Properties of Parallel Lines in Geometry

Parallel Lines are two lines that never meet or cross each other in a given plane. In other words, lines that do not cross each other and do not have any intersection points are known as parallel lines.

The parallel lines help us understand where the objects and sides of different shapes are going. For example, we can observe the opposite sides of a square, rectangle, and parallelogram as they are equal and parallel to each other.

Representation of Parellel Lines

If we represent two lines that are parallel to each other as \[\overleftrightarrow{AB}\parallel \overleftrightarrow{CD}\] which shows that \[\overleftrightarrow{AB}\]is parallel to \[\overleftrightarrow{CD}\], then the perpendicular distance between those two parallel lines is constant.

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The figure above shows us that two straight lines that stay away from each other and are distant at the same length are parallel lines. We can even draw an infinite parallel line between two existing parallel lines \[\overline{PQ}\] and \[\overline{RS}\].

Real-Life Examples

We often use parallel lines in professions like architecture and construction. We can see buildings that are constructed with walls that are parallel to each other. Even the ceilings are parallel to the floors and neighbouring buildings also have parallel lines when they are adjacent to one another.

Notebooks are another well-known example of parallel lines as they are filled with them. These lines are even parallel from page to page.

Other examples include railroad tracks, edge of the sidewalks, street markings, zebra crossing on a road, staircases and railings on those staircases.

Angles and Parallel Lines

When a line interacts with two or more parallel lines by intersecting them, it creates a series of angles. After this, we give these angles specific names based on where they exist on the lines regarding their side.

Some of the types of angles that get created are:

• Corresponding angles

• Alternate interior angles

• Alternate exterior angles

• Interior angles on same-side interior angles 

• Linear pair

Let us look at all these angles in detail:

Types of Angles

As discussed in the topic above, there are different types of angles that get created when an interesting line occurs on two lines that are parallel to each other.

Let us take the help of a figure to explain all these different pairs of angles:

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In the figure above, a and b are parallel lines which, when intersected by another line creates two points, P and Q.

This intersection causes eight angles to get created. In order of their positions from top to bottom let us label these angles as:

∠1,∠2,∠7 and ∠8 that can be observed as the exterior angles and ∠3,∠4,∠5 and ∠6 that denote the interior angles.

Now these angles will have different types of relationships with each other, their pairs will be of different types,

• Corresponding Angles: Angles ∠1 and ∠6; ∠4 and ∠8; ∠2 and ∠5; ∠3 and ∠7 are the corresponding pair of angles because they are in a similar position vertically.

• Alternate Interior Angles: ∠4 and ∠5 ; ∠3 and ∠6 can be observed as pairs of alternate interior angles as they face each other at an intersecting position and are interior angles.

• Alternate Exterior Angles: ∠1 and ∠7; ∠2 and ∠8 are the alternate exterior angles as they are at an intersecting position from each other but they are also exterior angles.

• Same-side Interior Angles: ∠3 and ∠5; ∠4 and ∠6 are same side interior angles because they are on the same side of the line that is intersecting the two parallel lines. They are also known as co-interior or consecutive interior angles.

• Linear Pair: ∠1 and ∠2, ∠3 and ∠4, ∠5 and ∠6, & ∠7 and ∠8 will be considered as linear pairs as they lie on the same plane. 

Parallel Line Theorems

There are two parallel line theorems, these theorems are:

1. Theorem 1: When two parallel lines get intersected by another line, then the pair of the alternate angles that get created are equal.

2. Theorem 2: When two parallel lines get intersected by another line, then the pair of the interior angles that are on the same side of the intersecting line are supplementary.

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. 

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.

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 page also.

Parallel line examples in real life are railroad tracks, the edges of  sidewalks,  markings 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