How to Draw Parallel Lines Through a Point

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Parallel lines are lines that are equidistant at all points and would never touch if they went on forever.^[1] Sometimes you may be presented with one line and need to create another line parallel to it through a given point. You might be tempted to simply take a straight edge and draw a line that seems right; however, you could not be sure that the line you constructed was technically parallel. Using geometry and a compass, you can plot additional points that will ensure the line you construct is truly parallel.

1. 1

   Locate the given line and the given point. The point will not be on the given line, and can be above or below it. Label the line ${\displaystyle m}$ and the point ${\displaystyle A}$ .

2. 2

   Draw an arc that intersects the given line at two different points. To do this, place the compass tip on point ${\displaystyle A}$ . Open the compass so that it is wide enough to reach beyond line ${\displaystyle m}$ , then draw an arc that sweeps across the line at points ${\displaystyle B}$ and ${\displaystyle C}$ .

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3. 3

   Draw a small arc opposite the given point. To do this, open the compass a little wider. Set the compass tip on point ${\displaystyle B}$ , and draw an arc that sweeps directly across from point ${\displaystyle A}$ .

   • If the given point is above the line, you should draw this arc below the line. If the given point is below the line, you should draw this arc above the line.
   • The arc does not have to be very long, as long as part of it falls directly under the given point.

4. 4

   Draw a another small arc intersecting the previous one. To do this, keep the compass set to the same width. Set the compass tip on point ${\displaystyle C}$ and draw an arc that intersects the previous small arc. Label this point ${\displaystyle D}$

5. 5

6. 6

   Draw an arc that intersects the perpendicular line at two different points. To do this, place the compass tip on point ${\displaystyle A}$ , then draw an arc that sweeps across line ${\displaystyle n}$ at points ${\displaystyle E}$ and ${\displaystyle F}$ .

7. 7

   Draw a small arc opposite the given point. To do this, open the compass a little wider. Set the compass tip on point ${\displaystyle E}$ , and draw an arc that sweeps directly across from point ${\displaystyle A}$ .

8. 8

   Draw a another small arc intersecting the previous one. To do this, keep the compass set to the same width. Place the compass tip on point ${\displaystyle F}$ and draw an arc that intersects the previous small arc. Label this point ${\displaystyle G}$ .

9. 9

   Draw a line connecting the given point to this new point. This line is perpendicular to line ${\displaystyle n}$ . ^[3] Thus, it is parallel to line ${\displaystyle m}$ and passes through the given point, point ${\displaystyle A}$ .

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1. 1

   Locate the given line and the given point. The point will not be on the given line, and can be above or below it. Think of this point as one vertex of a rhombus. Since opposite sides of a rhombus are parallel, by drawing a rhombus we can construct a parallel line. ^[4]

2. 2

   Draw the second vertex of the rhombus. To do this, set the compass tip on the given point and draw an arc that intersects the given line at some point. Do not change the width of the compass.

3. 3

   Draw the third vertex of the rhombus. Using the same compass width, set the compass tip at the second vertex, and draw an arc that intersects the given line at a new point. Do not change the width of the compass.

4. 4

   Draw the fourth vertex of the rhombus. Using the same compass width, set the compass at the third vertex, and draw an arc that intersects the first arc you drew (through the second vertex).

5. 5

   Draw a line through the first and fourth vertices of the rhombus. This line will pass through the given point and be parallel to the given line, as the two lines form two opposite sides of a rhombus.

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1. 1

   Locate the given line and the given point. The point will not be on the given line, and can be above or below it.

2. 2

   Draw a line through the given point and any point on the given line. This is the transverse line you will use to draw two corresponding angles, which will help you form the parallel line.^[5]

3. 3

   Prepare the compass. Set the compass to a width that is less than half of the line segment you constructed.

   • The exact width of the compass does not matter, as long as it is less than half the width of the line segment.
   • For example, you should set the width of the compass so that it is less than half the width of line segment ${\displaystyle AB}$ .

