Curves can be parallel

Two parallel straight lines

Straight lines or stretches can be in special positions to each other. This is about "parallel".

These two straight lines are parallel to each other. That means: They have the same distance from one another everywhere.



Straight lines are infinitely long. You can imagine that the straight lines are still parallel even at infinity. That never changes.

Two straight lines $$ g $$ and $$ h $$ are parallel to one another if they always have the same distance from one another.

Short notation: $$ g $$ $$ || $$ $$ h $$.

A donkey bridge for the writing area $$ || $$ is that the $$ || $$ also occurs in the word “parael”.

If you draw your parallel lines somewhere in your exercise book, in your imagination they will run parallel to infinity.



If two straight lines are not parallel, you write:.
Two straight lines are not parallel if they have a common point of intersection.

Two parallel routes

Not only straight lines can be parallel to one another, but also lines.



Here the segment $$ bar (AB) $$ is parallel to the segment $$ bar (CD) $$.

Short notation: $$ bar (AB) $$ $$ || $$ $$ bar (CD) $$.

Draw straight lines parallel to each other

How do you draw parallel lines in your exercise book?

Possibility Number 1

You use the parallel lines of your set square. They are on every set square. In the picture you can see them in.



The distances are each 0.5 cm = 5 mm. With the pink lines you draw parallels at a distance of 0.5 cm, 1 cm, 1.5 cm, ..., 4 cm.

If you want to draw parallels at a distance of, for example, 2.3 cm, you can also use the set square. Use the little ticks.



Example:
Draw a parallel to the blue line at a distance of 2.3 cm.
You put the set square at the right distance ...



and then draw the.


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Draw parallels - option 2

For the second option, you use the vertical and the distance as an aid. How it works???

1. Place the set square with the center line (90 °) on the straight line. Then you draw the vertical line and mark the required distance. Here it is 2.3 cm.




2. You draw another vertical line at a second, slightly distant point and mark the required distance.




3. Connect the two marking points.




Option 2 allows you to draw more precisely.

More than a parallel

There are always two parallel straight lines that are at the same distance from a given straight line. Figuratively speaking, one lies above the given straight line. The other is below the given straight line.



The two red straight lines are at the same distance from straight line g.

Most of the time you just need to draw exactly one parallel straight line. Then you can choose which ones to draw.

Parallels more than 8 cm apart

The length of 8 cm is the highest that your set square has to offer. But there are also tasks in which you should draw a parallel that is more than 8 cm apart.

Method 1

You draw auxiliary lines that are parallel to each other. If you need a distance of 10 cm, you draw an auxiliary parallel at 4 cm, then another at 4 cm and then the required parallel at a distance of 2 cm from the last auxiliary parallel. 4 cm + 4 cm + 2 cm = 10 cm.


Method 2

Work with an extension of the vertical. You can use a long ruler to help. Draw very carefully as you extend the vertical.


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Draw a parallel straight line through a point

Sometimes you didn't specify the distance, but a point through which the parallel should go.
Then the task is: Draw a parallel to the straight line through the given point P.


Here you have the two options again.

Possibility Number 1

You place the set square on the starting line and move it parallel until you reach the point. Shifting parallel means that you use the lines drawn parallel to each other on the set square.


Possibility 2

You draw a vertical line through the point. Then you draw another perpendicular to the first auxiliary line (the first perpendicular). Then that is the parallel.


If you draw a perpendicular $$ s_1 $$ to a straight line $$ g $$ and then another perpendicular $$ s_2 $$ to the perpendicular $$ s_1 $$, then $$ s_2 $$ and $$ g $$ are parallel to each other.

special cases

Distance = 0

You can draw a straight line parallel to another straight line that has a distance of 0. This parallel is then not really visible, because it is identical to the straight line.

In 3D

Straight lines can lie in space in such a way that they never intersect, but are also not parallel. These straight lines are called crooked. This is not possible on the plane, i.e. on paper.

In the plane, straight lines are always either parallel (special case identical) or they have exactly one point of intersection.

Draw distant parallels through a point P.

If your task is to draw quite distant parallel through a point, you can use a trick. You will need a long ruler for this.

The straight line and a distant point are given.




1. You place the set square with the edge on the straight line.




2. You place the long ruler precisely on one leg of the set square.




3. You hold the ruler and move the set square to any position parallel to the starting line.




4. You slide until you reach point P.




5. Draw the parallel line through P.




This is what your result looks like:


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Parallels in everyday life

Parallel straight lines or lines are very common.

Railroad tracks

Railway tracks are parallel. Otherwise the train would derail.

What is interesting about the train tracks is that they look to the observer as if they converge at one point at the end of the field of view. But that's just an optical illusion. You know it's not like that.



Image: Panther Media GmbH (Helmut Knab)

Building construction



Image: fotolia.com (Uwe Kantz)


All the lines that strive upwards are parallel to each other.

When you look at the train tracks, it becomes clear to you that curves can also be parallel. But you only need the parallelism of straight lines and lines. And they have no curvature.



Image: TopicMedia Service (Bühler)

Parallels in everyday life

Urban construction

There are tons of parallel streets in the Manhattan neighborhood of New York.
All streets from north to south are parallel to each other.
All streets from east to west are parallel to each other.



Image: Joachim Zwick

packaging

The properties “parallel” and “vertical” play a role in sales and storage. Many goods are in boxes that are cuboid. The cardboard edges are perpendicular or parallel to each other. That is why the boxes are stackable.



Image: fotolia.com

Parallels in mathematics

You probably already know parallel sides of special rectangles:

rectangle


Trapezoid


Parallels also occur in bodies. You can find parallel edges in cubes, cuboids or prisms, for example.

Cuboid

All parallel edges are marked in color.



Do you already know the oblique image? This is the name of this type of 3D view. The advantage of oblique images is that the parallel edges are also parallel on the image.

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