The methods for constructing parallel lines using various tools are based on the signs of parallel lines.
Constructing parallel lines with a compass and straightedge
Consider principle of constructing a parallel line passing through a given point, using a compass and ruler.
Let a line be given, and some point A, which does not belong to the given line.
It is necessary to construct a line passing through the given point $A$ parallel to the given line.
In practice, it is often required to construct two or more parallel lines without a given line and point. In this case, it is necessary to draw a line arbitrarily and mark any point that will not lie on this line.
Consider steps for constructing a parallel line:
In practice, the method of constructing parallel lines using a drawing square and a ruler is also used.
Construction of parallel lines using a square and a ruler
For constructing a line that will pass through the point M parallel to the given line a, necessary:
- Attach the square to the straight line $a$ with a diagonal (see the figure), and attach a ruler to its larger leg.
- Move the square along the ruler until the given point $M$ is on the diagonal of the square.
- Draw the desired line $b$ through the point $M$.
We have obtained a line passing through a given point $M$ parallel to a given line $a$:
$a \parallel b$, i.e. $M \in b$.
The parallelism of the lines $a$ and $b$ is evident from the equality of the corresponding angles, which are marked in the figure by the letters $\alpha$ and $\beta$.
Construction of a parallel line at a given distance from a given line
If it is necessary to construct a straight line parallel to a given straight line and spaced from it at a given distance, you can use a ruler and a square.
Let a line $MN$ and a distance $a$ be given.
- We mark an arbitrary point on the given line $MN$ and call it $B$.
- Through the point $B$ we draw a line perpendicular to the line $MN$ and call it $AB$.
- On the line $AB$ from the point $B$ we plot the segment $BC=a$.
- With the help of a square and a straightedge, draw the line $CD$ through the point $C$, which will be parallel to the given line $AB$.
If we postpone the segment $BC=a$ on the line $AB$ from the point $B$ to the other side, then we get one more line parallel to the given one, spaced from it by the given distance $a$.
Other ways to draw parallel lines
Another way to build parallel lines is to build with a T-square. Most often, this method is used in drawing practice.
When performing carpentry work for marking and building parallel lines, a special drawing tool is used - a bevel - two wooden planks that are fastened with a hinge.
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Class: 7
Lesson type: lesson in the application of knowledge.
Lesson Form: a lesson in the study of an object, posing a problem and solving it.
Goals: To introduce students to various ways of constructing parallel lines;
Tasks:
educational
- formulate the definition of parallel lines, rays and segments; find them on the drawing and build them using drawing tools;
- Learn to draw parallel lines using a ruler, square, square and ruler, compass and ruler.
- Learn to draw parallel lines using tools;
developing
- develop the ability to compare, analyze, generalize, draw a conclusion, transfer knowledge and skills in a new non-standard situation;
- develop the ability to analyze information
- develop spatial representations and skills, teach how to use a geometric language;
- create conditions for the development of cognitive interest in mathematics
educational
- to cultivate a conscious attitude to work, to broaden one's horizons;
- to cultivate accuracy, independence, interest in the subject;
- education of mathematical culture and speech
Forms of student work: frontal, individual, group.
Subject students learn to build parallel lines using various methods, based on previously studied material.
Personal: show cognitive interest in the study of mathematics, ways to solve educational problems; give a positive assessment and self-assessment of educational activities; adequately perceive the assessment of the teacher and peers; understand the reasons for success in educational activities.
Metasubject:
determine the goal of educational activity with the help of a teacher and independently, search for a means to achieve it;
Forms of work in the lesson:
- frontal,
- steam room,
- Individual.
Organization of student activities in the classroom:
- independently come to the problem and solve it;
- independently determine the topic, objectives of the lesson;
- work with the text of the textbook;
- work with the assessment sheet when completing assignments;
- answer questions;
- solve problems independently;
- evaluate themselves and each other;
- reflect.
Equipment: computer, projector, presentation for the lesson: teacher's presentation, student's worksheet, ruler, square, pencil, Geometry textbook grades 7-9, Atanasyan L.S. etc. - M .: Education, 2009.
