Definition 1
The straight line $c$ is called secant for lines $a$ and $b$ if it intersects them at two points.
Consider two lines $a$ and $b$ and a secant line $c$.
When they intersect, angles appear, which we denote by numbers from $1$ to $8$.
Each of these angles has a name that is often used in mathematics:
- pairs of angles $3$ and $5$, $4$ and $6$ are called lying crosswise;
- pairs of angles $1$ and $5$, $4$ and $8$, $2$ and $6$, $3$ and $7$ are called relevant;
- pairs of angles $4$ and $5$, $5$ and $6$ are called unilateral.
Signs of parallel lines
Theorem 1
The equality of the pair of crosswise lying angles for the lines $a$ and $b$ and the secant $c$ says that the lines $a$ and $b$ are parallel:
Proof.
Let the cross-lying angles for the lines $а$ and $b$ and the secant $с$ be equal: $∠1=∠2$.
Let us show that $a \parallel b$.
Provided that the angles $1$ and $2$ are right, we get that the lines $a$ and $b$ are perpendicular to the line $AB$, and therefore parallel.
Provided that the angles $1$ and $2$ are not right, we draw from the point $O$, the midpoint of the segment $AB$, the perpendicular $ON$ to the line $a$.
On the line $b$ we set aside the segment $BH_1=AH$ and draw the segment $OH_1$. We get two equal triangles $OHA$ and $OH_1B$ on two sides and the angle between them ($∠1=∠2$, $AO=BO$, $BH_1=AH$), so $∠3=∠4$ and $ ∠5=∠6$. Because $∠3=∠4$, then the point $H_1$ lies on the ray $OH$, so the points $H$, $O$ and $H_1$ belong to the same line. Because $∠5=∠6$, then $∠6=90^(\circ)$. Thus, the lines $а$ and $b$ are perpendicular to the line $HH_1$ and are parallel. The theorem has been proven.
Theorem 2
The equality of the pair of corresponding angles for the lines $a$ and $b$ and the secant $c$ means that the lines $a$ and $b$ are parallel:
if $∠1=∠2$, then $a \parallel b$.
Proof.
Let the corresponding angles for the lines $а$ and $b$ and the secant $с$ be equal: $∠1=∠2$. Angles $2$ and $3$ are vertical, so $∠2=∠3$. So $∠1=∠3$. Because angles $1$ and $3$ are crosswise, then lines $a$ and $b$ are parallel. The theorem has been proven.
Theorem 3
If the sum of two one-sided angles for lines $a$ and $b$ and secant $c$ is equal to $180^(\circ)C$, then lines $a$ and $b$ are parallel:
if $∠1+∠4=180^(\circ)$ then $a \parallel b$.
Proof.
Let the one-sided angles for the lines $a$ and $b$ and the secant $c$ add up to $180^(\circ)$, for example
$∠1+∠4=180^(\circ)$.
Angles $3$ and $4$ are adjacent, so
$∠3+∠4=180^(\circ)$.
It can be seen from the obtained equalities that the cross-lying angles are $∠1=∠3$, which implies that the lines $a$ and $b$ are parallel.
The theorem has been proven.
Parallelism of straight lines follows from the considered signs.
Examples of problem solving
Example 1
The intersection point bisects segments $AB$ and $CD$. Prove that $AC \parallel BD$.
Given: $AO=OB$, $CO=OD$.
Prove: $AC\parallel BD$.
Proof.
From the conditions of the problem $AO=OB$, $CO=OD$ and the equality of vertical angles $∠1=∠2$ according to the I-th triangle equality criterion it follows that $\bigtriangleup COA=\bigtriangleup DOB$. Thus, $∠3=∠4$.
Angles $3$ and $4$ are crosswise at two lines $AC$ and $BD$ and secant $AB$. Then, according to the I-th criterion of parallelism of lines $AC \parallel BD$. The assertion has been proven.
Example 2
Given an angle $∠2=45^(\circ)$, and $∠7$ is $3$ times the given angle. Prove that $a \parallel b$.
Given: $∠2=45^(\circ)$, $∠7=3∠2$.
Prove: $a \parallel b$.
