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Completed by a student of grade 7 "G" MBOU "OK "Lyceum No. 3" Gavrilov Dmitry
Axiom
Comes from the Greek “axios”, which means “valuable, worthy”. A position accepted without logical proof due to immediate persuasiveness is the true starting position of the theory. (Soviet encyclopedic dictionary)
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Axiom of parallel lines Completed by a student of grade 7 "G" MBOU "OK "Lyceum No. 3" Gavrilov Dmitry 2015-2016 academic year (teacher Konareva T.N.)
Known definitions and facts. Finish the sentence. 1. Line x is called a transversal in relation to lines a and b if... 2. When two straight lines intersect, a transversal forms... undeveloped angles. 3. If lines AB and C D are intersected by line B D, then line B D is called... 4. If points B and D lie in different half-planes relative to the secant AC, then angles BAC and DCA are called... 5. If points B and D lie in one half-plane relative to the secant AC, then the angles BAC and DCA are called... 6. If the interior angles of one pair are equal, then the interior angles of the other pair are equal... D C A C B D A B
Checking the task. 1 . ...if it intersects them at two points 2. 8 3. ... secant 4. ... lying crosswise 5. ... one-sided 6. ... equal
Match a) a b m 1) a | | b, since internal crosswise angles are equal b) 2) a | | b, since the corresponding angles are equal c) a b 3) a | | b, since the sum of internal one-sided angles is equal to 180° 50 º 130 º 45 º 45 º m a b m a 150 º 150º
About the axioms of geometry
Axiom Comes from the Greek "axios", which means "valuable, worthy." A position accepted without logical proof due to immediate persuasiveness is the true initial position of the theory. Soviet encyclopedic dictionary
A straight line passes through any two points, and only one. How many straight lines can be drawn through any two points lying on a plane?
On any ray, from its beginning, one can lay off a segment equal to the given one, and, moreover, only one. How many segments of a given length can be laid off from the beginning of the ray?
From any ray in a given direction it is possible to plot an angle equal to a given undeveloped angle, and only one. How many angles equal to a given one can be plotted from a given ray to a given half-plane?
axioms theorems logical reasoning famous essay “Principia” Euclidean geometry Logical construction of geometry
Axiom of parallel lines
M a Let us prove that through the point M it is possible to draw a line parallel to the line a c b a ┴ c b ┴ c a II c
Is it possible to draw another line through point M parallel to line a? a M in 1 Is it possible to prove this?
Many mathematicians, since ancient times, have tried to prove this statement, and in Euclid’s Elements this statement is called the fifth postulate. Attempts to prove Euclid's fifth postulate were unsuccessful, and only in the 19th century it was finally clarified that the statement about the uniqueness of a line passing through a given point parallel to a given line cannot be proven on the basis of the rest of Euclid's axioms, but is itself an axiom. The Russian mathematician Nikolai Ivanovich Lobachevsky played a huge role in solving this issue.
Fifth postulate of Euclid 1792-1856 Nikolai Ivanovich
“Through a point not lying on a given line, only one line parallel to the given line passes.” “Through a point not lying on a given line, one can draw a line parallel to the given one.” Which of these statements is an axiom? How are the above statements different?
Through a point not lying on a given line there passes only one line parallel to the given one. Statements that are derived from axioms or theorems are called corollaries. Corollary 1. If a line intersects one of two parallel lines, then it also intersects the other. a II b , c b ⇒ c a Axiom of parallelism and consequences from it. a A Corollary 2. If two lines are parallel to a third line, then they are parallel. a II c, b II c a II b a b c c b
Consolidation of knowledge. Test Mark correct statements with a “+” sign and erroneous statements with a “-” sign. Option 1 1. An axiom is a mathematical statement about the properties of geometric figures that requires proof. 2. A straight line passes through any two points. 3. On any ray, from the beginning, you can plot segments equal to the given one, and as many as you like. 4. Through a point not lying on a given line, only one line parallel to the given line passes. 5. If two lines are parallel to a third, then they are parallel to each other. Option 2 1. An axiom is a mathematical statement about the properties of geometric figures, accepted without proof. 2. A straight line passes through any two points, and only one. 3. Through a point not lying on a given line, only two lines parallel to the given line pass through. 4. If a line intersects one of two parallel lines, then it is perpendicular to the other line. 5. If a line intersects one of two parallel lines, then it also intersects the other.
