Two parallel lines can intersect.

Ave.Con asserts that I have not provided any definitions for parallel lines. This is not a factual assertion, and anyone who reads the first round would know that I provided definitions for the resolution. Two parallel lines was defined as two lines that have the same slope. Intersection was defined as a point that is common to two lines or surfaces that intersect. Therefore, my burden of proof is to demonstrate that two lines with the same slope can have a point that is common between them. I have provided all of the relevant definitions for this debate. Con's definitions are superfluous and unnecessary, and they should be ignored because all relevant terms have been defined already.Con is really just trying to define his way to a victory. You can't enter a debate about parallel lines intersecting and then define parallel lines as lines that never touch. It's like entering a debate called 'Public health care is better than private health care' and defining public health care as 'the system of health care that is inferior to private health care'. It's a semantic trick called the 'definist fallacy' [1]. I'm not suggesting that Con did this on purpose, as I don't think there is any evidence of that, but it's a good reason to ignore the definitions he offered.Now that the unpleasant business is out of the way, we can talk about geometry. There's a problem with trying to define the word geometry. It's rather like talking about sports. Saying 'I play sports' doesn't tell us much about what you're actually doing. There are sports like football, soccer, swimming, etc. Which sport do you play? Geometry is similar to this. Saying 'geometry' is like saying 'sports'. There are types of geometry in the same way that there are different types of sports. There's Euclidean geometry, hyperbolic geometry, sphreical geometry, absolute geometry, and others [2]. In the first round, I indicated that we wouldn't be focusing on any specfic geometry. So, taking an example from one geometry is good enough for my case. Most people learn in high school geometry that two parallel lines never touch. It's so intuitively obvious that nobody who learns it ever questions it. You just imagine those two lines going on forever, never touching. However, geometry seems to be a lot more imaginative than we are. The idea that two parallel lines will never intersect is an aspect of Euclidean geometry. It's called the fifth postulate. The problem is that there are many geometries that reject this postulate. Spherical and hyperbolic, geometries I named earlier, are examples of this. My third reference talks about the implications of hyperbolic geometry, or 'saddle geometry'. (All pictures and text are from the website, I claim no credit.) In hyperbolic geometry, the sum of the angles of a triangle is less than 180°. In hyperbolic geometry, triangles with the same angles have the same areas. There are no similar triangles in hyperbolic geometry. In hyperbolic space, the concept of perpendicular to a line can be illustrated as seen in the picture at the right. Lines can be drawn in hyperbolic space that are parallel (do not intersect). Actually, many lines can be drawn parallel to a given line through a given point. It demonstrates that two parallel lines can touch each other quite nicely, which is due to the positive and negative curvature of the 'saddle'. Here's a picture. [4] As the above diagram demonstrates, hyperbolic geometry allows two parallel lines to eventually diverge and intersect. The fifth postulate isn't a necessary truth, and it's only for true Euclidean geometry. Once you step outside of Euclidean geometry, two parallel lines can clearly touch. (It's also interesting that this diagram shows a triangle with an angle sum that is less than 180, which shows that another Euclidean postulate doesn't always have to be true.)I'll turn this debate over to Con. I thank him for his wishes of good luck, and hope the best for him as well.Vale.References1: http://www.logicallyfallacious.com...2: http://soler7.com...3: http://regentsprep.org...4: http://www.debate.org...