Proof:
- Find the value of the angle $7$:
$∠7=3 \cdot 45^(\circ)=135^(\circ)$.
- Vertical angles $∠5=∠7=135^(\circ)$, $∠2=∠4=45^(\circ)$.
- Find the sum of the interior angles $∠5+∠4=135^(\circ)+45^(\circ)=180^(\circ)$.
According to the III-th criterion of parallelism of lines $a \parallel b$. The assertion has been proven.
Example 3
Given: $\bigtriangleup ABC=\bigtriangleup ADB$.
Prove: $AC \parallel BD$, $AD \parallel BC$.
Proof:
The considered drawings have a common side $AB$.
Because triangles $ABC$ and $ADB$ are equal, then $AD=CB$, $AC=BD$, and the corresponding angles are $∠1=∠2$, $∠3=∠4$, $∠5=∠6 $.
The pair of angles $3$ and $4$ are cross-lying for the lines $AC$ and $BD$ and the corresponding secant $AB$, therefore, according to the I-th criterion of parallelism of the lines $AC \parallel BD$.
The pair of angles $5$ and $6$ are cross-lying for the lines $AD$ and $BC$ and the corresponding secant $AB$, therefore, according to the I-th criterion of parallelism of the lines $AD \parallel BC$.
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Question 1. Prove that two lines parallel to the third are parallel.
Answer. Theorem 4.1. Two lines parallel to a third are parallel.
Proof. Let lines a and b be parallel to line c. Assume that a and b are not parallel (Fig. 69). Then they do not intersect at some point C. Hence, two lines pass through the point C and are parallel to the line c. But this is impossible, since through a point that does not lie on a given line, at most one line parallel to the given line can be drawn. The theorem has been proven.
Question 2. Explain what angles are called internal one-sided. What angles are called internal cross-lying?
Answer. Pairs of angles that are formed when lines AB and CD intersect AC have special names.
If the points B and D lie in the same half-plane relative to the straight line AC, then the angles BAC and DCA are called internal one-sided (Fig. 71, a).
If the points B and D lie in different half-planes with respect to the line AC, then the angles BAC and DCA are called internal crosswise lying (Fig. 71, b).
Rice. 71
Question 3. Prove that if the internal cross-lying angles of one pair are equal, then the internal cross-lying angles of the other pair are also equal, and the sum of the internal one-sided angles of each pair is 180°.
Answer. The secant AC forms with lines AB and CD two pairs of internal one-sided and two pairs of internal cross-lying angles. The internal cross lying corners of one pair, for example, angle 1 and angle 2, are adjacent to the internal cross lying angles of another pair: angle 3 and angle 4 (Fig. 72).
Rice. 72
Therefore, if the internal cross-lying angles of one pair are equal, then the internal cross-lying angles of the other pair are also equal.
A pair of interior cross-lying corners, such as angle 1 and angle 2, and a pair of interior one-sided corners, such as angle 2 and angle 3, have one common angle, angle 2, and two other adjacent angles, angle 1 and angle 3.
Therefore, if the interior cross-lying angles are equal, then the sum of the interior angles is 180°. And vice versa: if the sum of interior cross-lying angles is equal to 180°, then the interior cross-lying angles are equal. Q.E.D.
Question 4. Prove the criterion for parallel lines.
Answer. Theorem 4.2 (test for parallel lines). If interior cross-lying angles are equal or the sum of interior one-sided angles is 180°, then the lines are parallel.
Proof. Let the lines a and b form equal internal crosswise lying angles with the secant AB (Fig. 73, a). Suppose the lines a and b are not parallel, which means they intersect at some point C (Fig. 73, b).
Rice. 73
The secant AB splits the plane into two half-planes. Point C lies in one of them. Let's construct triangle BAC 1 , equal to triangle ABC, with vertex C 1 in the other half-plane. By condition, the internal cross-lying angles for parallel a, b and secant AB are equal. Since the corresponding angles of triangles ABC and BAC 1 with vertices A and B are equal, they coincide with the internal cross-lying angles. Hence, line AC 1 coincides with line a, and line BC 1 coincides with line b. It turns out that two different lines a and b pass through the points C and C 1. And this is impossible. So lines a and b are parallel.
