Parallel and perpendicular lines (Algebra 1, Formulating linear equations) – Mathplanet
Use slope to identify parallel and perpendicular lines . What about the relationship between the slopes of the two lines? [Figure3]. To find the. The only difference between the two lines is the y-intercept. We can determine from their equations whether two lines are parallel by comparing their slopes. This tutorial looks at the relationship between the slopes of parallel lines as Note that two lines are parallel if their slopes are equal and they.
Another property that also involves measurement is that lines parallel to each other have the same gradient slope. History[ edit ] The definition of parallel lines as a pair of straight lines in a plane which do not meet appears as Definition 23 in Book I of Euclid's Elements.
Proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein.
Simplicius also mentions Posidonius' definition as well as its modification by the philosopher Aganis.
The traditional treatment of geometry was being pressured to change by the new developments in projective geometry and non-Euclidean geometryso several new textbooks for the teaching of geometry were written at this time. A major difference between these reform texts, both between themselves and between them and Euclid, is the treatment of parallel lines.
According to Wilhelm Killing  the idea may be traced back to Leibniz. Wilson edited this concept out of the third and higher editions of his text. The main difficulty, as pointed out by Dodgson, was that to use them in this way required additional axioms to be added to the system.
The equidistant line definition of Posidonius, expounded by Francis Cuthbertson in his text Euclidean Geometry suffers from the problem that the points that are found at a fixed given distance on one side of a straight line must be shown to form a straight line. This can not be proved and must be assumed to be true.
Cooley in his text, The Elements of Geometry, simplified and explained requires a proof of the fact that if one transversal meets a pair of lines in congruent corresponding angles then all transversals must do so. Again, a new axiom is needed to justify this statement. So it has the same y-intercept, but its slope is 3, so if x goes up by 1, y will go up by 3. So this line is going to look something like this. Trying my best to connect the dots.
It has a steeper slope, and you see that when x increases, this blue line increases by more in the y direction. So that is line B-- and notice, they do intersect, there's definitely not two parallel lines. And then finally, let's look at line C.Deciding if Lines Coincide, Are Skew, Are Parallel or Intersect in 3D
The y-intercept is 5. The point 0, 5, its y-intercept. And its slope is 2.
So you increase by 1 in the x direction, you're going to go up by 2 in the y direction. If you decrease by 1, you're going to go down 2 in the y direction. If you increase by, well, you're going to go to that point, you're going to have a bunch of these points. And then if I were to graph the line-- let me do it one more time-- if I were to decrease by two, I'm going to have to go down by 4, right?
Parallel lines from equation | Analytic geometry (video) | Khan Academy
Negative 4 over negative 2 still a slope of 2, so 1, 2, 3, 4. And I can do that one more time, get right over there. And then you'll see the line.
The line will look like that, it will look just like that. And notice that line C and line A never intersect. They have the exact same slope. Different y-intercepts, same slope, so they're increasing at the exact same rate, but they're never going to intersect each other. So line A and line C are parallel.