intersects line a.

8. Angle. Degree measure of angle.

An angle is a figure that consists of a point - the vertex of the angle and two different half-lines emanating from this point - the sides of the angle (Fig. 14). If the sides of an angle are complementary half-lines, then the angle is called a developed angle.

An angle is indicated either by indicating its vertex, or by indicating its sides, or by indicating three points; vertex and two points on the sides of the angle. The word "angle" is sometimes replaced by the symbol Z.

The angle in Figure 14 can be designated in three ways:

A ray c is said to pass between the sides of an angle if it comes from its vertex and intersects some segment with endpoints on the sides of the angle.

In Figure 15, the ray c passes between the sides of the angle since it intersects the segment AB.

In the case of a straight angle, any ray emanating from its vertex and different from its sides passes between the sides of the angle.

Angles are measured in degrees. If you take a straight angle and divide it into 180 equal angles, then the degree measure of each of these angles is called a degree.

The basic properties of angle measurement are expressed in the following axiom:

Each angle has a certain degree measure greater than zero. The rotated angle is 180°. The degree measure of an angle is equal to the sum of the degree measures of the angles into which it is divided by any ray passing between its sides.

This means that if a ray c passes between the sides of an angle, then the angle is equal to the sum of the angles

The degree measure of an angle is found using a protractor.

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 obtuse.

Let us formulate the main property of setting aside corners.

From any half-line, into a given half-plane, you can put an angle with a given degree measure less than 180°, and only one.

Consider the half-line a. Let us extend it beyond the starting point A. The resulting straight line divides the plane into two half-planes. Figure 16 shows how, using a protractor, to plot an angle with a given degree measure of 60° from a half-line to the upper half-plane.

If two angles from a given half-line are put into one half-plane, then the side of the smaller angle, different from the given half-line, passes between the sides of the larger angle.

Let the angles plotted from a given half-line a into one half-plane, and let the angle be less than the angle . Theorem 1. 2 states that ray b passes between the sides of the angle (ac) (Fig. 17).

The bisector of an angle is the ray that emanates from its vertex, passes between its sides and divides the angle in half. In Figure 18, the ray OM is the bisector of the angle AOB.

In geometry there is the concept of a plane angle. A plane angle is a part of a plane bounded by two different rays emanating from one point. These rays are called sides of the angle. There are two plane angles with given sides. They are called additional. In Figure 19, one of the plane angles with sides a and b is shaded.

If a plane angle is part of a half-plane, then its degree measure is the degree measure of an ordinary angle with the same sides. If a plane angle contains a half-plane, then its degree measure is 360° - a, where a is the degree measure of an additional plane angle.

Example. Ray a passes between the sides of an angle equal to 120°. Find the angles if their degree measures are in the ratio 4:2.

Solution. Ray a passes between the sides of the angle, which means, according to the main property of measuring angles (see paragraph 8)

Since degree measures have a ratio of 4:2, then

9. Adjacent and vertical angles.

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 20 the angles are adjacent.

The sum of adjacent angles is 180°.

The following properties follow from Theorem 1.3:

1) if two angles are equal, then their adjacent angles are equal;

2) an angle adjacent to a right angle is a right angle;

3) an angle adjacent to an acute one is obtuse, and an angle adjacent to an obtuse one is acute.

Two angles are called vertical if the sides of one angle are complementary half-lines of the sides of the other. In Figure 21, the corners are vertical.

Vertical angles are equal.

Obviously, two intersecting lines form adjacent and vertical angles. Adjacent angles complement each other up to 180°. The angular measure of the smaller of them is called the angle between the lines.

Example. In Figure 21, b, the angle is 30.° What are the angles AOC and

Solution. Angles COD and AOK are vertical, therefore, by Theorem 1.4 they are equal, i.e. Angle TYUK adjacent to angle SOD means, by Theorem 1.3

10. Central and inscribed angles.

A central angle in a circle is a plane angle with a vertex at its center. The part of the circle located inside a plane angle is called the circular arc corresponding to this central angle. The degree measure of an arc of a circle is the degree measure of the corresponding central angle.

In Figure 22, angle AOB is the central angle of a circle, its vertex O is the center of the circle, and sides OA and OB intersect the circle. Arc AB is the part of the circle located inside the central angle.

The degree measure of the arc AB in Figure 22 is equal to the degree measure of the angle AOB. The degree measure of the arc AB is designated AB.

An angle whose vertex lies on a circle and whose sides intersect this circle is called inscribed in the circle. Figure 23 shows inscribed angles.

An angle inscribed in a circle, the sides of which pass through two given points on the circle, is equal to half the angle between the radii drawn at these points, or complements this half to 180°.

When proving Theorem 1.5, it is necessary to consider three different cases, which are shown in Figure 23: one of the sides of the inscribed angle passes through the center of the circle (Figure 23, c); the center of the circle lies inside the inscribed angle (Fig. 23, b); the center of the circle lies outside the inscribed angle (Fig. 23, c).

