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Parallel Lines

Definition

There can be many lines in a plane, some of which may intersect each other while some may not intersect when produced in either direction. Thus we can define parallel lines as – “Two lines l and m in the same plane are said to be parallel lines of they do not intersect when produced indefinitely in either direction.”

Properties

The following are the properties of parallel lines – 

  1. The distance between a pair of parallel lines always remains the same. This means that parallel lines are always the same distance apart from each other.
  2. No matter how much we extend the parallel lines in each direction, they would never meet.

Symbol

Symbolically, two parallel lines l and m are written as l || m.

It should be noted that if two lines are not parallel, they will intersect each other. For instance, below we have the lines l and m as intersecting lines as they are not parallel.

Equation

For obtaining the equation of parallel lines, let us recall what we mean by the slope intercept form of the equation of a line. The Slope Intercept forms of a straight line is given by  y = m x + c, where ‘m’ is the slope and ‘c’ is the y-intercept. The steepness of the line is determined by the slope or the gradient of the lone which is represented by the value m. 

It should be noted that the slope of any two parallel lines is always the same.

Gradient

To find the gradient of a parallel line, let us first recall what we mean by the slope of a line. The trigonometrical tangent of the angle that a line makes with the positive direction of the x-axis in an anticlockwise sense is called the slope or the gradient of a line. 

The slope of a line is generally denoted by m. Thus m = tantan θ

Since a line parallel to the x-axis makes an angle of 0o with the x-axis, therefore, its slope is tan 00 = 0

A line parallel to the y-axis, i.e. a line that is perpendicular to the x-axis makes an angle of 90o with the x-axis, so its slope is tan $\frac{\pi}{2}$ = ∞. Also, the slope of a line equally inclined with axes is 1 or -1 as it makes an angle of 45o or 135o with the x-axis.

The angle of inclination of a line with the positive direction of the x-axis in an anticlockwise sense always lies between 00 and 1800.

Let us now understand the slop using some examples.

Example

What can be said regarding a line is its slope is zero ?

Solution

Let θ be the angle of inclination of the given line with the positive direction of the x-axis in an anticlockwise sense.  Then, its slope is given by m = tan θ.

If the slope of a line is zero, then,

m = tan θ = 0 ⇒ θ = 00 

This means that either the line is x-axis or it is parallel to the x-axis.

Thus the line of zero slope is parallel to the x-axis.

Parallel Lines and Transversal

Before we move to understand how a transversal affects a pair of parallel lines, let us the basic definition of a transversal.

A transversal is defined as a line intersecting two or more given lines in a plane at different points.  For example, in the figure below, the lines l and m are parallel while the line p is intersecting both the line l and the line m. Hence, the line p is a transversal to the lines l and m.

Now, as we can see in the above figure, a transversal makes some angles with the lines it intersects. 

Let l and m be two lines and let n be the transversal intersecting them at P and Q respectively as shown below – 

Clearly, lines l and m make eight angles with the transversal n, four at P and four at Q. we have labelled them 1 to 8 for the sake of convenience and shall now classify them in the following groups – 

Exterior Angles 

The angles whose arms do not include the line segment PQ are called exterior angles. Therefore, in the above figure, angles 1, 2, 7 and 8 are exterior angles.

Interior Angles 

The angles whose arms include the line segment PQ are called interior angles. Therefore, in the above figure, angles 6, 4, 5 and 6 are interior angles.

Corresponding Angles

A pair of angles in which one arm of both the angles is on the same side of the transversal and their other arms are directed in the same sense is called a pair of corresponding angles. In the above figure, there are four pairs of corresponding angles, ∠1 and ∠ 5, ∠ 2 and ∠ 6, ∠ 3 and ∠ 7, ∠ 4 and ∠ 8.

We can also say that two angles on the same side of the transversal are known as corresponding angles if both lie either above the lines or below the two lines.

Alternate Interior Angles

A pair of angles in which one arm of each of the angles is on opposite sides of the transversal and whose other arms include segment PQ is called a pair of alternate interior angles. In other words, Alternate interior angles are angles formed when two parallel or non-parallel lines are intersected by a transversal. The angles are positioned at the inner corners of the intersections and lie on opposite sides of the transversal.

In the above figures, ∠ 3 and ∠ 5 form a pair of alternate interior angles. Another pair of alternate interior angles in this figure is ∠ 4 and ∠ 6.

Alternate Exterior Angles in a Transversal

A pair of angles in which one arm of each of the angles is on opposite sides of the transversal and whose other arms are directed in opposite directions and do not include segment PQ is called alternate exterior angles in a transversal.

In the above figure, ∠ 2 and ∠ 8 form a pair of alternate exterior angles. Another pair of alternate exterior angles in this figure is ∠ 1 and ∠ 7.

Now, we shall learn about the alternate interior angle theorem that is also one of the basic properties of alternate interior angles.

Rules for Parallel Lines Intersected by a Transversal

The following rules define the angles that are formed when a pair of parallel lines is intersected by a transversal – 

  1. The pairs of corresponding angles formed on the parallel lines will be equal.
  2. The pairs of alternate interior angles formed on the parallel lines will be equal.
  3. The pairs of alternate exterior formed on the parallel lines will be equal.
  4. The sum of consecutive interior angles on the same side of the transversal is 180o
  5. The sum of consecutive exterior angles on the same side of the transversal is 180o

Let us understand this by an example.

Example

Consider the following figure where a pair of parallel lines has been intersected by a transversal – 

Now, let us observe the 8 angles formed in the above figure.

