Parallel Lines
A pair of lines that never cross each other and share the same plane are said to be parallel. They are always equal distances apart and share the same slope. It is worthwhile to investigate the properties of parallel lines as well as the angles that result from a transversal’s intersection.
How do Parallel Lines work?
No matter how far they are extended, parallel lines are a particular kind of straight line that never intersect. In the image below, lines ‘a’ and ‘b’ are parallel to each other, while lines ‘p’ and ‘q’ are parallel to each other.
Parallel Lines and Transversal
When a transversal intersects two parallel lines, it creates several pairs of angles. Some of these angles are congruent, while others are supplementary. The figure below illustrates parallel lines intersected by a transversal, with L1 and L2 being the parallel lines. As a result of the intersection, eight distinct angles are formed and labeled alphabetically.
Parallel Lines Properties
When two lines are discovered to be parallel, we are aware that they do not meet on a shared plane. However, four angles are created at each place where a transversal joins two parallel lines at two separate locations. As a result, the following are the characteristics of parallel lines with respect to transversals:
- Congruent angles are those that correspond.
- Congruent angles are those that are vertical, or vertically opposite angles.
- Congruent alternate interior angles exist.
- Congruent external angles exist between the two.
- On the same side of the transversal, the pair of internal angles are additional.
Parallel Lines Equation
The equation of a straight line is frequently expressed in the slope-intercept form, y = mx + b, where ‘m’ stands for the slope and ‘b’ for the y-intercept. The slope of a line is its slope, which is measured by the value of ‘m’.
It is significant to remember that the slope of two parallel lines is constant. For instance, any line parallel to a line with a slope of 4 in the equation y = 4x + 3 will also have a slope of 4. However, parallel lines don’t cross each other and have unique y-intercepts.
Parallel Lines Symbol
Lines that never cross one other and stretch endlessly are known as parallel lines. The sign || is used to denote them. For instance, line AB II PQ indicates that line AB and line PQ are parallel. On the other side, the non-parallel lines are represented by the symbol.
Lines that never cross one other and stretch endlessly are known as parallel lines.
Parallel Line Real Life Example
Train tracks are an actual-world illustration of parallel lines. The tracks are two parallel lines that go far apart from one another without ever coming together. The layout of a standard ruled notebook or sheet of graph paper, which has vertical and horizontal lines that are parallel to one another, is another example of parallel lines in the real world.
Parallel Lines Axioms and Theorem
Review the axioms and theorems for the parallel lines below.
Congruent Angle Axiom
The pair of comparable angles are equal when a transversal intersects two parallel lines.
The inverse of this axiom, which states that if two matching angles are equal, then the specified lines are parallel to one another, is likewise true.
Theorem
When a transversal cuts between two parallel lines, the interior angles on the same side of the transversal are additional.
∠3+ ∠5=180° and ∠4+∠6=180°
Since (Alternate Interior Angles) 4=5 and 3=6,
The linear pair axiom states that 3+4=180° and 5+6=180°.
⇒∠3+ ∠5=180° and ∠4+∠6=180°
The reverse of the aforementioned theorem, which asserts that the given lines are parallel to one another if the two co-interior angles are supplementary, is likewise valid.
Applications of Parallel Lines in Real Life
Parallel lines can also be observed in real life if one is patient and has good observational abilities. Railroads provide a good illustration of this since their rails really form parallel lines. These two tracks are there to make it easier for train wheels to move. Mathematicians may visualise parallel lines on flat surfaces and on paper, but railway tracks must travel over a variety of terrains, including hills, slopes, and mountains, as well as bridges, making them different from those used in railroads.
Two parallel lines must always have the same angle or slope when graphed, according to mathematics.
Frequently Asked Questions – FAQs
Parallel lines are always equidistant apart from each other. They do not intersect each other. When cut by a transversal, parallel lines form a pair of angles. Hence, corresponding angles are equal, alternate interior angles are equal, alternate exterior angles are equal, vertical angles are equal and sum of interior angles on the same side of transversal are supplementary.
If one of the angle pairs is 108 degrees, then the angle that is 108 degrees on the vertical opposite is also 108 degrees.