If lines a and b and secant AB have the sum of internal one-sided angles equal to 180°, then, as we know, the internal cross-lying angles are equal. Hence, by what was proved above, the lines a and b are parallel. The theorem has been proven.
Question 5. Explain what angles are called corresponding. Prove that if interior cross-lying angles are equal, then the corresponding angles are also equal, and vice versa.
Answer. If a pair of internal cross-lying angles has one angle replaced by a vertical one, then a pair of angles will be obtained, which are called the corresponding angles of the given lines with a secant. Which is what needed to be explained.
From the equality of internal cross-lying angles follows the equality of the corresponding angles, and vice versa. Let's say we have two parallel lines (because by condition the internal cross-lying angles are equal) and a secant, which form angles 1, 2, 3. Angles 1 and 2 are equal as internal cross-lying. And angles 2 and 3 are equal as vertical. We get: \(\angle\)1 = \(\angle\)2 and \(\angle\)2 = \(\angle\)3. By the property of transitivity of the equal sign, it follows that \(\angle\)1 = \(\angle\)3. The converse assertion is proved similarly.
This results in a sign of parallel lines at the corresponding angles. Namely, lines are parallel if the corresponding angles are equal. Q.E.D.
Question 6. Prove that through a point not lying on a given line, it is possible to draw a line parallel to it. How many lines parallel to a given line can be drawn through a point not on this line?
Answer. Problem (8). Given a line AB and a point C not lying on this line. Prove that through point C it is possible to draw a line parallel to line AB.
Solution. The straight line AC divides the plane into two half-planes (Fig. 75). Point B lies in one of them. From the half-line CA, let us plot the angle ACD equal to the angle CAB into the other half-plane. Then lines AB and CD will be parallel. Indeed, for these lines and the secant AC, the angles BAC and DCA are interior crosswise. And since they are equal, lines AB and CD are parallel. Q.E.D.
Comparing the statement of problem 8 and axiom IX (the main property of parallel lines), we come to an important conclusion: through a point that does not lie on a given line, one can draw a line parallel to it, and only one.
Question 7. Prove that if two lines intersect with a third line, then the interior cross-lying angles are equal, and the sum of interior one-sided angles is 180°.
Answer. Theorem 4.3(converse to Theorem 4.2). If two parallel lines intersect with a third line, then the interior cross-lying angles are equal, and the sum of interior one-sided angles is 180°.
Proof. Let a and b be parallel lines and c be the line that intersects them at points A and B. Draw a line a 1 through point A so that the internal cross-lying angles formed by the secant c with lines a 1 and b are equal (Fig. 76).
By the criterion of parallelism of lines, lines a 1 and b are parallel. And since only one line passes through the point A, parallel to the line b, then the line a coincides with the line a 1 .
This means that the internal cross-lying angles formed by the secant with
parallel lines a and b are equal. The theorem has been proven.
Question 8. Prove that two lines perpendicular to a third are parallel. If a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.
Answer. It follows from Theorem 4.2 that two lines perpendicular to a third are parallel.
Assume that any two lines are perpendicular to the third line. Hence, these lines intersect with the third line at an angle equal to 90°.
From the property of the angles formed at the intersection of parallel lines by a secant, it follows that if a line is perpendicular to one of the parallel lines, then it is also perpendicular to the other.
Question 9. Prove that the sum of the angles of a triangle is 180°.
Answer. Theorem 4.4. The sum of the angles of a triangle is 180°.
Proof. Let ABC be the given triangle. Draw a line through vertex B parallel to line AC. Mark a point D on it so that the points A and D lie on opposite sides of the line BC (Fig. 78).
Angles DBC and ACB are equal as internal crosswise, formed by the secant BC with parallel lines AC and BD. Therefore, the sum of the angles of the triangle at the vertices B and C is equal to the angle ABD.
And the sum of all three angles of a triangle is equal to the sum of angles ABD and BAC. Since these angles are internal one-sided for parallel AC and BD and secant AB, their sum is 180°. The theorem has been proven.
Question 10. Prove that any triangle has at least two acute angles.
