Math 444, October 29: Parallels

1. Given two distinct points A and B, and line AB.  Let AX and BY be two other lines with X and Y on the same side of line AB.  Also, angle ABY = angle BAX = right angle.  Now suppose lines AX and BY intersect at a point C.  What can you say about the angle sum of triangle ABC?  Why is this impossible?  What do you conclude about lines AX and BY?

2. Given two distinct points A and B, and line AB.  Let AX and BY be two other lines with X and Y on the same side of line AB.  Also, angle ABY + angle BAX = 180 degrees. Now suppose lines AX and BY intersect at a point C.  What can you say about the angle sum of triangle ABC?  Why is this impossible?  What do you conclude about lines AX and BY?

3. Given points A and B, with lines a and b through A and B perpendicular to line AB, suppose that m is a line through A not perpendicular to line AB.  Pick any point X on m and let Y be the foot of the perpendicular from X to line AB. Now suppose that the sides of triangle AXY are AX = h, XY = u, YA = v.  Also, let AB = d.  If C is the intersection of m and b, what is the distance BC and what is the distance AC?

4. Now re-examine 3 from a slightly subtler point of view.  In 3 we assumed that the intersection C exists.  How can you use your distance calculations to PROVE that C exists?

5. Given a point B on line b and a point A not on b, what things you say about lines through a which are parallel to b? Is there an “if and only if” here?