Test answers Option 1 1. “-” 2. “-” 3. “-” 4. “+” 5. “+” Option 2 “+” “+” “-” “-” “+”
“Geometry is full of adventure because behind every problem lies an adventure of thought. Solving a problem means experiencing an adventure.” (V. Proizvolov)
§ 1 Axiom of parallel lines
Let's find out which statements are called axioms, give examples of axioms, formulate the axiom of parallel lines and consider some of its consequences.
When studying geometric figures and their properties, the need arises to prove various statements - theorems. When proving them, they often rely on previously proven theorems. The question arises: what are the proofs of the very first theorems based on? In geometry, some initial assumptions are accepted, and on their basis the following theorems are proved. Such initial provisions are called axioms. The axiom is accepted without proof. The word axiom comes from the Greek word "axios", which means "valuable, worthy."
We are already familiar with some axioms. For example, an axiom is the statement: through any two points there passes a straight line, and only one.
When comparing two segments and two angles, we superimposed one segment on the other, and superimposed the angle on the other angle. The possibility of such an imposition follows from the following axioms:
· on any ray from its beginning it is possible to plot a segment equal to the given one, and only one;
· from any ray in a given direction you can put off an angle equal to a given undeveloped angle, and, moreover, only one.
Geometry is an ancient science. For almost two millennia, geometry was studied according to the famous work “Elements” by the ancient Greek scientist Euclid. Euclid first formulated the starting points - postulates, and then, based on them, through logical reasoning he proved other statements. The geometry presented in the Principia is called Euclidean geometry. In the scientist’s manuscripts there is a statement called the fifth postulate, around which controversy flared up for a very long time. Many mathematicians have attempted to prove Euclid's fifth postulate, i.e. derive it from other axioms, but each time the proofs were incomplete or reached a dead end. Only in the 19th century was it finally clarified that the fifth postulate cannot be proven on the basis of the remaining axioms of Euclid, and is itself an axiom. The Russian mathematician Nikolai Ivanovich Lobachevsky (1792-1856) played a huge role in solving this issue. So, the fifth postulate is the axiom of parallel lines.
Axiom: through a point not lying on a given line there passes only one line parallel to the given one.
§ 2 Corollaries from the axiom of parallel lines
Statements that are derived directly from axioms or theorems are called corollaries. Let's consider some corollaries from the axiom of parallel lines.
Corollary 1. If a line intersects one of two parallel lines, then it also intersects the other.
Given: lines a and b are parallel, line c intersects line a at point A.
Prove: line c intersects line b.
Proof: if line c did not intersect line b, then two lines a and c would pass through point A, parallel to line b. But this contradicts the axiom of parallel lines: through a point not lying on a given line, only one line parallel to the given line passes. This means that line c intersects line b.
Corollary 2. If two lines are parallel to a third line, then they are parallel.
Given: lines a and b are parallel to line c. (a||c, b||c)
Prove: line a is parallel to line b.
Proof: let’s assume that lines a and b are not parallel, i.e. intersect at some point A. Then two lines a and b pass through point A, parallel to line c. But according to the axiom of parallel lines, through a point not lying on a given line, only one straight line passes through it, parallel to the given one. This means that our assumption is incorrect, therefore, lines a and b are parallel.
List of used literature:
- Geometry. Grades 7-9: textbook. for general education organizations / L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev et al. - M.: Education, 2013. - 383 p.: ill.
- Gavrilova N.F. Lesson developments in geometry grade 7. - M.: “VAKO”, 2004, 288 p. - (To help the school teacher).
- Belitskaya O.V. Geometry. 7th grade. Part 1. Tests. – Saratov: Lyceum, 2014. – 64 p.
Images used:
1. If two lines are parallel to a 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 when two straight lines intersect with 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 internal crosswise angles are equal, then the lines are parallel:
If ∠1 = ∠3, then a||b.
Properties of parallel lines
Statements inverse to the properties of parallel lines are their properties. They are based on the properties of angles formed by the intersection of two parallel lines with a third line.
1. When two parallel lines intersect a third line, the sum of the internal one-sided angles formed by them is equal to 180°:
If a||b, then ∠1 + ∠2 = 180°.
2. When two parallel lines intersect a third line, the corresponding angles formed by them are equal:
If a||b, then ∠2 = ∠4.
3. When two parallel lines intersect a third line, the crosswise angles they form are equal:
If a||b, then ∠1 = ∠3.
The following property is a special case for each previous one:
4. If a line on a plane is perpendicular to one of 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.