Signs of parallelism of two lines
Theorem 1. If, when two lines intersect with a secant:
crossed angles are equal, or
corresponding angles equal, or
the sum of one-sided angles is 180°, then
lines are parallel(Fig. 1).
Proof. We limit ourselves to proving case 1.
Let the intersecting lines a and b be crosswise and the angles AB be equal. For example, ∠ 4 = ∠ 6. Let us prove that a || b.
Suppose that lines a and b are not parallel. Then they intersect at some point M and, therefore, one of the angles 4 or 6 will be the external angle of triangle ABM. For definiteness, let ∠ 4 be the external angle of the triangle ABM, and ∠ 6 the internal one. From the theorem about external angle triangle it follows that ∠ 4 is greater than ∠ 6, and this contradicts the condition, which means that lines a and 6 cannot intersect, so they are parallel.
Corollary 1. Two different lines in a plane perpendicular to the same line are parallel(Fig. 2).
Comment. The way we just proved case 1 of Theorem 1 is called the method of proof by contradiction or reduction to absurdity. This method received its first name because at the beginning of the argument an assumption is made that is contrary (opposite) to what needs to be proven. It is called leading to absurdity due to the fact that, reasoning on the basis of the assumption made, we come to an absurd conclusion (to the absurd). Receiving such a conclusion forces us to reject the assumption made at the beginning and accept the one that needed to be proven.
Task 1. Construct a line passing through a given point M and parallel to a given line a, not passing through the point M.
Solution. We draw a straight line p through the point M perpendicular to the straight line a (Fig. 3).
Then we draw a line b through point M perpendicular to the line p. Line b is parallel to line a according to the corollary of Theorem 1.
An important conclusion follows from the problem considered:
through a point not lying on a given line, it is always possible to draw a line parallel to the given one.
The main property of parallel lines is as follows.
Axiom of parallel lines. Through a given point that does not lie on a given line, there passes only one line parallel to the given one.
Let us consider some properties of parallel lines that follow from this axiom.
1) If a line intersects one of two parallel lines, then it also intersects the other (Fig. 4).
2) If two different lines are parallel to a third line, then they are parallel (Fig. 5).
The following theorem is also true.
Theorem 2. If two parallel lines are intersected by a transversal, then:
crosswise angles are equal;
corresponding angles are equal;
the sum of one-sided angles is 180°.
Corollary 2. If a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other(see Fig. 2).
Comment. Theorem 2 is called the inverse of Theorem 1. The conclusion of Theorem 1 is the condition of Theorem 2. And the condition of Theorem 1 is the conclusion of Theorem 2. Not every theorem has an inverse, i.e. if this theorem is true, then converse theorem may be incorrect.
Let us explain this using the example of the theorem on vertical angles. This theorem can be formulated as follows: if two angles are vertical, then they are equal. The converse theorem would be: if two angles are equal, then they are vertical. And this, of course, is not true. Two equal angles don't have to be vertical at all.
Example 1. Two parallel lines are crossed by a third. It is known that the difference between two internal one-sided angles is 30°. Find these angles.
Solution. Let Figure 6 meet the condition.
Which lie in the same plane and either coincide or do not intersect. In some school definitions, coincident lines are not considered parallel; such a definition is not considered here.
Properties
- Parallelism is a binary equivalence relation, therefore it divides the entire set of lines into classes of lines parallel to each other.
- Through any point you can draw exactly one straight line parallel to the given one. This is a distinctive property of Euclidean geometry; in other geometries the number 1 is replaced by others (in Lobachevsky geometry there are at least two such lines)
- 2 parallel lines in space lie in the same plane.
- When 2 parallel lines intersect, a third one, called secant:
- The secant necessarily intersects both lines.
- When intersecting, 8 angles are formed, some characteristic pairs of which have special names and properties:
- Lying crosswise the angles are equal.
- Relevant the angles are equal.
- Unilateral the angles add up to 180°.
In Lobachevsky geometry
In Lobachevsky geometry in the plane through a point C outside this line AB there are an infinite number of straight lines that do not intersect AB. Of these, parallel to AB only two are named. Straight CE called an equilateral (parallel) line AB in the direction from A To B, If:
- points B And E lie on one side of a straight line AC ;
- straight CE does not intersect a line AB, but every ray passing inside an angle ACE, crosses the ray AB .
A straight line is defined similarly AB in the direction from B To A .
All other lines that do not intersect this one are called ultraparallel or divergent.
see also
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This theorem is about parallel lines. For an angle based on a diameter, see another theorem. Thales' theorem is one of the theorems of planimetry. If you lay out several equal segments in succession on one of two straight lines and draw through their ends... ... Wikipedia The Russian Order of St. Anna was established by the sovereign Duke of Schleswig of Holstein, Karl Frederick, in 1736 in honor of his wife, Tsarevna Anna Petrovna (daughter of Peter the Great) and included in the Russian orders by the Emperor Peter III
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Question 1. What angles are called adjacent?
