How to draw a line parallel to another. Parallel lines. Complete Lessons - Knowledge Hypermarket
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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;
- Teach to build 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 conclusions, transfer knowledge and skills in a new unusual 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 learn to build parallel lines using various ways based on previously learned material.
Personal: show cognitive interest in the study of mathematics, ways to solve educational problems; provide positive feedback and self-esteem learning 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 usual for all of you school notebook and model 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. |
| To each modern man you need to know how parallel lines are constructed. | |
| 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 construction of 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. The task. 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 |
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| 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. |
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| Examples are given: Malka is a tool for transferring angular dimensions when marking parts, for building parallel lines. (slide 19) |
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| Thickness gauge - a tool for drawing marking lines on the workpiece parallel to the selected baseline (slide 20) |
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| Staple - For simultaneous drawing more lines (slide 21) | |
| 6. Consolidation of the studied material | |
| The task. 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 |
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| 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
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 the principle of constructing a parallel line passing through 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.
The construction of a straight line parallel to a given plane is based on
following position known from geometry: a straight line is parallel to a plane,
if this line is parallel to any line in the plane.
Through a given point in space, you can draw innumerable
set of straight lines parallel to a given plane: To obtain
the only solution requires some additional condition.
For example, through a point (Fig. 180) it is required to draw a straight line,
parallel to the plane defined by the triangle ABC and the plane of projections!
(additional condition).
Obviously, the desired line must be parallel to the line of intersection
both planes, i.e. should be parallel to the horizontal track
plane defined by triangle ABC. To determine the direction of this
trace, you can use the horizontal plane given by the triangle
ABC. On fig. 180 a horizontal line DC is drawn and then through point M a
straight line parallel to this horizontal line.
Let us pose the inverse problem: draw a plane through a given point,
parallel to a given straight line. planes passing through some
point A parallel to some straight line BC, form a bundle of planes, the axis
which is a line passing through the point A parallel to the line BC.
To obtain a unique solution, some additional
For example, it is necessary to draw a plane parallel to the straight line CD, not through
point, but through the straight line AB (Fig. 181). Straight AB and CD - crossing. If
through one of the two intersecting lines it is required to draw a plane,
parallel-
Rice. 180 Fig. 181
another, then the problem has a unique solution. through point B
a straight line is drawn parallel to the straight line CD; straight lines AB and BE define
a plane parallel to the straight line CD.
How can you tell if a given line is parallel to a given plane?
You can try to draw a straight line parallel to this plane
this straight line. If such a line in the plane cannot be constructed, then
the given line and plane are not parallel to each other.
You can also try to find the point of intersection of the given line with the given
plane. If such a point cannot be found, then the given line and
plane are mutually parallel.
§ 28. Construction of mutually parallel planes
Let a point K be given through which a plane must be drawn,
parallel to some plane given by intersecting lines AF and BF
Obviously, if we draw lines SK and DK through the point K, respectively
parallel to the straight lines AF and BF, then the plane defined by the straight lines CK and DK,
will be parallel to the given plane.
Another example of construction is given in Fig. 183 right. Through point A
held pl. parallel to sq. but. First, a straight line is drawn through point A,
obviously parallel square. . This is a horizontal with projections "" and "",
where A"N"\\h" o. So
Rice. 182 Fig. 183
since the point N is the front trace of the horizontal AN, then through this
the trace f "o% f" o will pass through the point, and the trace h "o || h" o will pass through X. planes
and are mutually parallel, since their similarly intersecting traces are mutually
are parallel.
On fig. 184 shows two planes parallel to each other - one
one of them is given by the triangle LBC, the other by the parallel lines DE and FG.
What establishes the parallelism of these planes? The one in the plane
given by the straight lines DE and FG, it turned out to be possible to draw two intersecting
lines KN and KM, respectively, parallel to intersecting lines AC and
Sun another plane.
Of course, one could try to find the intersection point at least
line DE with plane of triangle ABC. Failure would confirm
plane parallelism.
QUESTIONS To §§ 27-28
1. What is the basis for the construction of a straight line, which should be
parallel to some plane?
2. How to draw a plane through a line parallel to a given line?
3. What determines the mutual parallelism of two planes?
4. How to draw a plane parallel to a given plane through a point?
5. How to check in the drawing whether the given ones are parallel to each other
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 right connections, marking the drawing, creating 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 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 discussed 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.