During the classes
| Teacher activity | Student activities |
| 1. Organizational moment | |
| The teacher greets the students, explains the work of the lesson (worksheets) | Students listen attentively to the teacher |
| 2. Motivation for learning activities | |
| Guys, what do you think is common between the school notebook familiar to all of you and the model of the railway ( show notebook and rails)? | The children make their guesses. Give arguments in defense of their version ( All these items are united by the concept of parallelism: notebooks are lined with parallel lines, the railway track consists of sleepers and rails). |
| Did you know that the topic of parallel lines has worried people since ancient times. The first who systematized the knowledge of parallel lines was the ancient Greek scientist - Euclid. | The students listen to the historical background. Student message. |
What do you think, are parallel lines so important in our life? What would the world be like if there were no parallels in it?
|
A) During the construction of buildings, parallelism is strictly taken into account. (plumb line). B) railroad tracks. B) an escalator If they were not parallel, then they were in contact with each other, and this led to a short circuit, breakdowns, in which the electrical circuit opens and the current is turned off. If the rails were not parallel, they would converge somewhere and the train would crash. |
| Every modern person needs to know how parallel lines are built. | |
| Where you and I may need to build parallel lines? | On the board, in the notebook On the computer In production At home, in the country, on the street |
| What do we need to draw parallel lines? | Instruments Knowledge: theoretical material |
| What tools will we use? | Ruler, square, compass Special tools improvised means |
| Guys, let's try to formulate the topic of the lesson. | Practical Ways to Draw Parallel Lines (slide 1) |
| What should we learn in the lesson? | Students state the objectives of the lesson. (slide 2) |
| 3. Updating knowledge | |
| Guys, let's remember the theoretical material related to the term parallelism (slide 3-11):
What else do you know about parallel lines? |
Students answer.
1. What lines are called parallel? Two lines in a plane are called parallel if they do not intersect . 2. What two line segments are called parallel? The two segments are called parallel if they lie on parallel lines. 3. What is a secant? Direct called secant, if it intersects two lines at two points. 4. What are the main signs of parallel lines. 1. If, at the intersection of two lines of a secant, the lying angles are equal, then the lines are parallel. 2. If at the intersection of two lines of a secant, the corresponding angles are equal, then the lines are parallel. 3. If, at the intersection of two straight lines, a secant sum of one-sided angles is 180 °, then the lines are parallel. |
| What do you think, is it possible to use these signs when constructing parallel lines? | Yes |
| 4. Physical education . | |
| 5. Learning new material. Practical ways to construct parallel lines on the blackboard, in a notebook |
|
| Guys, look what instruments we have in the desk? Does anyone know how to draw parallel lines using a ruler? Explain the fact of parallelism. |
Students answer questions: ruler, drawing triangle, compasses. |
| A) Construction of parallel lines using a square and a ruler | |
| On fig. 103 ( slide 12
) shows a method for constructing parallel lines on paper, a board. Guys, which of the tools depict a secant? (ruler) Which tool represents an angle? (drawing triangle) Is one square and one ruler enough to draw parallel lines? Explain how to build. What is the method based on?
Task 2. Use a square and a ruler to construct parallel lines m and n. Slide 13-15. |
Textbook, p. 57. Look for answers to the teacher's questions. |
| B) Construction of parallel lines using a compass and ruler | |
| See how you can draw parallel lines with a compass and straightedge Slide 16-17 Circle Two diameters Straight lines are parallel. Exercise. Construct parallel lines c and b with a compass and straightedge. |
Students build parallel lines according to the algorithm. |
| In production Slide 18 C) Construction of parallel lines using a T-square |
|
| The inventive thought of mankind does not stand still, and for a more convenient construction of a drawing and drawing parallel lines, a special drawing tool was invented - T-series ( slide 18). Reisshina - a drawing device for drawing parallel lines, which consists of a ruler with a transverse bar. |
|
| Examples are given: Malka is a tool for transferring angular dimensions when marking parts, for building parallel lines. (slide 19) |
|
| Thickness gauge - a tool for drawing marking lines on the workpiece parallel to the selected baseline (slide 20) |
|
| Staple - For drawing more lines at the same time (slide 21) | |
| 6. Consolidation of the studied material | |
| Exercise. Through the vertices of triangle ABC draw lines parallel to line l. slide 22. | Build in a notebook. |
| 7. Summary of the lesson | |
| Guys, let's remember with what tools we learned to build parallel lines? | Students answer the question |
| 8. Homework | |
| Page 57 p. 26 to consider ways of building. Page 58-59, #194, #195 |
|
| Reflection | |
Our lesson is coming to an end. Please share your thoughts on today's session with us. The following words will help you to do this:
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Literature.
- Atanasyan L.S. Textbook for educational institutions. Geometry grades 7-9.
- http://www.mathvaz.ru
In any design training course, they learn to use thin auxiliary lines when creating drawings. Previously, they were applied on a drawing board, and then wiped off from the finished document. Now they use electronic programs for drawing, but the need for auxiliary lines is not even discussed. Although in Compass 3D it is even easier to work with them than on a classic drawing board. Auxiliary lines are used to form the necessary links, mark up the drawing, and create certain boundaries.