4. 4

   Draw the first angle. Place the tip of the compass on the point where the transverse line intersects the given line. Draw an arc that intersects the transverse line and the given line. Do not change the width of the compass.

5. 5

   Draw an arc. Using the same compass width, place the tip of the compass on the given point. Draw an arc that intersects the transverse line above the given point, extending to just below the given point.

6. 6

   Prepare the compass. Set the width of the compass to the width of the first angle you created.

7. 7

   Draw the corresponding angle. Using the width of the first angle, set the tip of the compass at the point on the transverse line above the given point, and draw an arc that intersects the arc you created before.

8. 8

   Draw a line through the given point and the point created by the two intersecting arcs. This line is parallel to the given line through the given point.

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Add New Question

• Question

  How do I construct a line parallel to a given line if the point is from a second given line?

  

  You can use any of the methods described above. It doesn't matter if the point is on another line; technically, every point is on an infinite number of lines, even if no line is shown. So use the given point and ignore the second given line.

• Question

  Why only one parallel line can be drawn through a given point?

  

  This is more a matter of intuition than proof. It's the subject of Euclid's Parallel Postulate. It's considered an axiom and has been the subject of much controversy for centuries.

• Question

  This is a little confusing. Will constructing a 90 degree angle and then taking the newly constructed line and constructing another 90 degree angle help?

  

  Yes, your suggestion will work. You also may find Method 3 to be less confusing than the other methods shown. Method 3 is just a variation of the method you're suggesting. It uses an acute angle rather than a right angle.

• Question

  How would I construct an isosceles triangle whose base is 6 cm and altitude 4 cm?

  

  Use a ruler to draw the 6 cm base. Use the ruler and a compass to construct the perpendicular bisector of the base. Mark a point on the perpendicular bisector 4 cm in either direction from the base. Draw lines connecting the 4 cm point on the bisector to each end of the base. That's the triangle. (In an isosceles triangle, the altitude to the base is a perpendicular bisector.)

• Question

  How would I do this with a triangle?

  

  Given a set square or drafting triangle ABC and a straight line drawn on a piece of paper: place any side of the triangle (say, AB) along the given line, and draw another line along either of the other sides of the triangle (say, BC). Flip the triangle over so that the face that had lain on the paper is now facing up away from the paper. Place AB along the second drawn line (BC). Draw a third line along side BC. This third line is parallel to the original line. If these instructions are hard to follow, just experiment with the triangle, and you'll figure it out.

• Question

  "Set the compass to a width that is less than half of the line segment you constructed." This should be: "Set the compass to a width that is more than half of the line segment you constructed," shouldn't it?

  

  It actually doesn't matter. The compass can be set at any convenient width, although a longer width makes it easier to accurately transfer angle CBD along the transverse to form angle PAQ.

• Question

  Can you draw a parallel line using a ruler?

  

  Yes. Locate two points on the given line that are apart from each other slightly less than the length of the ruler. Place the ruler perpendicular to the line at the first point. Measure a convenient distance from the line, and mark a third point at that distance. Do the same thing at the second point, thus marking a fourth point on the same side of and at the same distance from the line. With the ruler draw a line passing through the third and fourth points. That line will be parallel to the original line.

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Things You'll Need

• Pen or Pencil
• Straight edge or ruler
• Compass

About This Article

Article Summary X

To construct a line parallel to a given line through a point, locate the given line and the given point, labeling the line "m" and the point "A." Then, place the tip of a compass on point A, and draw a large arc that intersects the line at some point, Without changing the width of the compass, set one tip on the intersection point, and make another intersection point along the line, creating a third vertex for an invisible rhombus. Move the compass so that it's the same width with one tip on the third vertex, and intersect the first arc that you drew. Finally, connect point A to the final vertex of the rhombus to draw the line. For tips on using perpendicular lines and corresponding angles to draw parallel lines, read on!

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How to Draw Parallel Lines Through a Point

Source: https://www.wikihow.com/Construct-a-Line-Parallel-to-a-Given-Line-Through-a-Given-Point

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