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Constructing Parallel Lines - Problem 1

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

The corresponding angles theorem states that the corresponding angles of two parallel lines with a transversal are congruent. So, in order to create a parallel line, first draw a transversal through the given point p and the given line to create an angle. Placing the compass end at the vertex of the angle, swing an arc through the angle. Place the compass at point p and swing the same arc. Then, measure the distance between the two points of intersection at the angle with the compass. With the compass at the the intersection of arc around point p and the transversal, mark the location on the arc where the compass intersects it. By connecting point p to this point of intersection, a line parallel to line l has been created.

When you’re asked to construct a line parallel through a point to a given line, you have three options; one option is using corresponding angles. So in this problem it says construct a line parallel to line L through point P using corresponding angles. The other 2 options are alternate interior angles and alternate exterior angles. So before we start this construction, let’s make a little game plan.

So over here I have a little sketch of our problem and what I’m going to do is I’m going to create a transversal that passes through my point P thereby creating this angle and I know that the corresponding angle up here if it’s congruent will create two parallel lines because the converse of the parallel line Conjecture says if you have two corresponding angles that are congruent, then the two parallel lines cut by the transversal must be parallel. So let’s get started.

First thing you’re going to do is you’re going to grab your straightedge, and you’re going to make an angle through line P, excuse me through point P through line L. So it doesn’t matter what type of angle you create here. I think an acute angle is easiest to duplicate. So now that we have created this angle right here, we need to duplicate it up at point p.

So grab your compass and we’re going to follow the steps of duplicating an angle. So the first thing I’m going to do I’m going to swing an arc with my compass centered at that vertex, so compass centered at the vertex, I’m going to swing an arc. I’m going to come up to point P and I’m going to swing that same arc.

Okay the second step here is to measure this opening, the distance between these two points of intersection. So you’re going to use your compass to do that and once you have that distance, you’re going to make a little mark right there that’s what your teacher is looking for.

Now the question is where does the sharp end of my compass go? Well I only have one point of intersection that’s right up here so that’s the only place I can put the sharp end of my compass. So I’ve made a mark thereby duplicating this angle which will create corresponding angles. So I’m going to grab my straightedge and I’m going to connect point P with that point of intersection.

Again these might not look perfectly parallel and that’s because there’s a little bit of human error here. So the main step here was realizing that if we have corresponding angles that are congruent, then we will have created two parallel lines.

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