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What are lines that are also called parallel lines?
They could be lines with the same gradient (or slope).
If S equals straight lines in the plane and ab if a and b are parallel Verify that the relation is an equivalence relation on the set S given?
Establishing equivalence depends on the definition of parallel lines. If they are defined as lines which cannot ever meet (have no point in common), then the relation is not reflexive and so cannot be an equivalence relation.However, if the lines are in a coordinate plane and parallel lines are defined as those which have the same gradient then:the gradient of a is the gradient of a so the relationship is reflexive ie a ~ a.if the gradient of a is m then b is parallel to a if gradient of b = m and, if the gradient of b is m then b is parallel to a. Thus the relation ship is symmetric ie a ~ b b ~ a.If the gradient of a is m then b is parallel to a if and only if gradient of b = gradient of a, which is m. Also c is parallel to b if and only if gradient of c = gradient of b which is m. Therefore c is parallel to a. Thus the relation is transitive, that is a ~ b and b ~ c => a ~ c.The relation is reflexive, symmetric and transitive and therefore it is an equivalence relationship.
How do you calculate transversal lines?
Transversal lines are not parallel and so have a gradient that is different to that of the given lines.
What is it called when lines have the same slop?
If two lines have the same gradient, then this means they are travelling in the same direction. Lines travelling in the same direction will never touch or cross each other even if extended to infinity. Lines with such a property are called "parallel lines." This is assuming they are not just exactly the same line.
What lines are parallel?
Parallel lines are lines which share the same gradient. In Euclidean geometry (the geometry used in standard mathematics and day-to-day physics), parallel lines will never meet at a point, but will share every point along their (infinite) lengths if 1 point is observed to coincide with both. The parallel postulate, which is a geometric axiom of Euclid's geometry, defines these properties. However, by moving into elliptical and hyperbolic geometries, parallel lines can be allow to intersect at points (where parallel lines are defined as 2 lines having the same gradient), whilst still retaining logically consistent geometrical definitions. Parallel lines are the opposite of perpendicular lines which meet at right angles.