A corollary follows from Theorem 1.5: all angles inscribed in a circle, the sides of which pass through two given points of the circle, and the vertices lie on the same side of the straight line connecting these points, are equal; inscribed angles whose sides pass through the ends of the diameter of a circle are right angles.

In Figure 24, the sides of the inscribed angle ABC pass through the ends of the diameter AC, so

Example. Points A at B and C lie on a circle with center O. Find angle AOC if

Solution. Angle ABC, inscribed in a circle, rests on arc AC, and the central angle of this circle (Fig. 25). , which means, by Theorem 1.5, and since the angle AOS is central, its degree measure is equal to the degree measure of the arc AC, i.e.

11. Parallel lines.

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

Figure 26 shows how to use a square and ruler to draw through this point To line 6, parallel to given line a.

To indicate the parallelism of lines, the symbol II is used. The entry reads: “Line a is parallel to line b.”

The axiom of parallelism expresses the basic property of parallel lines.

Through a point not lying on a given line, it is possible to draw on the plane at most one straight line parallel to the given one.

Two lines parallel to a third are parallel to each other.

In Figure 27, lines a and b are parallel to line c. Theorem 1.6 states that .

It can be proven that through a point that does not belong to a line, one can draw a line parallel to the given one. In Figure 28, through point A, which does not belong to b, a line a is drawn parallel to line b.

Comparing this statement and the axiom of parallels, we come to an important conclusion: on a plane, through a point not lying on a given line, you can draw a line parallel to it, and only one.

The axiom of parallelism in Euclid’s book “Elements” was called the “fifth postulate.” Ancient geometers tried to prove the uniqueness of the parallel. These unsuccessful attempts continued for more than 2000 years, until the 19th century.

The great Russian mathematician N. I. Lobachevsky and, independently of him, the Hungarian mathematician J. Bolyai showed that, by accepting the assumption that it is possible to draw through a point several straight lines parallel to a given one, it is possible to construct another, equally “correct” non-Euclidean geometry. This is how Lobachevsky's geometry was born.

An example of a theorem that uses the concept of parallelism, and its proof is based on the axiom of parallels, is Thales' theorem. Thales of Miletus - ancient Greek mathematician who lived in 625-547. BC e.

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

Let the points of intersection of parallel lines with one of the sides of the angle lie between (Fig. 29). Let the corresponding points of intersection of these lines be with the other side of the angle. Theorem 1.7 states that if then

Example 1. Can seven lines intersect at eight points?

Solution. They can. For example, Figure 30 shows seven such lines, three of which are parallel.

Example 2. Divide an arbitrary segment AC by 6 equal parts.

Solution. Let's draw a line segment AC. Let us draw a ray AM from point A that does not lie on the straight line AC. On the ray AM from point A we will sequentially lay out 6 equal segments (Fig. 31). Let's give designations to the ends of the segments. Connect the point with a segment to point C and draw lines parallel to the line through the points. The points of intersection of these lines with the segment AC will divide it into 6 equal parts (according to Theorem 1.7).

12. Signs of parallel lines.

Let AB and CD be two straight lines. Let AC be the third straight line intersecting straight lines AB and CD (Fig. 32, c). The straight line AC in relation to the straight lines AB and CD is called a secant. The angles formed by these right angles are often considered in pairs. Pairs of angles received special names. So, if points B and D lie in the same half-plane relative to straight line AC, then angles BAC and DCA are called internal one-sided (Fig. 32, c). If points B and D lie in different half-planes relative to straight line AC, then angles BAC and DCA are called internal cross-lying ones (Fig. 32, b).

The secant AC forms with straight lines AB and CD two pairs of internal one-sided two pairs of internal cross-lying angles Fig. 32, c).

If internal crosswise angles are equal or the sum of internal one-sided angles is equal to 180°, then the lines are parallel.

In Figure 32, c, four pairs of angles are indicated by numbers. Theorem 1.8 states that if or then the lines c and b are parallel. Theorem 1.8 also states that if or , then the lines a and b are parallel.

Theorems 1.6 and 1.8 are tests for the parallelism of lines. The theorem is also true, converse of the theorem 1.8.

If two parallel lines are intersected by a third line, then the intersecting internal angles are equal, and the sum of the internal one-sided angles is 180°.

Example. One of the interior one-sided angles formed by the intersection of two parallel lines with a third line is 4 times larger than the other. What are these angles equal to?

Solution. By Theorem 1.9, the sum of internal one-sided angles for two parallel lines and a transversal is equal to 180°. Let us denote these angles by the letters a and P, then a is known that a is 4 times greater, which means then So,

13. Perpendicular lines.

Two lines are called perpendicular if they intersect at right angles (Fig. 33).

The perpendicularity of lines is written using the symbol. The entry reads: “Line a is perpendicular to line b.”

A perpendicular to a given line is a segment of a line perpendicular to a given line, whose end is their point of intersection. This end of the segment is called the base of the perpendicular.

