Download A student asks you if parallel lines meet at infinity

Jacque Boyle
MQFC #1
11/20/08
Parallel Lines
A student asks you if parallel lines meet at infinity. What is your response? What
if the student follows up with, “How do you know?”
First, I would ask other members of the class what they thought and why or how
they know. Then my initial answer would be: No, parallel lines do not meet at infinity. If
the student follows up with: How do you know? I would answer this way, No one has
ever been to infinity so we don’t know. But if parallel lines meet at infinity – infinity
must be a very noisy place with all those lines crashing together!
On a more serious note: I will have the students explain to me what an isosceles
triangle is, while I draw it on the board. I will explain that the if the two lines that are the
same increase in length the angle between then gets smaller and smaller. In the mean time
the other congruent angles in the triangle get larger. Since we know that the sum of the
angles in a triangle are 180 what happens if the angle between the two equal sides goes to
0? … the other angles are both 90 degrees. Then we have two lines that never meet, but
as long as this figure remains a triangle, we don not have parallel lines.
We can go back to the history of geometry to Euclid Geometry. There is a
Euclidean Parallel Postulate that states “For every line l and every point P that does not
lie on l, there is exactly one line m such that P lies on m and m is parallel to l.” This
means that these lines cannot intersect. Also Euclid’s fifth postulate says: “If l and l’ are
two lines cut by a transversal t such that the sum of the measures of the two interior
angles on one side of t is less than 180 degrees, then l and l’ intersect on that side of t.”
There for again we see that these two lines do not intersect.
Standards:
9-12.G.1.1. Students are able to apply the properties of triangles and quadrilaterals to find
unknown parts.
9-12.G.1.2. Students are able to identify and apply relationships among triangles.