Answer. Indeed, suppose that a triangle has only one acute angle or no acute angles at all. Then this triangle has two angles, each of which is at least 90°. The sum of these two angles is no less than 180°. But this is impossible, since the sum of all the angles of a triangle is 180°. Q.E.D.
1. The first sign of parallelism.
If, at the intersection of two lines with a third, the interior angles lying across are equal, then these lines are parallel.
Let lines AB and CD be intersected by line EF and ∠1 = ∠2. Let's take the point O - the middle of the segment KL of the secant EF (Fig.).
Let us drop the perpendicular OM from the point O to the line AB and continue it until it intersects with the line CD, AB ⊥ MN. Let us prove that CD ⊥ MN as well.
To do this, consider two triangles: MOE and NOK. These triangles are equal to each other. Indeed: ∠1 = ∠2 by the hypothesis of the theorem; OK = OL - by construction;
∠MOL = ∠NOK as vertical angles. Thus, 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; therefore, ΔMOL = ΔNOK, and hence ∠LMO = ∠KNO,
but ∠LMO is direct, hence ∠KNO is also direct. Thus, the lines AB and CD are perpendicular to the same line MN, therefore, they are parallel, which was to be proved.
Note. The intersection of the lines MO and CD can be established by rotating the triangle MOL around the point O by 180°.
2. The second sign of parallelism.
Let's see if the lines AB and CD are parallel if, at the intersection of their third line EF, the corresponding angles are equal.
Let some corresponding angles be equal, for example ∠ 3 = ∠2 (Fig.);
∠3 = ∠1 as vertical angles; so ∠2 will be equal to ∠1. But angles 2 and 1 are internal crosswise angles, and we already know that if at the intersection of two lines by a third, the internal crosswise lying angles are equal, then these lines are parallel. Therefore, AB || CD.
If at the intersection of two lines of the third the corresponding angles are equal, then these two lines are parallel.
The construction of parallel lines with the help of a ruler and a drawing triangle is based on this property. This is done as follows.
Let us attach a triangle to the ruler as shown in Fig. We will move the triangle so that one side of it slides along the ruler, and draw several straight lines along any other side of the triangle. These lines will be parallel.
3. The third sign of parallelism.
Let us know that at the intersection of two lines AB and CD by the third line, the sum of any internal one-sided angles is equal to 2 d(or 180°). Will the lines AB and CD be parallel in this case (Fig.).
Let ∠1 and ∠2 be one-sided interior angles and add up to 2 d.
But ∠3 + ∠2 = 2 d as adjacent angles. Therefore, ∠1 + ∠2 = ∠3+ ∠2.
Hence ∠1 = ∠3, and these interior angles are crosswise. Therefore, AB || CD.
If at the intersection of two lines by a third, the sum of the interior one-sided angles is equal to 2 d (or 180°), then the two lines are parallel.
Signs of parallel lines:
1. If at the intersection of two straight lines by a third, the internal cross lying angles are equal, then these lines are parallel.2. If at the intersection of two lines of the third, the corresponding angles are equal, then these two lines are parallel.
3. If at the intersection of two lines of the third, the sum of the internal one-sided angles is 180 °, then these two lines are parallel.
4. If two lines are parallel to the third line, then they are parallel to each other.
5. If two lines are perpendicular to the third line, then they are parallel to each other.
Euclid's axiom of parallelism
Task. Through a point M taken outside the line AB, draw a line parallel to the line AB.
Using the proven theorems on the signs of parallelism of lines, this problem can be solved in various ways,
Solution. 1st s o s o b (Fig. 199).
We draw MN⊥AB and through the point M we draw CD⊥MN;
we get CD⊥MN and AB⊥MN.
Based on the theorem ("If two lines are perpendicular to the same line, then they are parallel.") we conclude that СD || AB.
2nd s p o s o b (Fig. 200).
We draw a MK intersecting AB at any angle α, and through the point M we draw a straight line EF, forming an angle EMK with a straight line MK, equal to the angle α. Based on the theorem () we conclude that EF || AB.
Having solved this problem, we can consider it proved that through any point M, taken outside the line AB, it is possible to draw a line parallel to it. The question arises, how many lines parallel to a given line and passing through a given point can exist?
