How To Construct Parallel Lines
How to construct a parallel line through a point
Step 1
Begin by drawing a line (or ray or line segment) horizontally on your paper, relative to you. Draw points at each end of your line. Label the points of your line anything you like; the letters are unimportant except to identify your line.
For our example, we will construct line LD.
Draw a single point above your line, some distance away (like 3 inches) and give it a label.
We will call ours Point U.
Step 2
Next, we will use our straightedge to construct a transverse, a line intersecting your original line and going through your point above the line. Try to make it at an angle not 90°. This will make your work clearer to you.
Label the intersection of your transverse and your original line with another letter not already used. We will call ours Point Point E.
So far, we have line LD intersected by transverse UE.
Step 3
Use your compass to scribe an arc. An arc is a section of a circle. Open the compass legs so that they are more than half the distance from the two points on your transverse.
In our example, the compass is spread slightly more than halfway between Point U and Point E. Put the point of the compass on Point E and scribe an arc that goes through the transverse line and the horizontal line (in our example, lines UE and LD).
Keep the compass legs the same distance apart and repeat the arc with the compass's sharp point on Point U. Scribe another arc to look similar to the one you just drew.
Step 4
Lift the compass and do not worry about the distance between the legs. You will put the compass's sharp point on the intersection of the first arc you drew and the transverse. Open or close the compass leg to match the distance from that intersection to the arc's other intersection, where it crosses the horizontal line (LD in our example).
Lift the compass, being careful to keep the legs the same distance apart. Put the point down on the intersection of the second arc and the transverse (Point U in our example). Swing the pencil leg of the compass to make a tiny mark through that second arc.
Where you swung the compass and passed through the second drawn arc, you have a new point of intersection. Label that point. In our example, we call it Point M.
Step 5
Use your straightedge to construct a line that passes through the original point above your first line and through the newly labeled point. In our example, that means a line through Point U and Point M.
Put endpoints on that line. Label the endpoints. In our example, we used Point J on the left and Point B on the right.
We have now constructed line JB passing through Points U and M and parallel to line LD (which passes through Point E). Put it all together, and it may feel JUMBLED but it really is not!
You have constructed a line parallel to your original line, without measuring anything!
Constructing parallel lines
Geometry is hands-on mathematics. One skill you may need is the ability to construct parallel lines. This will show you how to do it, using the simplest of tools (and no measuring!).
What are parallel lines?
Two lines, line segments, or rays (or any combination of those) are parallel if they never meet and are always the same distance apart. Both lines have to be in the same plane (be coplanar).
You encounter parallel lines in geometry, of course, but also in everyday life. The lines of notebook paper are parallel. Sides of doors, edges of cereal boxes, and the floorboards of a home are parallel.
Classic examples of parallel lines that fool your eye are railroad tracks and roads, the two lines of which seem to meet in the distance. You know they cannot meet, because then a train could not move or cars could not fit on the road.
Tools of the geometrician
A geometrician is a mathematician who studies geometry. When you construct figures in geometry, you are a geometer. To construct parallel lines, you need these four simple tools:
Paper
Pencil
Straightedge (like a ruler or any straight, thin, smooth object)
Compass (not the kind for direction; the kind with two legs, one with a point and one with a pencil)
A good-quality compass will hold the position of its legs as you adjust them. If your compass legs slip, get a better compass. You are depending on the legs to stay exactly the same distance apart for several steps in constructing your parallel lines.
For checking your work, you may want an accurate ruler, but it is not necessary.
Checking your work
Open your compass to spread the legs so the point is on one parallel line and the pencil is on the other. Lift your compass, being careful not to disturb the legs, and check anywhere along the two lines. If the lines are parallel, the distance will be the same anywhere you check. As a less elegant way to check, you can put a ruler to the distance, so long as you measure perpendicular to the two lines.
Lesson summary
Using only a pencil, straightedge and compass, you have now learned how to draw parallel lines. You also know what parallel lines are, you know some examples from real life, and you know how to check to see if your drawn lines are parallel. You should also be able to explain the steps to someone else. Notice you did not need to measure anything! Geometry is a powerful part of mathematics.