In Figure 34, a perpendicular A B is drawn from point A to straight line a. Point B is the base of the perpendicular.

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

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

The length of a perpendicular drawn from a given point to a straight line is called the distance from the point to the straight line.

The distance between parallel lines is the distance from any point on one line to another line.

Let BA be a perpendicular dropped from a point to straight line a, and C be any point of straight line c, different from A. The segment BC is called inclined, drawn from point B to straight line a (Fig. 35). Point C is called the base of the inclined point. The segment AC is called an oblique projection.

A straight line passing through the middle of a segment perpendicular to it is called a perpendicular bisector.

In Figure 36, straight line a is perpendicular to segment AB and passes through point C - the middle of segment AB, i.e. a is the bisector perpendicular.

Example. Equal segments AD and CB, enclosed between parallel lines AC and BD, intersect at point O. Prove that.

Solution. Let us draw perpendiculars to straight line BD from points A to C (Fig. 37). AK=CM as the distance between parallel lines, ZAKD and DSLYAV are rectangular, they

equal in hypotenuse and leg (see T. 1.25), and therefore isosceles (T. 1.19), and therefore, From the equality of triangles ACT) and CTAB it follows that , and then, i.e. A. AOS isosceles , which means

14. Tangent to a circle. Touching circles.

A straight line passing through a point on a circle perpendicular to the radius drawn to this point is called a tangent. In this case, this point on the circle is called the point of tangency. In Figure 38, straight line a is drawn through point A of the circle perpendicular to the radius OA. The line c is tangent to the circle. Point A is the point of contact. We can also say that the circle touches line a at point A.

Two circles having a common point are said to touch at that point if they have a common tangent at that point. The tangency of circles is called internal if the centers of the circles lie on the same side of their common tangent. The tangency of circles is called external if the centers of the circles lie on opposite sides of their common

tangent. In Figure 39, c, the contact of the circles is internal, and in Figure 39, b - external.

Example 1. Construct a circle of given radius tangent to a given line at a given point.

Solution. A tangent to a circle is perpendicular to the radius drawn to the point of tangency. Therefore, the center of the desired circle lies on the perpendicular to the given line passing through the given point, and is located from the given point at a distance equal to the radius. The problem has two solutions - two circles, symmetrical to each other relative to a given line (Fig. 40).

Example 2. Two circles with diameters of 4 and 8 cm touch externally. What is the distance between the centers of these circles?

Solution. The radii of circles OA and O, A are perpendicular to the common tangent passing through point A (Fig. 41). Therefore see

15. Triangles.

A triangle is a figure that consists of three points that do not lie on the same line and three segments connecting these points in pairs. The points are called the vertices of the triangle, and the segments are its sides. A triangle is indicated by its vertices. Instead of the word “triangle” the symbol D is used.

Figure 42 shows triangle ABC; A, B, C are the vertices of this triangle; A B, BC and AC are its sides.

The angle of triangle ABC at vertex A is the angle formed by half lines A B and AC. The angles of the triangle at vertices B to C are also determined.

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.

The altitude of a triangle dropped from a given vertex is the perpendicular drawn from this vertex to the line containing the opposite side of the triangle. In Figure 43, c, segment AD is the height of the acute-angled A. ABC, and in Figure 43, b, the base of the height of the obtuse-angled segment - point D - lies on the continuation of side BC.

The bisector of a triangle is the bisector segment of an angle of a triangle connecting a vertex to a point on the opposite side. In Figure 44, segment AD is the bisector of triangle ABC.

The median of a triangle drawn from a given vertex is the segment connecting this vertex with the middle

the opposite side of the triangle. In Figure 45, segment AD is the median of the triangle

The midline of a triangle is the segment connecting the midpoints of its two sides.

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

Let DE be the midline of triangle ABC (Fig. 46).

The theorem states that .

The triangle inequality is the property of distances between three points, which is expressed by the following theorem:

Whatever the three points, the distance between any two of these points is not greater than the sum of the distances from them to the third point.

Let there be three given points. The relative position of these points can be different: a) two points out of three or all three coincide, in this case the statement of the theorem is obvious; b) the points are different and lie on the same line (Fig. 47, a), one of them, for example B, lies between the other two, in this case it follows that each of the three distances is no more than the sum of the other two; c) the points do not lie

on one straight line (Fig. 47, b), then Theorem 1.14 states that .

In case c) three points A, B, C are the vertices of the triangle. Therefore, in any triangle, each side is less than the sum of the other two sides.

Example 1. Is there a triangle ABC with sides: a) ; b)

Solution. The sides of triangle ABC must satisfy the following inequalities:

In case a) inequality (2) is not satisfied, which means that such an arrangement of points cannot exist; in case b) the inequalities are satisfied, i.e. the triangle exists.

Example 2. Find the distance between points A and separated by an obstacle.

Solution. To find the distance, we hang the basis CD and draw straight lines BC and AD (Fig. 48). Find point M - the middle of CD. We also carry out MPAD. It follows that PN is the middle line, i.e.