The practice of constructions allows us to assume that there is only one such line, since with a carefully executed drawing, lines drawn in various ways through the same point parallel to the same line merge.
In theory, the answer to this question is given by the so-called axiom of Euclid's parallelism; it is formulated like this:
Through a point taken outside a given line, only one line can be drawn parallel to this line.
In the drawing 201, a straight line SK is drawn through the point O, parallel to the straight line AB.
Any other line passing through the point O will no longer be parallel to the line AB, but will intersect it.
The axiom adopted by Euclid in his Elements, which states that on a plane through a point taken outside a given line, only one line can be drawn parallel to this line, is called Euclid's axiom of parallelism.
For more than two thousand years after Euclid, many mathematicians tried to prove this mathematical proposition, but their attempts were always unsuccessful. Only in 1826, the great Russian scientist, professor of Kazan University Nikolai Ivanovich Lobachevsky proved that, using all other Euclid's axioms, this mathematical proposition cannot be proved, that it really should be taken as an axiom. N. I. Lobachevsky created a new geometry, which, in contrast to the geometry of Euclid, was called the geometry of Lobachevsky.
1. If two lines are parallel to the third line, then they are parallel:
If a||c And b||c, That a||b.
2. If two lines are perpendicular to the third line, then they are parallel:
If a⊥c And b⊥c, That a||b.
The remaining signs of parallelism of lines are based on the angles formed at the intersection of two lines by a third.
3. If the sum of internal one-sided angles is 180°, then the lines are parallel:
If ∠1 + ∠2 = 180°, then a||b.
4. If the corresponding angles are equal, then the lines are parallel:
If ∠2 = ∠4, then a||b.
5. If the internal cross-lying angles are equal, then the lines are parallel:
If ∠1 = ∠3, then a||b.
Properties of parallel lines
Statements that are inverse to the signs of parallelism of lines are their properties. They are based on the properties of the angles formed by the intersection of two parallel lines by a third line.
1. When two parallel lines intersect with a third line, the sum of the internal one-sided angles formed by them is 180 °:
If a||b, then ∠1 + ∠2 = 180°.
2. When two parallel lines intersect with a third line, the corresponding angles formed by them are equal:
If a||b, then ∠2 = ∠4.
3. At the intersection of two parallel lines by a third line, the lying angles formed by them across are equal:
If a||b, then ∠1 = ∠3.
The following property is a special case of each previous one:
4. If a line on a plane is perpendicular to one of the two parallel lines, then it is also perpendicular to the other:
If a||b And c⊥a, That c⊥b.
The fifth property is the axiom of parallel lines:
5. Through a point not lying on a given line, only one line can be drawn parallel to the given line.
Parallel lines. Properties and signs of parallel lines1. Axiom of parallel. Through a given point, at most one straight line can be drawn parallel to the given one.
2. If two lines are parallel to the same line, then they are parallel to each other.
3. Two lines perpendicular to the same line are parallel.
4. If two parallel lines are intersected by a third, then the internal cross-lying angles formed at the same time are equal; corresponding angles are equal; interior one-sided angles add up to 180°.
5. If, at the intersection of two straight lines, the third one forms equal internal cross-lying angles, then the straight lines are parallel.
6. If at the intersection of two lines the third form equal corresponding angles, then the lines are parallel.
7. If at the intersection of two lines of the third, the sum of the internal one-sided angles is 180 °, then the lines are parallel.
Thales' theorem. If equal segments are laid out on one side of the angle and parallel straight lines are drawn through their ends, intersecting the second side of the angle, then equal segments will also be deposited on the second side of the angle.
Theorem on proportional segments. Parallel straight lines intersecting the sides of the angle cut proportional segments on them.
Triangle. Signs of equality of triangles.
1. 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 the triangles are congruent.
2. 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 the triangles are congruent.
3. If three sides of one triangle are respectively equal to three sides of another triangle, then the triangles are congruent.
