How to Construct a Line Parallel to a Given Line Through an External Point Step by Step with Diagram and Examples
We are basically going to learn how to construct parallel lines and the steps to construct parallel lines.
What is A Parallel Line?
Parallel lines are lines which do not have a common meeting point in the same plane, however far they are extended .The following figure represents parallel lines. Sometimes you may be presented with one line and you need to create another line parallel to it through any given point. You might think to simply take a straight edge and draw a line that might seem right; however, you could not be sure that the line you constructed is actually parallel. With the use of geometry and a compass, you can easily plot additional points that will ensure the line you construct is truly parallel.
Properties of Parallel Lines
1) The corresponding angles formed by parallel lines are equal.
2) The vertically opposite angles formed by parallel lines are equal.
3) The alternate interior angles formed by parallel lines are equal.
4) The alternate exterior angles formed by parallel lines are equal.
5) The pair of interior angles on the same side of the transversal are supplementary, that is they equal to 180 degrees.
Construction of Parallel Lines:
Given: We have been a point P .
Construct : We have to construct parallel lines.
The Steps for Constructing Parallel Lines Is Quite Simple! Here Are The Steps To Construct Parallel Lines-
Step 1) Use your straightedge, and draw a transversal through the given point . This is simply a straight line which passes through the given point P and intersects with the given line. Drawing the line slanted will make the construction easier than to try to make the line in a vertical manner. Be sure that you draw the line properly above point P.
Step 2) Using the construction you can copy an angle, construct a copy of the angle formed by the transversal and the given line such that the copy will be located UP at point P. The vertex of the copied angle will be located at the point P.
Step 3) When you draw the line to complete the angle copy, you will be able to draw a line parallel to the given line.
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Steps To Construct Parallel Lines
How To Construct Parallel Lines?
Let’s understand Step by Step to Construct Parallel Lines-
If we have to construct a line parallel to the other line from an external point all we require is a ruler and a compass and the following steps need to be kept in mind!
Given: A line segment named AB and a given point P that lies out of the line segment AB.
To construct a line that is parallel to line AB that passes through the given point P.
Step 1: You have to choose any point X on the given line segment AB and join it to point P as shown below in the diagram.
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Step 2: Considering X as the center and any suitable radius you need to draw an arc cutting the line segment PX at the point M and AB at point N respectively.
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Step 3: As you have P as the center and radius remains same as used in the previous step 2 you need to draw an arc EF cutting the line segment PX at point Q as shown in the figure below.
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Step 4: With Q as the center and same radius as we have used in Step 1, draw an arc cutting the arc EF at R as shown below in the given diagram.
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Step 5: Join the points R and P to draw a line segment CD as shown in the figure given below.
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The line segment CD is the line that needs to be parallel to the line segment AB and passes through the point P.
Note: Now you might think how to know whether a line is parallel to the other given line or not. To check whether or not two lines are parallel, we must compare their slopes which are denoted by m in the equation (y = mx+c). Two lines are parallel if and only if their slopes which are m are equal in measure.
FAQs on Construction of Parallel Lines Through an External Point in Geometry
1. What is the construction of parallel lines through an external point?
The construction of parallel lines through an external point is the geometric method of drawing a line through a given point that does not lie on a given line and is parallel to it. In geometry, this is done using a compass and ruler without measuring angles directly. The constructed line maintains the property that it never intersects the original line, even when extended infinitely in both directions.
2. How do you construct a parallel line through a point not on a given line?
To construct a parallel line through an external point, copy the corresponding angle formed by a transversal. Follow these steps:
- Draw a transversal from the given external point to intersect the given line.
- With the compass, draw an arc at the point of intersection cutting both the line and the transversal.
- Without changing the compass width, draw a similar arc at the external point.
- Measure the distance between the arc intersections on the given line.
- Transfer that distance to the arc at the external point.
- Join the external point to the marked point to form the new line.
3. Why does copying an angle help in constructing parallel lines?
Copying an angle ensures the formation of equal corresponding angles, which guarantees that the lines are parallel. When a transversal intersects two lines and the corresponding angles are equal, the lines are parallel according to the converse of the corresponding angles theorem. This geometric principle is the foundation of parallel line construction.
4. What tools are required to construct a parallel line from an external point?
The tools required to construct a parallel line are a compass and a straightedge (ruler). These tools allow you to copy angles and draw straight lines accurately. A protractor is not necessary because the construction relies on geometric properties rather than measuring degrees.
5. What is the theorem used in constructing parallel lines?
The main theorem used is the converse of the corresponding angles theorem. It states that if a transversal intersects two lines and the corresponding angles are equal, then the two lines are parallel. This theorem justifies the accuracy of the construction method.
6. Can you give an example of constructing a parallel line through an external point?
Yes, for example, given line AB and an external point P, you can construct a line through P parallel to AB by copying angle ∠APB formed by a transversal. Steps include:
- Draw a transversal from P to meet AB at C.
- Construct an arc at C cutting AB and the transversal.
- Draw the same arc at P.
- Transfer the measured arc distance.
- Join P to the transferred point.
7. What are the properties of parallel lines in geometry?
Parallel lines are lines in a plane that never intersect and remain equidistant. Their key properties include:
- Corresponding angles are equal.
- Alternate interior angles are equal.
- Co-interior angles are supplementary (sum = 180°).
8. Is there another method to construct a parallel line through an external point?
Yes, another method involves constructing perpendicular lines. Steps include:
- Draw a perpendicular from the external point to the given line.
- At the external point, construct another perpendicular to the first perpendicular.
9. What are common mistakes when constructing parallel lines?
Common mistakes include incorrect compass width adjustment and inaccurate arc intersections. Key points to remember:
- Do not change the compass width while copying arcs.
- Ensure arcs intersect clearly.
- Use a sharp pencil for precision.
10. Where is the construction of parallel lines used in real life?
The construction of parallel lines is used in engineering, architecture, and technical drawing to ensure structural alignment and design accuracy. Examples include:
- Designing railway tracks.
- Drawing building plans.
- Creating geometric diagrams.