By measuring PN, it is not difficult to find AB.

16. Equality of triangles.

Two segments are called equal if they have the same length. Two angles are said to be equal if they have the same angular measure in degrees.

Triangles ABC are said to be congruent if

This is briefly expressed in words: triangles are congruent if their corresponding sides and corresponding angles are equal.

Let us formulate the basic property of the existence of equal triangles (the axiom of the existence of a triangle equal to a given one):

Whatever a triangle is, there is an equal triangle in a given location relative to a given half-line.

Three conditions for equality of triangles are valid:

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 (a sign that triangles are equal along two sides and the angle between them).

If the side and adjacent angles of one triangle are equal, respectively, to the side and adjacent angles of another triangle, then such triangles are congruent (a sign of equality of triangles along a side and adjacent angles).

If three sides of one triangle are equal, respectively, to three sides of another triangle, then such triangles are congruent (a sign that triangles are equal on three sides).

Example. Points B and D lie in different half-planes relative to straight line AC (Fig. 49). It is known that Prove that

Solution. according to the condition, and since these angles are obtained by subtracting equal angles BC A and DAC from equal angles BCD and DAB. In addition, in these triangles the side AC is common. These triangles are equal in side and adjacent angles

17. Isosceles triangle.

A triangle is called isosceles if its two sides are equal. These equal sides are called the sides, and the third side is called the base of the triangle.

In a triangle it means ABC is isosceles with base AC.

In an isosceles triangle, the base angles are equal.

If two angles in a triangle are equal, then it is isosceles (the converse of Theorem T. 1.18).

In an isosceles triangle, the median drawn to the base is the bisector and the altitude.

It can also be proven that in an isosceles triangle the altitude drawn to the base is the bisector and the median. Similarly, the bisector of an isosceles triangle drawn from the vertex opposite the base is the median and altitude.

External angle of triangle ABC. In order not to confuse the angle of a triangle with a given straight line, the hypotenuse, NE and BA are legs.

For right triangles you can formulate your own signs of equality.

If the hypotenuse and acute angle of one right triangle are respectively equal to the hypotenuse and acute angle of another triangle, then such triangles are congruent (a sign of equality by hypotenuse and acute angle).

If the leg and the opposite angle of one right triangle are respectively equal to the leg and the opposite angle of another triangle, then such triangles are congruent (a sign of equality along the leg and the opposite angle).

If the hypotenuse and leg of one right triangle are respectively equal to the hypotenuse and leg of another triangle, then such triangles are congruent (a sign of equality by hypotenuse and leg).

In a right triangle with an angle of 30°, the leg opposite the atom is equal to half the hypotenuse.

In triangle ABC, shown in the drawing of a straight line, So, in this triangle .

In a right triangle, the Pythagorean theorem is valid, named after the ancient Greek scientist Pythagoras, who lived in the 6th century. BC e.

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs (Pythagorean theorem).

Let ABC be a given right triangle with right angle C, legs a and b and hypotenuse c (Fig. 56). The theorem states that

From the Pythagorean theorem it follows that in a right triangle any of the legs is less than the hypotenuse.

From the Pythagorean theorem it follows that if there is a perpendicular and an inclined line to a straight line from one point, then the inclined one is greater than the perpendicular; equal obliques have equal projections; Of the two inclined, the one with the larger projection is greater.

In Figure 57, from point O to straight line a, a perpendicular OA and inclined OB, OS and OD are drawn, and based on the above: a)

The perimeter of the KDMA rectangle is 18 cm

Example 3. In a circle with a radius of 25 cm, two parallel chords of length 40 and 30 cm are drawn on one side of its center. Find the distance between these chords.

Solution. Let's draw a radius OK perpendicular to the chords AB and CD, connect the center of the circle O with points C, A, D and B (Fig. 60). Triangles COD and AOB are isosceles, since (as radii); OM and ON are the heights of these triangles. By Theorem 1.20, each of the heights is simultaneously the median of the corresponding triangle, i.e., and

Triangles OSM and O AN are rectangular, in them . ON and OM will be found using the Pythagorean theorem.

20. Circles inscribed in a triangle and circumscribed about the triangle.

A circle is called circumscribed about a triangle if it passes through all its vertices.

The center of a circle circumscribed about a triangle is the point of intersection of the perpendicular bisectors to the sides of the triangle.

In Figure 61, a circle is circumscribed about triangle ABC. The center of this circle O is the intersection point of the bisectoral perpendiculars OM, ON and OJT, drawn respectively to the sides AB, BC and CA of A.

A circle is said to be inscribed in a triangle if it touches all its sides.

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

In Figure 62, the circle is inscribed in triangle ABC. The center of this circle O is the intersection point of the bisectors AO, BO and CO of the corresponding angles of the triangle.

Example. In a right triangle, the legs are equal to 12 and 16 cm. Calculate the radii of: 1) the circle inscribed in it; 2) circumscribed circle.