Signs of equality of right triangles
1. On two legs.
2. Along the leg and hypotenuse.
3. By hypotenuse and acute angle.
4. Along the leg and an acute angle.
The theorem on the sum of the angles of a triangle and its consequences
1. The sum of the interior angles of a triangle is 180°.
2. The external angle of a triangle is equal to the sum of two internal angles not adjacent to it.
3. The sum of the interior angles of a convex n-gon is
4. The sum of the external angles of a ga-gon is 360°.
5. Angles with mutually perpendicular sides are equal if they are both acute or both obtuse.
6. The angle between the bisectors of adjacent angles is 90°.
7. The bisectors of internal one-sided angles with parallel lines and a secant are perpendicular.
The main properties and signs of an isosceles triangle
1. The angles at the base of an isosceles triangle are equal.
2. If two angles of a triangle are equal, then it is isosceles.
3. In an isosceles triangle, the median, bisector and height drawn to the base are the same.
4. If any pair of segments from the triple - median, bisector, height - coincides in a triangle, then it is isosceles.
The triangle inequality and its consequences
1. The sum of two sides of a triangle is greater than its third side.
2. The sum of the links of the broken line is greater than the segment connecting the beginning
the first link with the end of the last.
3. Opposite the larger angle of the triangle lies the larger side.
4. Against the larger side of the triangle lies a larger angle.
5. The hypotenuse of a right triangle is greater than the leg.
6. If perpendicular and inclined are drawn from one point to a straight line, then
1) the perpendicular is shorter than the inclined ones;
2) a larger slope corresponds to a larger projection and vice versa.
The middle line of the triangle.
The line segment connecting the midpoints of the two sides of a triangle is called the midline of the triangle.
Triangle midline theorem.
The median line of the triangle is parallel to the side of the triangle and equal to half of it.
Triangle median theorems
1. The medians of a triangle intersect at one point and divide it in a ratio of 2: 1, counting from the top.
2. If the median of a triangle is equal to half of the side to which it is drawn, then the triangle is right-angled.
3. The median of a right triangle drawn from the vertex of the right angle is equal to half of the hypotenuse.
Property of perpendicular bisectors to the sides of a triangle. The perpendicular bisectors to the sides of the triangle intersect at one point, which is the center of the circle circumscribed about the triangle.
Triangle altitude theorem. The lines containing the altitudes of the triangle intersect at one point.
Triangle bisector theorem. The bisectors of a triangle intersect at one point, which is the center of the circle inscribed in the triangle.
Bisector property of a triangle. The bisector of a triangle divides its side into segments proportional to the other two sides.
Signs of similarity of triangles
1. If two angles of one triangle are respectively equal to two angles of another, then the triangles are similar.
2. If two sides of one triangle are respectively proportional to two sides of another, and the angles enclosed between these sides are equal, then the triangles are similar.
3. If the three sides of one triangle are respectively proportional to the three sides of another, then the triangles are similar.
Areas of Similar Triangles
1. The ratio of the areas of similar triangles is equal to the square of the similarity coefficient.
2. If two triangles have equal angles, then their areas are related as the products of the sides that enclose these angles.
In a right triangle
1. The leg of a right triangle is equal to the product of the hypotenuse and the sine of the opposite or the cosine of the acute angle adjacent to this leg.
2. The leg of a right triangle is equal to the other leg multiplied by the tangent of the opposite or the cotangent of the acute angle adjacent to this leg.
3. The leg of a right triangle lying opposite an angle of 30 ° is equal to half the hypotenuse.
4. If the leg of a right triangle is equal to half of the hypotenuse, then the angle opposite this leg is 30°.
5. R = ; g \u003d, where a, b are legs, and c is the hypotenuse of a right triangle; r and R are the radii of the inscribed and circumscribed circles, respectively.
The Pythagorean theorem and the converse of the Pythagorean theorem
1. The square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs.
2. If the square of a side of a triangle is equal to the sum of the squares of its other two sides, then the triangle is right-angled.
Mean proportionals in a right triangle.
The height of a right triangle, drawn from the vertex of the right angle, is the average proportional to the projections of the legs onto the hypotenuse, and each leg is the average proportional to the hypotenuse and its projection onto the hypotenuse.
Metric ratios in a triangle
1. Theorem of cosines. The square of a 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.
2. Corollary from the cosine theorem. The sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of all its sides.
3. Formula for the median of a triangle. If m is the median of the triangle drawn to side c, then m = 
where a and b are the remaining sides of the triangle.
