3 signs of the parallelity of two lines on the plane: proof

This article will provide information on the signs of the parallelism of lines on the plane. See evidence of straightness parallelism, presented examples and drawings for a visual explanation of this topic.

Content

From the textbook on geometry it follows that straight on the plane are considered parallel to the plane, which do not have common intersection points. If you interpret the rule in three-dimensional space, then two lines that are located on the same plane are considered parallel straight and, again, do not have common points.

The parallelity of lines has signs, axioms, properties. Next, we will study in more detail 3 signs of the parallelity of two lines on the plane.

Signs of the parallelism of two lines on the plane: what are signs, axioms, properties?

First, consider what is the difference between concepts: sign, property and axiom. This will not be confused in the future, which is very important for the exact sciences:

Signs - These are some facts, it is on the signs that it is possible to establish a true judgment about the objects of interest or not.
Properties - These are accurate formulations (rules) that cannot be refuted.
Axiom - This is a proper statement that completely does not require evidence. It is on axioms that, in particular, are built in geometry, evidence of signs and properties.

What are the terms: Askioma, theorem, investigation — What are the terms: axiom, theorem, investigation

As you can see, concepts have differences from each other. Then we will study more 3 signs of the parallelity of two lines on the plane, to prove the signs, you will have to use axioms, properties.

Signs of the parallelism of two lines on the plane: Determination

From the geometry it is known that there are 3 signs of the parallelity of two lines on the plane. This was studied in the seventh grade.

Signs of parallelism of two lines - Grade 7:

The first feature is about the fact that when two lines are perpendicular to the third, then they do not have any common points of intersection, and they parallel.
The second feature mentions the corners. More precisely, if two lines are crossed by a third, cross -line cornersformed as a result of the intersection equal, or the corresponding angles are equal - lines (||) parallel.
The sum of one -sided angles is 180º, then these lines (||) parallel.

IMPORTANT: There are reverse signs of the parallelity of the lines. They are interpreted in reverse order. More precisely, two lines are considered parallel. This will be discussed in the last paragraph.

The first sign of the parallelity of two lines on the plane is evidence

Signs of the parallelism of the two lines on the plane are very often used to solve a variety of geometric tasks, so you need not only to know how to formulate it, but also to be able and prove this statement.

Repeat again - the first sign sounds like this:

When two lines are perpendicular to the third, then they do not have common points of intersection and parallel. This saying should be added if the lines lie in one plane, since in three -dimensional space this statement is not entirely true.

Proof of the sign:

You can easily prove the sign. For clarity, the drawing is presented below:

The first feature drawing about the parallelity of two lines

There is an axiomthat to the line on the plane you can draw a perpendicular line from a given point, which does not belong to the line, and and only one.

Imagine that two lines from the other line can be drawn from one point. But then there will not be straight angles, respectively, the last statement is not true, and the sign is true.

The second sign of the parallelity of two lines is evidence

All signs of the parallelism of the two lines on the plane are not so difficult to remember, but the second is the most difficult in terms of evidence.

When two lines intersects oblique, cross -line corners equal, or the corresponding angles are equal, then the lines between themselves (||) parallel.

See the image further, it describes in detail what angles are formed when the line of two lines is crossing:

The names of the angles that are formed when the third line of two lines is crossed — The names of the corners

Proof:

Having studied the drawing above, now you can figure out which angles are the crossbow and which are appropriate. Below is the image according to which it is easy to prove, the second sign of parallel lines.

Let it be given: ∠ ACK=∠KDB ( cross lying corners∠ACK, ∠KDB equal), that line b.||a.

The second sign of the parallelity of two lines

So, points C, D are the intersections of the two lines A, b. First, on the segment by simple calculations, we find the middle point of the DC segment.
This will be k, it is necessary to draw a line ⊥ to b through the middle of the segment (through point K).
The corners at the top with point K will be equal to each other, because they are vertical, and according to the condition, it is set that ∠ACK \u003d ∠KDB. Also CK \u003d KD. From this it follows that the triangles formed as a result of the intersection of two lines are equal.
The CAK angle is 90º according to the condition, since the line AB is perpendicular to the line a. So the angles formed by the AB line with straight a, b are 90º and the triangles CAK and KBD are rectangular.
And on the first basis, perpendicular can only be drawn to two parallel lines.

Proof:

When the corresponding angles formed by the lines at the base are equal, the line a || b.

Again, the first thing to do perpendicular to the line a.
From the equality of triangles CAK and KBD, it follows that:
The angle at the base will be 90º according to the condition and the corresponding ∠KBD \u003d 90º.
So the BA line is a perpendicular for both line A and for the line b.

Conclusion: straight (||) parallel.

The third sign of the parallelity of two lines is evidence

The third statement is when the amount (∑) of one -sided angles is 180º, which means these lines (||) are parallel, It is very simple to prove.

It is necessary to draw a perpendicular line to the line a, the angles formed at the base on line A will be equal to 90º and 90º \u003d 180º.
The corners at the top with point K will be equal to each other, because they are vertical. Also CK \u003d KD by condition. From this it follows that the triangles formed as a result of the intersection of two lines are equal.
So the BA line is a perpendicular for both line A and for line b.

Signs of the parallelism of two lines on one surface

Based on the figure, ∠1 and ∠4 adjacent. As we already know, the sum of adjacent angles (∠1+∠4) is 180º. At the same time, ∠1 \u003d ∠2, as a delay lying.

Hence the conclusion: The sum of one -sided angles is 180º (∠2+∠4 \u003d 180º).

Reverse signs of the parallelity of two lines on the plane

There are also reverse signs of the parallelity of two lines on one plane. And their statements sound exactly the opposite:

Lines are considered (||) parallelwhen you can conduct One common perpendicular line.
Two lines on one surface parallelwhen they have contracts lying corners are equal to each other or they are straight.
Two lines on one surface are considered (||) parallelwhen the corresponding angles at the bases are equal.
Two lines on one surface (||) parallel, When the amount (∑) of one -sided angles is 180º.