Answer. 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 31, the angles (a 1 b) and (a 2 b) are adjacent. They have side b in common, and sides a 1 and a 2 are additional half-lines.
Question 2. Prove that the amount adjacent corners equal to 180°.
Answer. Theorem 2.1. The sum of adjacent angles is 180°.
Proof. Let angle (a 1 b) and angle (a 2 b) be given adjacent angles (see Fig. 31). Ray b passes between sides a 1 and a 2 of a straight angle. Therefore, the sum of the angles (a 1 b) and (a 2 b) is equal to the unfolded angle, i.e. 180°. Q.E.D.
Question 3. Prove that if two angles are equal, then their adjacent angles are also equal.
Answer.
From the theorem 2.1
It follows that if two angles are equal, then their adjacent angles are equal.
Let's say the angles (a 1 b) and (c 1 d) are equal. We need to prove that the angles (a 2 b) and (c 2 d) are also equal.
The sum of adjacent angles is 180°. It follows from this that a 1 b + a 2 b = 180° and c 1 d + c 2 d = 180°. Hence, a 2 b = 180° - a 1 b and c 2 d = 180° - c 1 d. Since the angles (a 1 b) and (c 1 d) are equal, we get that a 2 b = 180° - a 1 b = c 2 d. By the property of transitivity of the equal sign it follows that a 2 b = c 2 d. Q.E.D.
Question 4. What angle is called right (acute, obtuse)?
Answer. 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.
Question 5. Prove that an angle adjacent to a right angle is a right angle.
Answer. From the theorem on the sum of adjacent angles it follows that an angle adjacent to a right angle is a right angle: x + 90° = 180°, x = 180° - 90°, x = 90°.
Question 6. What angles are called vertical?
Answer. Two angles are called vertical if the sides of one angle are complementary half-lines of the sides of the other.
Question 7. Prove that vertical angles are equal.
Answer. Theorem 2.2. Vertical angles are equal.
Proof. Let (a 1 b 1) and (a 2 b 2) be the given vertical angles (Fig. 34). Angle (a 1 b 2) is adjacent to angle (a 1 b 1) and to angle (a 2 b 2). From here, using the theorem on the sum of adjacent angles, we conclude that each of the angles (a 1 b 1) and (a 2 b 2) complements the angle (a 1 b 2) to 180°, i.e. angles (a 1 b 1) and (a 2 b 2) are equal. Q.E.D.
Question 8. Prove that if, when two lines intersect, one of the angles is right, then the other three angles are also right.
Answer. Suppose lines AB and CD intersect each other at point O. Suppose angle AOD is 90°. Since the sum of adjacent angles is 180°, we get that AOC = 180° - AOD = 180° - 90° = 90°. Angle COB is vertical to angle AOD, so they are equal. That is, angle COB = 90°. Angle COA is vertical to angle BOD, so they are equal. That is, angle BOD = 90°. Thus, all angles are equal to 90°, that is, they are all right angles. Q.E.D.
Question 9. Which lines are called perpendicular? What sign is used to indicate perpendicularity of lines?
Answer. Two lines are called perpendicular if they intersect at right angles.
The perpendicularity of lines is indicated by the sign \(\perp\). The entry \(a\perp b\) reads: “Line a is perpendicular to line b.”
Question 10. Prove that through any point on a line you can draw a line perpendicular to it, and only one.
Answer. Theorem 2.3. Through each line you can draw a line perpendicular to it, and only one.
Proof. Let a be a given line and A a given point on it. Let us denote by a 1 one of the half-lines of the straight line a with the starting point A (Fig. 38). Let us subtract an angle (a 1 b 1) equal to 90° from the half-line a 1. Then the straight line containing the ray b 1 will be perpendicular to the straight line a.
Let us assume that there is another line, also passing through point A and perpendicular to line a. Let us denote by c 1 the half-line of this line lying in the same half-plane with the ray b 1 .
Angles (a 1 b 1) and (a 1 c 1), each equal to 90°, are laid out in one half-plane from the half-line a 1. But from the half-line a 1 only one angle equal to 90° can be put into a given half-plane. Therefore, there cannot be another line passing through point A and perpendicular to line a. The theorem has been proven.
Question 11. What is perpendicular to a line?
Answer. A perpendicular to a given line is a segment of a line perpendicular to a given line, which has one of its ends at their intersection point. This end of the segment is called basis perpendicular.
Question 12. Explain what proof by contradiction consists of.
Answer. The proof method we used in Theorem 2.3 is called proof by contradiction. This method of proof is that we first make an assumption opposite to what the theorem states. Then, by reasoning, relying on axioms and proven theorems, we come to a conclusion that contradicts either the conditions of the theorem, or one of the axioms, or a previously proven theorem. On this basis, we conclude that our assumption was incorrect, and therefore the statement of the theorem is true.
Question 13. What is the bisector of an angle?
Answer. The bisector of an angle is a ray that emanates from the vertex of the angle, passes between its sides and divides the angle in half.