4. Sine theorem. The sides of a triangle are proportional to the sines of the opposite angles.
5. Generalized sine theorem. The ratio of a side of a triangle to the sine of the opposite angle is equal to the diameter of the circle circumscribing the triangle.
Triangle area formulas
1. The area of a triangle is half the product of the base and the height.
2. The area of a triangle is equal to half the product of its two sides and the sine of the angle between them.
3. The area of a triangle is equal to the product of its semiperimeter and the radius of the inscribed circle.
4. The area of a triangle is equal to the product of its three sides divided by four times the radius of the circumscribed circle.
5. Heron's formula: S=, where p is the semiperimeter; a, b, c - sides of the triangle.
Elements of an equilateral triangle. Let h, S, r, R be the height, area, radii of the inscribed and circumscribed circles of an equilateral triangle with side a. Then
Quadrilaterals
Parallelogram. A parallelogram is a quadrilateral whose opposite sides are pairwise parallel.
Properties and features of a parallelogram.
1. The diagonal divides the parallelogram into two equal triangles.
2. Opposite sides of a parallelogram are equal in pairs.
3. Opposite angles of a parallelogram are equal in pairs.
4. The diagonals of the parallelogram intersect and bisect the point of intersection.
5. If the opposite sides of a quadrilateral are equal in pairs, then this quadrilateral is a parallelogram.
6. If two opposite sides of a quadrilateral are equal and parallel, then this quadrilateral is a parallelogram.
7. If the diagonals of a quadrilateral are bisected by the intersection point, then this quadrilateral is a parallelogram.
Property of the midpoints of the sides of a quadrilateral. The midpoints of the sides of any quadrilateral are the vertices of a parallelogram whose area is half the area of the quadrilateral.
Rectangle. A rectangle is a parallelogram with a right angle.
Properties and signs of a rectangle.
1. The diagonals of a rectangle are equal.
2. If the diagonals of a parallelogram are equal, then this parallelogram is a rectangle.
Square. A square is a rectangle all sides of which are equal.
Rhombus. A rhombus is a quadrilateral all sides of which are equal.
Properties and signs of a rhombus.
1. The diagonals of the rhombus are perpendicular.
2. The diagonals of a rhombus bisect its corners.
3. If the diagonals of a parallelogram are perpendicular, then this parallelogram is a rhombus.
4. If the diagonals of a parallelogram divide its angles in half, then this parallelogram is a rhombus.
Trapeze. A trapezoid is a quadrilateral in which only two opposite sides (bases) are parallel. The median line of a trapezoid is a segment connecting the midpoints of non-parallel sides (lateral sides).
1. The median line of the trapezoid is parallel to the bases and equal to their half-sum.
2. The segment connecting the midpoints of the diagonals of the trapezoid is equal to the half-difference of the bases.
Remarkable property of a trapezoid. The point of intersection of the diagonals of the trapezoid, the point of intersection of the extensions of the sides and the midpoints of the bases lie on the same straight line.
Isosceles trapezium. A trapezoid is called isosceles if its sides are equal.
Properties and signs of an isosceles trapezoid.
1. The angles at the base of an isosceles trapezoid are equal.
2. The diagonals of an isosceles trapezoid are equal.
3. If the angles at the base of the trapezoid are equal, then it is isosceles.
4. If the diagonals of a trapezoid are equal, then it is isosceles.
5. The projection of the lateral side of an isosceles trapezoid onto the base is equal to the half-difference of the bases, and the projection of the diagonal is half the sum of the bases.
Formulas for the area of a quadrilateral
1. The area of a parallelogram is equal to the product of the base and the height.
2. The area of a parallelogram is equal to the product of its adjacent sides and the sine of the angle between them.
3. The area of a rectangle is equal to the product of its two adjacent sides.
4. The area of a rhombus is half the product of its diagonals.
5. The area of a trapezoid is equal to the product of half the sum of the bases and the height.
6. The area of a quadrilateral is equal to half the product of its diagonals and the sine of the angle between them.
7. Heron's formula for a quadrilateral around which a circle can be described:
S \u003d, where a, b, c, d are the sides of this quadrilateral, p is the semi-perimeter, and S is the area.
