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Aug 11, 2011 · $\begingroup$...which means, using Euclid's definition 23 (thanks @GEdgar), that since a line intersects itself, it is not parallel with itself; but using a (barely significantly) different but more modern and convenient definition, it is parallel to itself. So check what is being used in the context you are reading. $\endgroup$
Apr 18, 2018 · $\begingroup$ If you consider a line to be parallel to itself, then you can say that any two lines in the plane with the same slope are parallel. Under the definition of "have no points in common" that statement is false, since you need to add an exception. $\endgroup$
Theorem 7.1.1 7.1. 1. For any point P P and any line ℓ ℓ there is a unique line m m that passes thru P P and is parallel to ℓ ℓ. The above theorem has two parts, existence and uniqueness. In the proof of uniqueness we will use the method of similar triangles. Proof.
Mar 12, 2021 · This is where the fun starts. Normally, definitions, like the concept of parallel lines in 2 dimensional Geometry, are created in order to facilitate solving problems. So, a hidden issue is the question of why the concept, in 3 dimensional Geometry, of a line being parallel to a plane, was created in the first place.
Lines that do not meet are said to be parallel. Euclid, long ago noticed the following things: Every pair of points determines a unique straight line containing both. Two lines are parallel if they never meet. In a plane, every line and a point not on it determine a unique line parallel to the former containing the latter.
This implies that two parallel lines are always a constant distance apart from each other, which is another important characteristic of parallel lines. Thinking more intuitively, this has to be true since if the lines were getting farther apart from each other, then on the opposite side of the lines would be getting closer (and eventually meeting), which contradicts the definition that two ...
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Non-intersecting or parallel lines are the lines that do not intersect each other. They are always at the same distance from one another. Hence, they never meet. Say you are given a line A that is parallel to a line. There is another line B which is parallel to the same line. Does this imply that lines A and B are parallel to each other?
