Properties of Parallel Lines

What are the Parallel Lines?

Two lines are said to be parallel to each other if they never meet or cross each other in a plane. Lines which do not have any common intersection point or cross each other are considered as parallel lines. Two parallel lines are represented as \[\overline{AB}\]|| \[\overline{CD}\] which implies that the line \[\overline{AB}\] is parallel to \[\overline{CD}\]. The perpendicular distance between any two parallel lines is always constant.


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In the above figure, line segments PQ and RS denote parallel lines as they have no common intersection point in a plane. You can draw an infinite parallel line between the two parallel lines \[\overline{PQ}\] and  \[\overline{RS}\]in the given plane.


Parallel Line Definition

Parallel lines are two straight lines that are always away from each at the same distance. No matter how far you extend the two parallel lines  they will never meet or intersect each other. 


Parallel lines help us to understand the path of the objects and sides of the various shapes. For example, opposite sides of squares, rectangles, and parallelograms are parallel to each other.


Parallel Line Examples in Real Life

The parallel line is widely used in the construction industry. Buildings are constructed with walls parallel to each other, ceilings are parallel to floors of the building and one building is usually constructed parallel to the other building on the same block.


Notebooks are huge collections of parallel lines. When you close your notebook, then you will see that lines are not only parallel on each page of the notebook but they are parallel from page to the page also.


Parallel line examples in real life are railroad tracks, the edges of  sidewalks,  marking on the streets, zebra crossing on the roads, the surface of pineapple and strawberry fruit, staircase and railings, etc.


Angles and Parallel Lines

When a transversal line intersects by two or more parallel lines in the same plane, the series of angles are drawn. Some specific names are given to these angles on the basis of their location in terms of their side.


Names given to the pairs of angles in parallel lines are

  • Alternate interior angles

  • Alternate exterior angles

  • Corresponding angles

  • Interior angles on the same side of the transversal 

  • Linear pair

If two given parallel lines are cut by transversal lines, then the alternate interior angles are equal.


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In the above figure m //n, ∠1 = ∠2 and ∠3 = ∠4 (Alternate interior angles)


If two straight lines that are parallel to each other are intersected by a transversal then the pair of alternate exterior angles will always be equal.


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In the above figure m //n, ∠1 = ∠2 and ∠3 = ∠4 (Alternate exterior angles)

If two straight lines that are parallel to each other are intersected by a transversal then the pair of corresponding angles will always be equal.


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In the above figure m //n, ∠1 = ∠2 , ∠3 = ∠4,  ∠5 =∠6, and ∠7 =8 (Corresponding angles)

If two straight lines that are parallel to each other are intersected by a transversal then the interior angles on the same side of the transversal are supplementary.


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In the above figure m //n, ∠1 +∠2 = 180 , and ∠3+ ∠4 = 180 ( interior angles on the same side of the tranvsersal).


If two straight lines that are parallel to each other are intersected by a transversal then the pair of vertical angles will always be equal.


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In the above figure m //n, ∠1 = ∠2 , ∠3 = ∠4,  ∠5 =∠6, and ∠7 =8 (vertical angles)

If two angles in a parallel line form a linear pair, they are supplementary.


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In the above figure m //n, the linear pairs are


∠1 + ∠4 = 180, ∠1 + ∠3 = 180, ∠2 + ∠4 = 180, ∠2 + ∠3 = 180, ∠5 + ∠8 = 180, ∠5 + ∠7 = 180, ∠6 + ∠8 = 180, and ∠6 + ∠7 = 180 ( Linear pairs)


Parallel Line Theorem 

The two parallel lines theorems are given below:


Theorem 1.

If two straight lines which are parallel to each other are intersected by a transversal then the pair of alternate interior angles are equal.


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In the above figure, you can see

∠4= ∠5 and ∠3=∠6


Proof: 

As, ∠1= ∠4 and ∠2= ∠3(Vertically Opposite Angles)

Also, ∠1=∠5 and ∠2=∠6 (Corresponding Angles)

➝ ∠4=∠5 and ∠3=∠6

The Converse of  Theorem 1  is also true which defines that if the pairs of alternate interior angles are equal then the given lines will always be parallel to each other.


Theorem 2

If two given straight lines which are parallel to each other and are intersected by a transversal then the pair of interior angles on the same side of transversal lines are supplementary.


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∠3+ ∠5=180° and ∠4+∠6=180°

As ∠4=∠5 and ∠3=∠6 (Alternate interior angles)

∠3+ ∠4=180° and ∠5+∠6=180° (Linear pair)

➝ ∠3 + ∠5 = 180° and ∠4 + ∠6 = 180°

The converse of Theorem 2 is also valid which defines that if the pair of interior angles on the same sides of transversal are supplementary then the given lines will always be parallel to each other.


Solved Examples

1. In figure M || N and ∠1 = 80°. find ∠ and ∠8


Solution:

As we know,

∠1 = ∠3 ( vertically opposite angles)

and ∠ 3= ∠ 8 ( corresponding angles)

∴ ∠1 = ∠8

∠8 = 80°   [ ∠1 = 80° ( given)

 Now, ∠5 + ∠8 = 180°

➝ ∠5 + 80°  = 180°

➝ ∠5 = 180 - 80 = 20

Hence ∠5° = 20 and ∠8 = 80°


2. In the below fig BC // AD and BA // CD , prove that ∠ABC = ∠ADC


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As AB // CD  and BC is a transversal line intersecting them at B and C respectively. Hence,


∠ABC + ∠ BCD = 180 ( Consecutive interior angles)      (1)


Again BC // DA and CD is a transversal line intersecting them at C and D respectively. Hence,


∠ BCD  + ∠ ADC = 180 ( Consecutive interior angles)     (2)

 

From equation (1) and (2) we get,


∠ABC +BCD = ∠ BCD +∠ ADC


∠ABC = ∠ADC


Quiz Time

1. The two straight lines drawn in the same plane and never meets is known as

  1. Transversal line’

  1. Parallel lines

  2. Perpendicular Line

  3. vertex

  

2. A line that intersects two lines in a distinct point is known as 

  1. Transversal line’

  2. Parallel lines

  3. Perpendicular Line

  4. vertex

      

3. If two parallel lines are cut by a transversal line, then their corresponding angles will be supplementary.

  1. True 

  2. False

FAQ (Frequently Asked Questions)

1. What are the Applications of Parallel Lines in Real Life?

Some of the applications of parallel lines in real-life can be seen if we observe it patiently. For example, consider the roadways. The railway tracks are parallel lines as they never met each other. The two lines or tracks are constructed for the wheels of the train to travel on. The difference between the Mathematician who defined parallel lines and the one who constructed railway tracks is that Mathematicians are liberal to image parallel lines over flat surfaces and paper while the train travels across all sorts of terrain, from hills, slopes and mountains to over bridges.


According to the Mathematician, when two parallel lines are drawn on a graph, they should always be at the same angles which means they will have a  similar slope or steepness.

2. What are the Necessary Conditions for the Lines to be Parallel?

Some of the conditions for the lines to be parallel are:


If two different straight lines are cut by the transversal lines

  1. the pair of corresponding angles are equal then, the two lines straight will be parallel to each other.

  2. the pair of alternate angles are equal then, the two lines straight will be parallel to each other.

  3. the pair of interior angles ion the same side of transversal is supplementary than , the two lines straight will be parallel to each other 


Therefore, in order to prove that the given straight lines are parallel, then either show corresponding angles equal, alternate angles equal, pair of interior angles on the same side of transversal is supplementary.