Solution. 1) Let triangle ABC be given, in which is the center of the inscribed circle (Fig. 63, a). The perimeter of triangle ABC is equal to the sum of the double hypotenuse and the diameter of the circle inscribed in the triangle (use the definition of a tangent to a circle and the equality of right triangles AOM and AOC, MOC and LOC by hypotenuse and leg).

Thus, from where, according to the Pythagorean theorem, i.e.

2) The center of the circumscribed circle around a right triangle coincides with the middle of the hypotenuse, where the radius of the circumscribed circle is cm (Fig. 63, b).

4. Triangles, quadrangles, polygons. Formulas for the areas of a triangle, rectangle, parallelogram, trapezoid.

5. Circle, circle.

Triangles

A triangle is one of the simplest geometric shapes. But its study gave birth to a whole science - trigonometry, which arose from practical needs in measuring land plots, drawing up maps of the area, designing various mechanisms.

Triangle called geometric figure, which consists of three points that do not lie on the same line and three pairwise segments connecting them.

Any triangle divides the plane into two parts: internal and external. A figure consisting of a triangle and its interior region is also called a triangle (or planar triangle).

In any triangle, the following elements are distinguished: sides, angles, altitudes, bisectors, medians, midlines.

The angle of a triangle ABC at vertex A is the angle formed by half lines AB and AC.

Height of a triangle dropped from a given vertex is called the perpendicular drawn from this vertex to the line containing the opposite side.

Bisector of a triangle is the bisector segment of an angle of a triangle connecting a vertex to a point on the opposite side.

Median of a triangle drawn from a given vertex is called a segment connecting this vertex with the midpoint of the opposite side.

Middle line of a triangle is the segment connecting the midpoints of its two sides.

Triangles are called congruent if their corresponding sides and corresponding angles are equal. In this case, the corresponding angles must lie opposite the corresponding sides.

In practice and in theoretical constructions, signs of equality of triangles are often used, which provide a faster solution to the question of the relationship between them. There are three such signs:

1. 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.

2. If the side and adjacent angles of one triangle are equal, respectively, to the side and adjacent angles of another triangle, then such triangles are congruent.

3. If three sides of one triangle are equal to three sides of another triangle, then such triangles are congruent.

The triangle is called isosceles, if its two sides are equal. These equal sides are called lateral, and the third side is called the base of the triangle.

Isosceles triangles have a number of properties, for example:

In an isosceles triangle, the median drawn to the base is the bisector and the altitude.

Let us note several properties of triangles.

1. The sum of the angles of a triangle is 180º.

From this property it follows that in any triangle at least two angles are acute.

2. The middle line of the triangle connecting the midpoints of the two sides is parallel to the third side and equal to its half.

3. In any triangle, each side is less than the sum of the other two sides.

For a right triangle, the Pythagorean theorem is true: the square of the hypotenuse is equal to the sum of the squares of the legs.

Quadrilaterals

Quadrangle is a figure that consists of four points and four consecutive segments connecting them, and no three of these points should lie on the same line, and the segments connecting them should not intersect. These points are called the vertices of the quadrilateral, and the segments connecting them are called its sides.

Any quadrilateral divides the plane into two parts: internal and external. A figure consisting of a quadrilateral and its interior region is also called a quadrilateral (or planar quadrilateral).

The vertices of a quadrilateral are called adjacent if they are the ends of one of its sides. Vertices that are not adjacent are called opposite. The segments connecting opposite vertices of a quadrilateral are called diagonals.

The sides of a quadrilateral emanating from the same vertex are called adjacent. Sides that do not have a common end are called opposite. In a quadrilateral ABCD, vertices A and B are opposite, sides AB and BC are adjacent, BC and AD are opposite; segments AC and ВD are the diagonals of this quadrilateral.

Quadrilaterals can be convex or non-convex. Thus, the quadrilateral ABCD is convex, and the quadrilateral KRMT is non-convex. Among convex quadrangles, parallelograms and trapezoids are distinguished.

A parallelogram is a quadrilateral whose opposite sides are parallel.

Let ABCD be a parallelogram. From vertex B we drop a perpendicular BE to line AD. Then the segment BE is called the height of the parallelogram corresponding to sides BC and AD. Segment

CM is the height of the parallelogram corresponding to sides CD and AB.

To simplify the recognition of parallelograms, consider the following sign: if the diagonals of a quadrilateral intersect and are divided in half by the intersection point, then this quadrilateral is a parallelogram.

A number of properties of a parallelogram that are not contained in its definition are formulated as theorems and proven. Among them:

1. The diagonals of a parallelogram intersect and are divided in half at the intersection point.

2. A parallelogram has opposite sides and opposite angles equal.

Let us now consider the definition of a trapezoid and its main property.

Trapeze is a quadrilateral whose only two opposite sides are parallel.

These parallel sides are called the bases of the trapezoid. The other two sides are called lateral.

The segment connecting the midpoints of the sides is called the midline of the trapezoid.

