is a line parallel to itself

Is a line parallel to itself?

Is a line parallel to itself?

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Parallel (geometry)

In geometry, parallel lines are coplanar straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called skew lines.

Parallel lines are the subject of Euclid’s parallel postulate. Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism.

Symbol

The parallel symbol is {parallel }. For example, {\ AB\parallel CD} indicates that line AB is parallel to line CD.

In the Unicode character set, the “parallel” and “not parallel” signs have codepoints U+2225 (∥) and U+2226 (∦), respectively. In addition, U+22D5 (⋕) represents the relation “equal and parallel to”.

The same symbol is used for a binary function in electrical engineering (the parallel operator). It is distinct from the double-vertical-line brackets that indicate a norm, as well as from the logical or operator (||) in several programming languages.

Parallel Lines (Geometry)

Parallel lines are lines in a plane which do not intersect. Like adjacent lanes on a straight highway, two parallel lines face in the same direction, continuing on and on and never meeting each other. In the figure in the first section below, the two lines {AB} and {CD} are parallel. The symbol for parallel lines is \parallel, so we can say that {AB}\parallel{CD} in that figure.

According to the axioms of Euclidean geometry, a line is not parallel to itself, since it intersects itself infinitely often. However, some authors allow a line to be parallel to itself, so that “is parallel to” forms an equivalence relation.

Basic Properties of Parallel Lines

Parallel lines never intersect. In the language of linear equations, this means that they have the same slope. In other words, for some change in the independent variable, each line will have identical change to each other in the dependent variable.

Is a line parallel to itself?
Is a line parallel to itself?

 

 

Is a line parallel to itself?
Is a line parallel to itself?

 

 

Is a line parallel to itself?
Is a line parallel to itself?

 

Now, imagine drawing a transversal (line {PQ}) that meets perpendicularly with the two parallel lines, as shown in the figure above. Then the length of \overline{XY} will be the shortest distance between the two parallel lines. Now we draw an additional transversal (line {P’Q’}) which also meets perpendicularly with the two parallel lines. Note that {PQ} and {P’Q’} are parallel (try to prove this by using the concept of corresponding angles and alternative angles!).

Now we also have the length of \overline{P’Q’} as the shortest distance between the two parallel lines, and hence it is true that \lvert\overline{PQ}\rvert=\lvert\overline{P’Q’}\rvert. This implies that two parallel lines are always a constant distance apart from each other, which is another important characteristic of parallel lines.

Thinking more intuitively, this has to be true since if the lines were getting farther apart from each other, then on the opposite side of the lines would be getting closer (and eventually meeting), which contradicts the definition that two parallel lines never meet. Note that the distance between two distinct lines can only be defined when the lines are parallel. If the lines are not parallel, then the distance will keep on changing.

The discussion just above, for your information, in fact accords to Euclid’s fifth postulate, or the parallel postulate. It states that if a line segment intersects two straight lines forming two interior angles on the same side that sum to less than 180 degrees, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than 180 degrees. In other words, two lines are parallel when the interior angles on the same side sum to exactly 180 degrees.

In summary,

  • The angles that fall on the same sides of a transversal and between the parallels (called corresponding angles) are equal. The converse is also true: if two lines have equal corresponding angles, the lines are parallel.
  • The angles that fall on alternate sides of a transversal and between the parallels (called alternate angles) are equal. The converse is also true: if alternate angles are equal, the lines are parallel.
  • The two parallels lines are a constant distance apart, so any pair of lines that intersects them at the same angle will make segments with the same length.

 

Why a line is parallel to itself?

Parallel lines are lines in a plane which do not intersect. According to the axioms of Euclidean geometry, a line is not parallel to itself, since it intersects itself infinitely often. However, some authors allow a line to be parallel to itself, so that “is parallel to” forms an equivalence relation.

Why are parallel lines important in geometry?

Parallel lines are important in mathematics because they are at the base of several conjectures involving angles in geometry. Each angle in a set is congruent to the other angles of the same type. Therefore, if the measure of one angle is known, it is possible to find the measure of other angles.

What does it mean when lines are parallel?

Parallel Lines: Definition: We say that two lines (on the same plane) are parallel to each other if they never intersect each other, ragardless of how far they are extended on either side. Pictorially, parallel lines run along each other like the tracks of a train.

What contradicts parallel lines?

Take two parallel lines that are cut by a transversal. Since the lines are parallel, the consecutive interior angles of intersection are supplementary. Specifically, if the two parallel lines intersect, we have formed a triangle with angles that add up to more than 180 degrees, which is a contradiction.

How can a line parallel to itself?

Theorem 3.1 In the set of all lines in the plane, the relation of being parallel is an equivalence relation. Proof. First, since a line has infinitely many points in common with itself, it is parallel to itself; hence the relation is reflexive (this is the point of the strange definition).

 

Is a line parallel to itself?

Yes, it is much more convenient to make definitions inclusive rather than exclusive.

Take, for example, the theorem that says parallelism is an equivalence relation on lines. It is a theorem if a line is considered to be parallel to itself, but it’s not a theorem if a line is considered not to be parallel to itself.

For parallelism to be an equivalence relation, three things are required.

  1. Reflexivity: For any line LL, it is the case that LLL‖L.
  2. Symmetry: If LML‖M, then MLM‖L.
  3. Transitivity: If LML‖M and MNM‖N, then LNL‖N.

If lines aren’t considered parallel to themselves, 1 is false, and 3 is false when L=NL=N.

 

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