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Constructing Parallel Lines - Problem 3

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 exterior angles theorem states that the alternate exterior 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 one of the exterior angles. Place the compass at point p and swing an arc on the opposite side as the first arc, outside of the parallel lines (since the aim is to draw alternate exterior 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.

In this problem we’re being asked to construct a line parallel to a given line L through a point P using alternate exterior angles. The other two methods are alternate interior angles and corresponding angles, but why does this work?

Well if you come over here where I’ve sketched this problem, the idea behind this is to create a transversal that passes through point P and then say well if I duplicate alternate exterior angles then we would have created two parallel lines. So I’m going to draw my transversal that’s going to be my first step, then I’m going to duplicate this angle out here this exterior angle up at point P. So let’s grab our compass and get started.

The first step is to draw a transversal, I was going to swing an arc, but I won’t have an angle to start with, so we’re going to draw a line through point p intersecting our given line. So this angle right here you could also choose this obtuse angle it doesn’t matter they’re both going to be exterior angles, I think it’s easier to duplicate an acute angle. So I’m going to swing an arc from that vertex and I’m going to come up to point P and I know that if this angle is on the left side of the transversal, it’s alternate exterior angle will be on the right side, so I’m going to swing another arc from point p a little mistake there, but I think it will work.

Next I need to measure that point of intersection using my compass and I’m going to make a mark right here which is what your teacher is looking for. Come back up here and I’m going to swing that identical arc so now we have two points of intersection. We have P and that point. Connect them with your straightedge and we’ll have our parallel line.

The reason why these two lines have to be parallel is because the alternate exterior angles are congruent, which is the converse of the parallel lines there, so I’m going to mark these two lines as parallel.

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