The midline of a trapezoid has the following property: it is parallel to the bases and equal to their half-sum.

Of the many parallelograms, rectangles and rhombuses are distinguished.

Rectangle is called a parallelogram in which all angles are right.

Based on this definition, it can be proven that the diagonals of a rectangle are equal.

Diamond is called a parallelogram in which all sides are equal.

Using this definition, we can prove that the diagonals of a rhombus intersect at right angles and are bisectors of its angles.

Squares are selected from many rectangles.

A square is a rectangle in which all sides are equal.

Since the sides of a square are equal, it is also a rhombus. Therefore, a square has the properties of a rectangle and a rhombus.

Polygons

A generalization of the concept of triangle and quadrilateral is the concept of polygon. It is defined through the concept of a broken line.

A broken line A₁A₂A₃...An is a figure that consists of points A₁, A₂, A₃, ..., An and the segments A₁A₂, A₂A₃, ..., An-₁An connecting them. Points А₁, А₂, А₃, …, Аn are called the vertices of the broken line, and the segments А₁А₂, А₂А₃, …, Аn-₁Аn are its links.

If a broken line has no self-intersections, then it is called simple. If its ends coincide, then it is called closed. About the broken lines shown in the figure we can say: a) – simple; b) – simple closed; c) is a closed broken line that is not simple.

A) b) c)

The length of a broken line is the sum of the lengths of its links.

It is known that the length of a broken line is not less than the length of the segment connecting its ends.

Polygon A simple closed broken line is called if its neighboring links do not lie on the same straight line.

The vertices of the broken line are called the vertices of the polygon, and its links are called its sides. Line segments connecting non-adjacent vertices are called diagonals.

Any polygon divides the plane into two parts, one of which is called the inner and the other - the outer region of the polygon (or planar polygon).

There are convex and non-convex polygons.

A convex polygon is called regular if all its sides and all angles are equal.

The correct one is equilateral triangle, a regular quadrilateral is a square.

The angle of a convex polygon at a given vertex is the angle formed by its sides converging at this vertex.

It is known that the sum of the angles of a convex n-gon is 180º (n – 2).

In geometry, in addition to convex and non-convex polygons, polygonal figures are also considered.

A polygonal figure is the union of a finite set of polygons.

A) b) c)

The polygons that make up a polygonal figure may not have common interior points, but they may also have common interior points.

A polygonal figure F is said to consist of polygonal figures if it is their union and the figures themselves do not have common interior points. For example, the polygonal figures shown in figures a) and c) can be said to consist of two polygonal figures, or that they are divided into two polygonal figures.

Circle and circle

Circumference is a figure that consists of all points of the plane equidistant from a given point, called center.

Any segment connecting a point on a circle to its center is called the radius of the circle. Radius also called the distance from any point on a circle to its center.

A line segment connecting two points on a circle is called chord. The chord passing through the center is called diameter.

The boundary of a circle is a circle with the same center and radius.

Let us recall some properties of the circle and circle.

A line and a circle are said to touch if they have a single point in common. Such a line is called a tangent, and the common point of the line and the circle is called a point of tangency. It has been proven that if a straight line touches a circle, then it is perpendicular to the radius drawn to the point of contact. The converse statement is also true (Fig. a).

A central angle in a circle is a plane angle with a vertex at its center. The part of the circle located inside the plane angle is called the arc of the circle corresponding to this central angle (Fig.b).

An angle whose vertex lies on a circle and whose sides intersect it is called inscribed in this circle (Fig. c).

An angle inscribed in a circle has the following property: it is equal to half the corresponding central angle. In particular, the angles based on the diameter are right angles.

A circle is called circumscribed about a triangle if it passes through all its vertices.

To describe a circle around a triangle, you need to find its center. The rule for finding it is justified by the following theorem:

The center of a circle circumscribed about a triangle is the point of intersection of perpendiculars to its sides drawn through the midpoints of these sides (Fig.a).

A circle is said to be inscribed in a triangle if it touches all its sides.

The rule for finding the center of such a circle is justified by the theorem:

The center of a circle inscribed in a triangle is the intersection point of its bisectors (Fig.b)

Thus, the perpendicular bisectors and bisectors intersect at one point, respectively. In geometry it is proven that the medians of a triangle intersect at one point. This point is called the center of gravity of the triangle, and the point of intersection of the altitudes is called the orthocenter.

Thus, in any triangle there are four remarkable points: the center of gravity, the centers of the inscribed and circumscribed circles, and the orthocenter.

A circle can be circumscribed around any regular polygon, and a circle can be inscribed into any regular polygon, and the centers of the circumscribed and inscribed circles coincide.

