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Parallel lines

 

The lines which do not intersect each other at any point are known as parallel lines. The lines may line in the same plane but must not touch each other. || is the symbol used to denote a set of parallel lines or relation of parallelism between two or more objects or things. ∦ is the symbol used to denote non-parallel lines or show non-parallelism between two or more objects.

The best example we see in our daily life of parallel lines are the railway tracks. They lie on the same plane, extend in either direction, but do not meet each other at any point.

Parallel lines angle

The angle formed by the parallel lines when they meet a third line intersecting them is called parallel lines angle. The intersecting line is called a transversal. The angles formed by the transversal and parallel lines are equal and opposite. These angles formed have unique names and special properties.

Consider the image above. Lines L1 and L2 are parallel to each other. A third line, transversal, intersects them and forms parallel line angles. The following angles are-

Name of the angle Example Equality
Corresponding Angles a and e, b and f, c and g, d and h a = e, b = f, c = g, d = h
Alternate Interior Angles c and e, d and f c = e, d = f
Alternate Exterior Angles a and g, b and h a = g, b = h
Vertically Opposite Angles a and c, b and d, e and g, f and h a = c, b = d, e = g, f = h

Properties of parallel lines-

  • Corresponding angles so formed are equal.
  • Vertically opposite angles so formed are equal.
  • Alternate interior angles so formed are equal, and the pair of interior angles on the same side of the transversal is supplementary.
  • Alternate exterior angles so formed are equal.

Equation of parallel lines

The equation representing any line is y = mx + b, where m is the slope of the line, x and y are the intercepts, and b is a constant. In parallel lines, the slope m is either equal or proportional to the other slope of the other line. x and y-intercepts can be different, but the slope must be equal or proportional. x and y-intercepts can be negative, positive, or zero in these lines. However, the value of slope must be positive or negative of both the parallel lines.

How do you find out whether the line is parallel or not?

Two lines can be called parallel lines, if-

  • Any pair of corresponding angles are equal
  • Any pair of alternate interior angles are equal
  • Any pair of alternate exterior angles are equal
  • Consecutive interior angles (co-interior) angles are supplementary

Are parallel lines always straight?

Parallel lines can be straight and curved. These lines should not meet each other at any point to be called parallel lines. They can travel in any direction or can be of any length. They should be placed at an equidistance.

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