Mathematics

Understanding Parallel Lines



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During the course of studying mathematics students will encounter the topic of parallel lines. During this study, students will need to know how to define parallel lines, their relationships with other parallel lines, perpendicular lines and planes. A student will also need to know the difference between parallel lines and intersecting lines and how parallel sides help form geometric figures.

A plane is a two dimensional surface that has an infinite length and width. One could think of a plane as a wall, that would be a verical plane. The length and wideth of that plane technically continues forever, upward towards the sky, downward through floor, and left and right. Another plane could be the ceiling, which is a horizontal plane, which also extends lengthwise and widthwise forever. Those two planes intersect, meaning they touch and cross each other. Lines on two intersecting planes will also intersect. Lines on two parallel planes might or might not be parallel, but they will never intersect. Lines on the same plane can be either parallel or perpendicular. The term parallel means, "never intersecting".

The algebraic relationship between two parallel lines is they are lines that have the same slope. The slope of a line is defined at the change in the y variable divided by the change in the x variable. It is often referred to as the "rise" over the "run" and is basically the degree to which the line slants upward (from lower left to upper right) or downward from upper left to lower right. For example, two lines could have the equations y=4x+6 and y=4x+9. Both of the lines have a slope of 4 (the coefficient of the x variable determines the slope). So in this case the lines are parallel. If the equations were y=4x+6 and y=3x+10, the lines are not parallel because the slopes are 4 and 3 respectively.

There is an interesting and important relationship between lines that have slopes multiplying to -1. These two lines will interesect at a right angle (90 degree angle) are said to be perpendicular to one another. An example to illustrate this would be the equations y=2x+3 and y= -(1/2)x-5. The slopes of these two lines are 2 and -1/2, respectively. Multiplying these two slopes together results in -1. Now add another line with a slope of 2 and another one with a slope of -1/2 and we have two pairs of perpendicular lines and also 2 pairs of parallel lines, thus forming a geometric figure, either a square or a rectangle.

A rectangle has 4 right angles, and two pairs of parallel lines forming the 4 sides. The two vertical sides have the same length and the two horizontal sides have the same length. If all four sides have the same length, the rectangle is a square. So some rectangles can be squares but all squares are rectangles. A rhombus is a 4 sides geometric figure with 2 pairs of parallel sides with no right angles. Think of a diamond and you'll get the idea of what a rhombus looks like. Other figures with opposide sides that are parallel include the octogon (a stop sign in an octogon), the pentagon (a baseball home plate) and the 6 sided hexagon.

Parallel lines can also be observed in 3 dimensional objects. Take a globe for example. On the globe one will notice the latitudinal and longitudinal lines which run vertically and horizontally, repectively. Any two latitudinal lines are parallel as are any two longitudinal lines.

The key point is that parallel lines are lines that never intersect, meaning they never cross each other. They are part of the formation of geometric figures, and may or may not lie in the same plane and will never lie in intersecting planes. Hope this clear up any questions about parallel lines.

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