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Constructing Parallel Lines - Problem 2

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 alternate interior angles theorem states that the alternate interior 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 an arc on the opposite side as the first arc (since the aim is to draw alternate interior angles).

Find the distance between the arc and the intersections on the first angle with the compass. Then, keeping the compass the same size, place it on the arc drawn around point p and mark the intersection. Connecting this point of intersection with point p creates a line parallel to line l.

When you’re asked to construct a parallel line through a point to given line you have three options; using corresponding angles, alternate interior angles, or alternate exterior angles. In this problem I’m going to ask you to construct using alternate interior angles. So let’s come up with a little game plan here.

If this is sketch of what we’re going to do, now I have point P, the first step is to draw a transversal through point P so it intersects our line.

The second step is to say well alternate interior angles. I can duplicate this angle right here on this side of point P. Let’s use our transversal, the alternate interior angle will be over here and if alternate interior angles are congruent, then we must have two parallel lines so let’s get started.

The first step is to grab your straightedge and your pencil and you’re going to construct a line through point P that intersects the line that’s given. It doesn’t matter what that angle is as long as you’re going to duplicate this angle right here.

So the first step in duplicating an angle is swinging an arc from the vertex, so the idea here is we’re going to duplicate an angle creating alternate interior angles, so we’re going to swing an arc with our sharp end centered at the vertex, come up to point P and we know we want our alternate interior angle to be over here. So I’m going to swing another arc right there.

Now we need to measure the distance between that point of intersection on either side of our angle, so I’m going to use a sharp end of my compass here and move my marker until I have measured the distance. Make a little mark right there and now I’m going to use this point of intersection and I’m going to make a mark somewhere up here which will duplicate my angle.

So now the last step is to connect this point of intersection with your point p using a straightedge and again this will not be perfect, but there will be an element of human error, but on your test or quiz which your teacher is looking for are these marks and if we’ve created two alternate interior angles that are congruent then these two lines are parallel.

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