§ 4. Mathematical proof

26. Schemes of deductive reasoning.

§5. Text problem and its solution process

29. Structure of a word problem

30. Methods and methods for solving word problems

31. Stages of solving a problem and techniques for their implementation

2. Search and drawing up a plan for solving the problem

3. Implementation of a plan to solve the problem

4. Checking the solution to the problem

5. Modeling in the process of solving word problems

Exercises

32. Solving problems “in parts”

Exercises

33. Solving motion problems

Exercises

34. Main conclusions.

§6. Combinatorial problems and their solutions

§ 7. Algorithms and their properties

Exercises

Chapter II. Elements of algebra

§ 8. Correspondences between two sets

41. The concept of compliance. Methods for specifying correspondences

2. Graph and correspondence graph. Correspondence is the inverse of the given one. Types of correspondences.

3. One-to-one correspondences

Exercises

42. One-to-one correspondences. The concept of a one-to-one mapping from a set x to a set y

2. Equivalent sets. Methods for establishing equal cardinality of sets. Countable and uncountable sets.

Exercises

43. Main conclusions § 8

§ 9. Numerical functions

44. Concept of function. Methods for specifying functions

2. Graph of a function. Property of monotonicity of a function

Exercises

45. Direct and inverse proportionality

Exercises

46. Main conclusions § 9

§10. Relationships on the set

47. The concept of a relation on a set

Exercises

48. Properties of relationships

R is reflexive on x ↔ x r x for any x € X.

R is symmetrical on x ↔ (x r y →yRx).

49. Equivalence and order relations

Exercises

50. Main conclusions § 10

§ 11. Algebraic operations on a set

51. The concept of algebraic operation

Exercises

52. Properties of algebraic operations

Exercises

53. Main conclusions § 11

§ 12. Expressions. Equations. Inequalities

54. Expressions and their identical transformations

Exercises

55. Numerical equalities and inequalities

Exercises

56. Equations with one variable

2. Equivalent equations. Theorems on the equivalence of equations

3. Solving equations with one variable

Exercises

57. Inequalities with one variable

2. Equivalent inequalities. Theorems on the equivalence of inequalities

3. Solving inequalities with one variable

Exercises

58. Main conclusions § 12

Exercises

Chapter III. Natural numbers and zero

§ 13. From the history of the emergence of the concept of natural number

§ 14. Axiomatic construction of a system of natural numbers

59. About the axiomatic method of constructing a theory

Exercises

60. Basic concepts and axioms. Definition of natural number

Exercises

61. Addition

62. Multiplication

63. Order of the set of natural numbers

Exercises

64. Subtraction

Exercises

65. Division

66. Set of non-negative integers

Exercises

67. Method of mathematical induction

Exercises

68. Quantitative natural numbers. Check

Exercises

69. Main conclusions § 14

70. Set-theoretic meaning of the natural number, zero and the “less than” relation

Exercises

Lecture 36. Set-theoretic approach to constructing a set of non-negative integers.

71. Set-theoretic meaning of sum

Exercises

72. Set-theoretic meaning of difference

Exercises

73. Set-theoretic meaning of a work

Exercises

74. Set-theoretic meaning of the quotient of natural numbers

Exercises

75. Main conclusions § 15

§16. Natural number as a measure of magnitude

76. The concept of a positive scalar quantity and its measurement

Exercises

77. The meaning of a natural number obtained as a result of measuring a quantity. The meaning of sum and difference

Exercises

78. The meaning of the product and quotient of natural numbers obtained as a result of measuring quantities

79. Main conclusions § 16

80. Positional and non-positional number systems

81. Writing a number in the decimal system

Exercises

82. Addition algorithm

Exercises

83. Subtraction algorithm

Exercises

84. Multiplication algorithm

Exercises

85. Division algorithm

86. Positional number systems other than decimal

87. Main conclusions § 17

§ 18. Divisibility of natural numbers

88. Divisibility relation and its properties

89. Signs of divisibility

90. Least common multiple and greatest common divisor

2. Basic properties of the least common multiple and greatest common divisor of numbers

3. Divisibility test for a composite number

Exercises

91. Prime numbers

92. Methods for finding the greatest common divisor and least common multiple of numbers

93. Main conclusions § 18

3. Distributivity:

§ 19. On the expansion of the set of natural numbers

94. The concept of a fraction

Exercises

95. Positive rational numbers

96. The set of positive rational numbers as an extension

97. Writing positive rational numbers as decimals

98. Real numbers

99. Main conclusions § 19

Chapter IV. Geometric shapes and quantities

§ 20. From the history of the emergence and development of geometry

1. The essence of the axiomatic method in theory construction

2. The emergence of geometry. Geometry of Euclid and geometry of Lobachevsky

3. The system of geometric concepts studied at school. Basic properties of belonging of points and lines, relative positions of points on a plane and a line.

§ 21. Properties of geometric figures on the plane

§ 22. Construction of geometric figures

1. Elementary construction tasks

2. Stages of solving the construction problem

Exercises

3. Methods for solving construction problems: transformations of geometric figures on a plane: central, axial symmetry, homothety, motion.