The program allows you to create auxiliary lines in several ways, again, this is very convenient, since sometimes one is used, and in another situation, a different method of drawing auxiliary lines.
1. Create a straight line using two points.
One of the most popular ways. To activate, you need to open the main menu Tools - Geometry - Auxiliary lines - Auxiliary line.
Or you can click in the panel Geometry-Auxiliary line.
Let's set our line by left-clicking on the sheet, so setting the first point, then specifying the end point of the line. At the same time, the program itself will generate the desired angle of inclination for the created straight line. However, you can change the angle by entering your own values in the box below, then just press Enter.
The auxiliary line is formed, now you need to click on the familiar icon Abort command, located in the properties panel. However, you can activate this command by completing the work with the line by simply right-clicking the mouse, and then selecting the appropriate item in the drop-down menu.
Using a base point, you can create an infinite number of straight lines at any angle. By the way, if you have coordinates or it is more convenient to work with a coordinate grid, then you can always set the desired values in the menu below. You will place a straight line, without any adjustments on the sheet. It is worth paying attention to the group Modes, it has two important switches. The first one is active on standard startup - Do not place intersection points, and the second you can choose yourself - Set intersection points. Using this setting, you can automatically put points at any intersection, without additional options and manual setting.
However, here you need to specify the style Auxiliary. By the way, to remove all auxiliary elements from the finished drawing, it is enough to activate the item in the main menu Editor-Delete-Auxiliary curves and points. The work with points on curves was considered in detail in lesson number 3.
2. Draw a horizontal line
Construction lines can be drawn using horizontal lines. Let's open the already familiar menu Tools-Geometry-Auxiliary lines-Horizontal line.
A faster option, using the compact panel, select Geometry - Horizontal line. However, the base panel will not be visible on the screen, to correct the position, press the auxiliary lines button and hold it for a while.
It remains to specify the desired point by clicking the left key, through which we will pass our straight line. You can create any number of horizontal lines. To complete your work, just press Abort command in the properties panel or in the drop-down menu, by right-clicking.
You also need to remember that the horizontal straight line is always parallel to the current x-axis. However, when setting horizontal lines using a rotated coordinate system, they will not be horizontal already on the sheet.
3. Draw a vertical straight line.
The general mechanism for calling the line drawing mechanism is absolutely identical to the one described above, with the exception of choosing vertical straight.
However, there are a few important things to keep in mind. The created vertical line is always parallel only to the current coordinate axis, here the case is identical with the horizontal straight line. Therefore, if you have a modified coordinate system, the vertical straight lines will not be parallel to the sheet.
4. Create a parallel straight line.
You can draw a parallel straight line only if there is any object on the sheet. It is to these lines that we will create a parallel. Moreover, absolutely any object can act as objects for binding, from straight and auxiliary lines to the faces of polygonal objects. So, let's within the framework of the lesson, for the main take the horizontal line that goes from the origin on our sheet.
Calling a parallel straight line is identical, open Tools - Geometry - Auxiliary lines - Parallel line.
Or use the compact panel, here you need to call Geometry-Parallel line.
Now let's specify the base object, to which we will draw a parallel line. As agreed, a horizontal straight line acts as an object, select it with the mouse. Then, you need to set the distance at which our parallel line will be. At the bottom, you can specify a numerical value, for example 30 mm, or drag the straight line with the mouse to the desired distance.
When setting the distance by numbers, the system will suggest two phantom lines at the same distance. This can be disabled if in the properties Number of straight lines - Two straight lines remove the activation by translating it into the creation of a single straight line. To fix the created line, just select the active phantom with the mouse and click the create object button. When you need to create both lines, click create object again, and then abort the command.
When you need to draw a new parallel line, but near another object, just click on the button Specify again. Now, you can specify a new object and build a line, in the way described in this chapter of the lesson.
That's all, in the lesson we revealed the basics of creating auxiliary straight lines.
Lessons on the KOMPAS program.
Lesson number 4. Auxiliary straight lines in Compass 3D.
When developing drawings on a drawing board, designers always use thin lines, their counterpart in Compass 3D are auxiliary straight lines. They are necessary for preliminary constructions and for setting projection relationships between views. Auxiliary straight lines when printing Auxiliary, it cannot be changed.
There are several ways to construct auxiliary lines. In this lesson, we will look at some of these methods.
1. Arbitrary straight line on two points.
In the main menu of the program, sequentially press the commands Tools-Geometry-Auxiliary lines-Auxiliary line.
Or in the compact panel, press the buttons Geometry-Auxiliary line.
