Geometric constructions: parallel line | Congruence | High school geometry | Khan Academy

Let's say that we have a line. I'm drawing it right over there, and our goal is to construct another line that is parallel to this line that goes through this point. How would we do that? Well, the way that we can approach it is by creating what will eventually be a transversal between the two parallel lines. So let me draw that.

So I'm just drawing a line that goes through my point and intersects my original line, doing that. So it's going to look like that, and then I'm really just going to use the idea of corresponding angle congruence for parallel lines. So what I can do is now take my compass and think about this angle right over here.

So I'll draw it like that and say, all right, if I have, if I draw an arc of the same radius over here, can I reconstruct that angle? And so where should the point be on this left end? Well, to do that, I can just measure the distance between these two points using my compass.

So I'm adjusting it a little bit to get the point, the distance between those two points, and then I can use that up over here to figure out—and got a little bit shaky—I could figure out that point right over there. And just like that, I now have two corresponding angles defined by transversal and parallel lines.

So what I could do is take my straight edge and make it go through those points that I just created. So let's see, make sure I'm going through them, and it would look like that. And I have just constructed two parallel lines.

And once again, how do I know that this line is parallel to this line? Because we have a transversal that intersects both of them, and these two angles, which are corresponding angles, are congruent. So these two lines must be parallel.