Key Findings

§24. Image of spatial figures on a plane

1. Properties of parallel design

2. Polyhedra and their image

Tetrahedron Cube Octahedron

Exercises

3. Sphere, cylinder, cone and their image

Key Findings

§ 25. Geometric quantities

1. Length of a segment and its measurement

1) Equal segments have equal lengths;

2) If a segment consists of two segments, then its length is equal to the sum of the lengths of its parts.

Exercises

2. Magnitude of an angle and its measurement Every angle has a magnitude. Special name for her in

1) Equal angles have equal magnitudes;

2) If an angle consists of two angles, then its value is equal to the sum of the sizes of its parts.

Exercises

1) Equal figures have equal areas;

2) If a figure consists of two parts, then its area is equal to the sum of the areas of these parts.

4. Area of a polygon

5. Area of an arbitrary flat figure and its measurement

Exercises

Key Findings

1. The concept of a positive scalar quantity and its measurement

1) The mass is the same for bodies balancing each other on scales;

2) Mass adds up when bodies are combined together: the mass of several bodies taken together is equal to the sum of their masses.

Conclusion

References

§ 21. Properties of geometric figures on the plane

Lecture 53. Properties of geometric figures on the plane

1. Geometric figures on a plane and their properties

2. Angles, parallel and perpendicular lines

3. Parallel and perpendicular lines

A geometric figure is defined as any set of points. A segment, a straight line, a circle, a ball are geometric shapes.

If all the points of a geometric figure belong to one plane, it is called flat. For example, a segment, a rectangle are flat figures. There are figures that are not flat. This is, for example, a cube, a ball, a pyramid.

Since the concept of a geometric figure is defined through the concept of a set, we can say that one figure is included in another (or contained in another), we can consider the union, intersection and difference of figures.

For example, the union of two rays AB and MK is the straight line KB, and their intersection is the segment AM.

There are convex and non-convex figures. A figure is called convex if, together with any two of its points, it also contains a segment connecting them.

The figures F₁ are convex, and the figure F₂ is non-convex.

Convex figures are a plane, a straight line, a ray, a segment, a point, and a circle.

For polygons, another definition is known: a polygon is called convex if it lies on one side of each straight line containing its side. Since the equivalence of this definition and the one given above for a polygon has been proven, we can use both.

Let's consider some concepts studied in the school geometry course, their definitions and properties, accepting them without proof.

Angles

Corner is a geometric figure that consists of a point and two rays emanating from this point. The rays are called the sides of the angle, and their common beginning is its vertex.

An angle is designated in different ways: either its vertex, or its sides, or three points are indicated: the vertex and points on the sides of the angle: A,(k,l), ABC.

The angle is called expanded, if its sides lie on the same straight line.

An angle that is half a straight angle is called direct. An angle less than a right angle is called sharp. An angle greater than a right angle but less than a straight angle is called stupid.

Flat angle- this is a part of the plane limited by two different rays emanating from one point.

There are two plane angles formed by two rays with a common origin. They are called additional.

ABOUT

The angles considered in planimetry do not exceed the unfolded angle.

The two angles are called adjacent, if they have one side in common, and the other sides of these angles are additional half-lines.

The sum of adjacent angles is 180º. The validity of this property follows from their definition of adjacent angles.

The two angles are called vertical, if the sides of one angle are complementary half-lines of the sides of the other.

Vertical angles are equal.

Parallel and perpendicular lines

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

If line a is parallel to line b, then write a║b.

Let's consider some properties of parallel lines, and first of all, the signs of parallelism.

Signs are theorems that establish the presence of any property of an object in a certain situation. In particular, the need to consider the signs of parallel lines is caused by the fact that in practice it is often necessary to resolve the issue of the relative position of two lines, but at the same time it is impossible to directly use the definition.

Consider the following signs of parallel lines:

1. Two lines parallel to a third are parallel to each other.

2. If internal crosswise angles are equal or the sum of internal one-sided angles is equal to 180º, then the lines are parallel.

It is a true statement the opposite the second sign of parallelism of lines: if two parallel lines are intersected by a third, then the internal angles lying across each other are equal, and the sum of one-sided angles is 180º.

An important property of parallel lines is revealed in theorem named after the ancient Greek mathematician Thales: if parallel lines intersecting the sides of an angle cut off equal segments on one side, then they cut off equal segments on the other side.

Two straight lines are called perpendicular if they intersect at right angles.

If line a is perpendicular to line b, then write ab.

The basic properties of perpendicular lines are reflected in two theorems:

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

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

A perpendicular to a given line is a segment of a line perpendicular to a given line and ending at their point of intersection. The end of this segment is called the base of the perpendicular.

The length of the perpendicular dropped from a given point to a straight line is called distance from a point to a straight line.

Distance between parallel lines is the distance from any point on one line to another.

Lecture 54. Properties of geometric figures on the plane

4. Triangles, quadrangles, polygons. Formulas for the areas of a triangle, rectangle, parallelogram, trapezoid.

5. Circle, circle.

Triangles

Triangle is a geometric figure that consists of three points that do not lie on the same line and three pairwise segments connecting them.