Properties of Parallel Lines



What are Parallel Lines?



Two lines in a plane are said to be parallel if they do not intersect, when extended infinitely in both the direction. Also, the distance between the two lines is the same throughout.

Parallel Line


                Parallel Lines
The symbol for denoting parallel lines is ∥ . If lines l and m are parallel to each other, we can write it as l∥m and which is read as ‘l is parallel to m’.


Properties Of Angles Associated with Parallel Lines:

angles associated with parallel lines



If two parallel lines are cut by a transversal, then

• the pair of corresponding angles is equal (∠2 = ∠6); (∠3 = ∠7); (∠1 = ∠5); (∠4 = ∠8).

• the pair of interior alternate angles is equal (∠4 = ∠6); (∠3 = ∠5).

• the pair of exterior alternate angles is equal (∠1 = ∠7); (∠2 = ∠8).

• interior angles on the same side of transversal are supplementary, i.e., ∠3 + ∠6 = 180° and ∠4 + ∠5 = 180°.

For example let us observe, the adjacent figure shows two parallel straight lines AB and CD.When two parallel lines AB and CD are cut by a transversal MN.

two parallel straight lines


(i) Interior and exterior alternate angles are equal.
i.e. ∠3 = ∠6 and ∠4 = ∠5 [Interior alternate angles]
∠1 = ∠8 and ∠2 = ∠7 [Exterior alternate angles]

(ii) Corresponding angles are equal.
i.e. ∠1 = ∠5; ∠2 = ∠6; ∠3 = ∠7 and ∠4 = ∠8

(iii) Co-interior or allied angles are supplementary.
i.e. ∠3 + ∠5 = 180° and ∠4 + ∠6 = 180°



Conditions of Parallelism:

If two straight lines are cut by a transversal, and if

• the pair of corresponding angles is equal, then the two straight lines are parallel to each other.

• the pair of alternate angles is equal, then the two straight lines are parallel to each other.

• the pair of interior angles on the same side of transversal is supplementary, then the two straight lines are parallel.

Therefore, in order to prove that the given lines are parallel; show either alternate angles are equal or, corresponding angles are equal or, the co-interior angles are supplementary.


Parallel Rays:

Two rays are parallel if the corresponding lines determined by them are parallel. In other words, two rays in the same plane are parallel if they do not intersect each other even if extended indefinitely beyond their initial points.

parallel rays


                Parallel Rays
Therefore, ray AB ∥ ray MN


Parallel Segments:

Two segments are parallel if the corresponding lines determined by them are parallel.
In other word, two segments which are in the same plane and do not intersect each other even if extended indefinitely in both directions are said to be parallel.

parallel segments


                Parallel Segments
Therefore, segment AB ∥ segment MN

One segment and one ray are parallel if the corresponding lines determined by them are parallel.

parallel


Therefore, segment AB ∥ ray PQ.
The opposite edge of a ruler is an example of parallel line segments.



Lines and Angles

  • Fundamental Geometrical Concepts
  • Basic Geometrical Concepts
  • Angles
  • Classification of Angles
  • Related Angles
  • Some Geometric Terms and Results
  • Complementary Angles
  • Supplementary Angles
  • Complementary and Supplementary Angles
  • Adjacent Angles
  • Linear Pair of Angles
  • Vertically Opposite Angles
  • Parallel Lines
  • Transversal Line
  • Parallel and Transversal Lines

  • 7th Grade Math Problems

    8th Grade Math Practice

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