Click the left mouse button to specify the first base point (for example, the origin). Now we specify the second point through which the line will pass. The angle of inclination between the straight line and the abscissa axis of the current coordinate system will be determined automatically. You can enter the angle through the properties panel. For example, enter an angle of 45º and press the key Enter.
To complete the build, click on the icon "Abort command" in the properties panel. This command can be implemented through the context menu, which is called by clicking the right mouse button.
Similarly, through the base point, you can build as many arbitrary lines as you like at any angle. You have probably already noticed that the coordinates of points can be entered from the keyboard using the properties panel. In addition, in the properties panel there is a group Modes, which has two switches: "Don't put intersection points"(active by default) and "Set intersection points". If you need to mark the intersection points of the line with other objects, activate the switch "Set intersection points", now the system will automatically put intersection points with all graphic objects in the current view.
Dot style will be- Auxiliary. To remove all auxiliary elements, use the commands of the main menu Editor-Delete-Auxiliary curves and points. How to mark intersection points not with all, but only with some objects is described in lesson No. 3.
2. Horizontal straight line.
Commands are called to construct a horizontal line Tools-Geometry-Auxiliary lines-Horizontal line.
Or via the compact panel by pressing the buttons: Geometry-Horizontal line. The toolbar for constructing auxiliary lines is not visible on the screen. To see it, press the auxiliary lines button, which is active at the time of construction, and hold for a few seconds.
Now it is enough, by clicking the left mouse button, to specify the point through which the horizontal line will pass. You can build as many lines as you want at the same time. To complete the build, click the button "Abort command" on the properties panel.
It must be remembered that the horizontal line is parallel to the x-axis of the current coordinate system. Horizontal, built in a coordinate system rotated relative to the absolute system, will not be parallel to the horizontal sides of the sheet.
3. Vertical straight line.
The construction is similar to the construction of horizontal lines, so you will figure it out yourself.
It must be remembered that the vertical line is parallel to the y-axis of the current coordinate system. Vertical, built in a coordinate system rotated relative to the absolute system, will not be parallel to the vertical sides of the sheet.
4. Parallel line.
To build a parallel line, we need an object parallel to which it will pass. Such objects can be: auxiliary lines, segments, polyline links, sides of polygons, dimension lines, etc. Let's construct a parallel line for the horizontal line passing through the origin.
Calling commands Tools-Geometry-Auxiliary lines-Parallel line.
The methods for constructing parallel lines using various tools are based on the signs of parallel lines.
Constructing parallel lines with a compass and straightedge
Consider principle of constructing a parallel line passing through a given point, using a compass and ruler.
Let a line be given, and some point A, which does not belong to the given line.
It is necessary to construct a line passing through the given point $A$ parallel to the given line.
In practice, it is often required to construct two or more parallel lines without a given line and point. In this case, it is necessary to draw a line arbitrarily and mark any point that will not lie on this line.
Consider steps for constructing a parallel line:
In practice, the method of constructing parallel lines using a drawing square and a ruler is also used.
Construction of parallel lines using a square and a ruler
For constructing a line that will pass through the point M parallel to the given line a, necessary:
- Attach the square to the straight line $a$ with a diagonal (see the figure), and attach a ruler to its larger leg.
- Move the square along the ruler until the given point $M$ is on the diagonal of the square.
- Draw the desired line $b$ through the point $M$.
We have obtained a line passing through a given point $M$ parallel to a given line $a$:
$a \parallel b$, i.e. $M \in b$.
The parallelism of the lines $a$ and $b$ is evident from the equality of the corresponding angles, which are marked in the figure by the letters $\alpha$ and $\beta$.
Construction of a parallel line at a given distance from a given line
If it is necessary to construct a straight line parallel to a given straight line and spaced from it at a given distance, you can use a ruler and a square.
Let a line $MN$ and a distance $a$ be given.
- We mark an arbitrary point on the given line $MN$ and call it $B$.
- Through the point $B$ we draw a line perpendicular to the line $MN$ and call it $AB$.
- On the line $AB$ from the point $B$ we plot the segment $BC=a$.
- With the help of a square and a straightedge, draw the line $CD$ through the point $C$, which will be parallel to the given line $AB$.
If we postpone the segment $BC=a$ on the line $AB$ from the point $B$ to the other side, then we get one more line parallel to the given one, spaced from it by the given distance $a$.
Other ways to draw parallel lines
Another way to build parallel lines is to build with a T-square. Most often, this method is used in drawing practice.
When performing carpentry work for marking and building parallel lines, a special drawing tool is used - a bevel - two wooden planks that are fastened with a hinge.