Similar figures
1. The ratio of the corresponding linear elements of similar figures is equal to the similarity coefficient.
2. The ratio of the areas of similar figures is equal to the square of the similarity coefficient.
regular polygon.
Let a n be the side of a regular n-gon, and r n and R n be the radii of the inscribed and circumscribed circles. Then
Circle.
A circle is the locus of points in a plane that are at the same positive distance from a given point, called the center of the circle.
Basic properties of a circle
1. The diameter perpendicular to the chord divides the chord and the arcs it subtracts in half.
2. A diameter passing through the middle of a chord that is not a diameter is perpendicular to that chord.
3. The median perpendicular to the chord passes through the center of the circle.
4. Equal chords are removed from the center of the circle at equal distances.
5. The chords of a circle that are equidistant from the center are equal.
6. The circle is symmetrical with respect to any of its diameters.
7. Arcs of a circle enclosed between parallel chords are equal.
8. Of the two chords, the one that is less distant from the center is larger.
9. Diameter is the largest chord of a circle.
Tangent to circle. A line that has a single point in common with a circle is called a tangent to the circle.
1. The tangent is perpendicular to the radius drawn to the point of contact.
2. If the line a passing through a point on the circle is perpendicular to the radius drawn to this point, then the line a is tangent to the circle.
3. If the lines passing through the point M touch the circle at points A and B, then MA = MB and ﮮAMO = ﮮBMO, where the point O is the center of the circle.
4. The center of a circle inscribed in an angle lies on the bisector of this angle.
tangent circle. Two circles are said to touch if they have a single common point (tangent point).
1. The point of contact of two circles lies on their line of centers.
2. Circles of radii r and R with centers O 1 and O 2 touch externally if and only if R + r \u003d O 1 O 2.
3. Circles of radii r and R (r
4. Circles with centers O 1 and O 2 touch externally at point K. Some straight line touches these circles at different points A and B and intersects with a common tangent passing through point K at point C. Then ﮮAK B \u003d 90 ° and ﮮO 1 CO 2 \u003d 90 °.
5. The segment of the common external tangent to two tangent circles of radii r and R is equal to the segment of the common internal tangent enclosed between the common external ones. Both of these segments are equal.
Angles associated with a circle
1. The value of the arc of a circle is equal to the value of the central angle based on it.
2. An inscribed angle is equal to half the angular magnitude of the arc on which it rests.
3. Inscribed angles based on the same arc are equal.
4. The angle between intersecting chords is equal to half the sum of opposite arcs cut by the chords.
5. The angle between two secants intersecting outside the circle is equal to the half-difference of the arcs cut by the secants on the circle.
6. The angle between the tangent and the chord drawn from the point of contact is equal to half the angular value of the arc cut on the circle by this chord.
Properties of circle chords
1. The line of centers of two intersecting circles is perpendicular to their common chord.
2. The products of the lengths of the segments of the chords AB and CD of the circle intersecting at the point E are equal, that is, AE EB \u003d CE ED.
Inscribed and circumscribed circles
1. The centers of the inscribed and circumscribed circles of a regular triangle coincide.
2. The center of a circle circumscribed about a right triangle is the midpoint of the hypotenuse.
3. If a circle can be inscribed in a quadrilateral, then the sums of its opposite sides are equal.
4. If a quadrilateral can be inscribed in a circle, then the sum of its opposite angles is 180°.
5. If the sum of the opposite angles of a quadrilateral is 180°, then a circle can be circumscribed around it.
6. If a circle can be inscribed in a trapezoid, then the lateral side of the trapezoid is visible from the center of the circle at a right angle.
7. If a circle can be inscribed in a trapezoid, then the radius of the circle is the average proportional to the segments into which the tangent point divides the side.
8. If a circle can be inscribed in a polygon, then its area is equal to the product of the semiperimeter of the polygon and the radius of this circle.
The tangent and secant theorem and its corollary
1. If a tangent and a secant are drawn from one point to the circle, then the product of the entire secant by its outer part is equal to the square of the tangent.
2. The product of the entire secant by its outer part for a given point and a given circle is constant.
The circumference of a circle of radius R is C= 2πR
