Understanding the Mathematical Subject Parallel Lines

A straight line is a line segment extended infinitely in both directions.  Two straight lines can be either parallel, skew, or intersect. Parallel lines are straight lines that do not intersect and are always spaced an equal distance apart.   

Skew lines also do not intersect, but they are not spaced an equal distance apart. Perpendicular lines intersect at 90 degree angles. The equation for straight line is a linear equation. Linear equations have an exponent of 1.  It is also called degree 1.  Examples are y=3x+1, y=5x-7, y=x+3, y=3x+10, etc. 

Lines that are parallel have equal slopes.  The slope can be found by putting the equation in the form y=mx+b, where m is the slope and b is the y-intercept.  The slope is the angle the line makes with the horizontal.  The y-intercept is the value for y when x=0 because the y-intercept is the point (0,y) of the equation.  Using the examples above that are already in the y=mx+b form, the slope of the line y=3x+1 is 3, the slope of the line y=5x-7 is 5, the slope of the line y=x+3 is 1, and the slope of the line y=3x+10 is 3.

The lines y=3x+1 and y=3x+10 are parallel because both lines have slopes equal to 3. The other lines are not parallel because they do not have equal slopes. Another way to find the slope is change in y divided by change in x equals the slope.  It can also be expressed as m=(y2-y1)/(x2-x1).  The x1, x2, y1, y2 are the points (x1, y1) and (x2, y2) in the coordinate plane.  An example is (0, 1), (-9, 100), where x1=0, y1=1, x2=-9, y2=100.  Using the equation, m= (y2-y1)/(x2-x1)= (100-1)/(-9- 0) = 99/-9= -11.  The slope is -11.  Parallel lines lie in the same plane in plane geometry.

Squares, rectangles, and parallelograms all have two parallel lines.  A square is a four-sided shape that is four line segments equal in length connected at four right (90 degree) angles.  A rectangle is four line segments with equal widths and lengths connected at four right angles.  If the length and width of the rectangle are equal, then it is also a square.  A parallelogram is four segments with two pairs of parallel segments connected to each other.  The rectangle and square are special parallelograms.   

If two lines intersect at a point, then they are not parallel.  Two lines intersect if they have one point in common.  For example, y=3x+1 and y=10x+1 have the common solution x=0, y=1.  This means the intersection is at the point (0, 1).  Another example is y=5x-7 and y=39x-41 have the common solution x=1, y=-2 and so intersect at the point (1,-2).  This can be determined by substituting the second equation y=39x-41in the first equation y=5x-7 for y as follows: 39x-41=5x-7, 34x=34, x=1.  Then plugging x=1 back in either equation and solving for y will generate -2.

If two lines do not have any points in common, they are either parallel lines or skew lines. Perpendicular lines are not parallel because they intersect at an angle of 90 degrees. Perpendicular lines have slopes that are negative reciprocals of each other.  Lines perpendicular to the lines already mentioned are y=-1/3x+1 is perpendicular to y=3x+1, y=-1/5x-7 is perpendicular to y=5x-7, y=-x+3 is perpendicular to y=x+3, and y=-1/3x+10 is perpendicular to y=3x+10.  This is because -1/3, -1/5, -1, and -1/3 are negative reciprocals of the slopes of the previous lines: 3, 5, 1, and 3.  The y-intercept does not change the slope of the line, so y=-1/3x+9998, y=-1/3x-989899, y=-1/3x+25363565, and y=-1/3x-33 are all also perpendicular to y=3x+1.

Parallel lines are always separated by the same distance wherever they are measured. An infinite number of parallel lines will always exist in a plane.  A plane is determined by a point and a line or three points.

Euclid’s Fifth Postulate states that any point not on a line has only one line parallel with that line through the point.  It has recently been proved true. Non-Euclidean geometry uses curved lines to eliminate the parallel postulate which is based on plane geometry that uses both curved and straight lines.  Hyperbolic geometry is an example of Non-Euclidean geometry.  Curved lines are more practical than straight lines because the Earth is round in shape.

There are some important postulates and theorems concerning parallel lines. If a line intersects two lines and the corresponding

angles are equal, then those two lines are parallel. It is also true that the corresponding angles formed by a line that intersects

two parallel lines will be equal. Also the alternate interior angles and the alternate exterior angles formed by the line intersecting the 

parallel lines are equal.