  1. The pairs of corresponding angles formed on the above parallel lines are ∠ 1 and ∠ 5, ∠ 2 and ∠ 6, ∠ 3 and ∠7, ∠ 4 and ∠ 8. All these four pairs of angles will be equal to each other.
  2. The pairs of alternate exterior angles formed on the above parallel lines are ∠ 1 and ∠ 8, ∠ 2 and ∠ 7, These two pairs of angles will be equal to each other.
  3. The pairs of alternate interior angles formed on the above parallel lines are ∠ 3 and ∠ 6, ∠ 4 and ∠ 5, These two pairs of angles will be equal to each other.
  4. The pairs of consecutive interior angles on the same side of the transversal are ∠ 4 and ∠ 6, ∠ 3 and ∠ 5. The sum of each of these pairs will be 180o.
  5. The pairs of consecutive exterior angles on the same side of the transversal are ∠ 2 and ∠ 8, ∠ 1 and ∠ 7. The sum of each of these pairs will be 180o.

The above rules are also used to prove that the two lines are parallel to each other.

Real Life Applications of Parallel Lines

Observe our surroundings. Do you see any use of parallel lines in real life? Here are some real life instances, where we make use of parallel lines and their concepts – 

Roads and Railways Tracks – Have you ever noticed that the roads and the railway tracks in any region of the world are always parallel lines? Although they lie in the same direction, yet they never meet.

Notebooks – Notice the lines marked for your notebooks for marking the space for writing. They are parallel lines that allow you to write neatly and in uniformity.

Pedestrian crossings – The painted lines that define the pedestrian crossings are always parallel lines.

Examples

Example 1 Are the two lines cut by the transversal line parallel?  What property can you use to justify your answer?

Solution We have been given a figure where two lines have been cut by a transversal. We need to confirm whether the given lines are parallel. Let us observe the angles marked on the transversal intersecting the two lines. We have been that – 

∠ a = 96o and ∠ c = 96o

We can clearly see that ∠ a and ∠ c are alternate interior angles. Now, we have learned that if the alternate interior angles formed by a transversal intersecting two lines are equal, this means that the two lines are parallel. 

Now, here we have been given that these two alternate interior angles are equal. Hence, we can say that by the property of transversal intersecting parallel lines, the two lines are parallel to each other.

Example 2 In the given figure, the lines l and m are parallel. n is a transversal and ∠ 1 = 40o. Find all the angles marked in the figure.

Solution We have been given that lines l and m are parallel. n is a transversal and ∠ 1 = 40o. we need to find the remaining angles.

Let us start with each angle one by one.

First, let us find ∠ 2. We can clearly see that ∠ 1 and ∠ 2 form a supplementary pair of angles. This means that the sum of ∠ 1 and ∠ 2 should be equal to 180o. Hence, we have,

 ∠ 1 + ∠ 2 = 180o

⇒ 40o + ∠ 2 = 180o

⇒ ∠ 2 = 180o – 40o 

2 = 140o

Now, that we have found the value of ∠ 2, we will find the value of ∠ 6

Note that ∠ 2 and ∠ 6 form a pair of corresponding angles. Since the lines that have been intersected by a transversal are parallel, therefore, the pair of corresponding angles should be equal. Hence, we have,

2 = 6 =  140o

Similarly, 1 = 5 =  40o

Note that ∠ 3 and ∠ 5 form a pair of alternate interior angles. Since the lines that have been intersected by a transversal are parallel, therefore, the pair of alternate interior angles should be equal. Hence, we have,

3 = 5 =  40o

Similarly, 4 = 6 =  140o

Similarly, we can see that ∠ 5 and ∠ 8 form a supplementary pair of angles. This means that the sum of ∠ 5 and ∠ 8 should be equal to 180o. Hence, we have,

∠ 5 + ∠ 8 = 180o

⇒ 40o + ∠ 8 = 180o

⇒ ∠ 8 = 180o – 40o 

8  =  140o

Again, we can see that ∠ 6 and ∠ 7 form a supplementary pair of angles. This means that the sum of ∠ 6 and ∠ 7 should be equal to 180o. Hence, we have,

∠ 6 + ∠ 7 = 180o

⇒ 140o + ∠ 7 = 180o

⇒ ∠ 7 = 180o – 140o 

7 = 40o

Hence, we have, 1 = 3 = 5 = 7 = 40o and 2 = 4 = 6 = 8 = 140o

Example 3 Determine the equation of the line through the point (4, -3) and parallel axis.

Solution We have been given that the straight line passes through the point ( 4, -3 ).

This means we have been given that x1 = 4 and y1 =  -3

Also, the straight line is parallel to the x-axis.

Now, recall that we have learned that the line of zero slope is parallel to the x-axis. This means that is a line is parallel to the axis if the slope of a line is zero, or,

m = tan θ = 0 ⇒ θ = 00

So, we have m = 0

We also know that, 

Now, we know that the equation of the line in point-slope forms I 

y-y1 = m (x-x1)

Substituting the given values in the above equation, we have

y-( -3 ) = 0 (x-2)

⇒ y  + 3 = 0

Hence, the equation of the line through the point (4, -3) and parallel axis will be given by y  + 3 = 0

Key Facts and Summary

  1. Two lines l and m in the same plane are said to be parallel lines of they do not intersect when produced indefinitely in either direction.
  2. Symbolically, two parallel lines l and m are written as l || m.
  3. The slope of any two parallel lines is always the same.
  4. The trigonometrical tangent of the angle that a line makes with the positive direction of the x-axis in an anticlockwise sense is called the slope or the gradient of a line.
  5. A pair of angles in which one arm of both the angles is on the same side of the transversal and their other arms are directed in the same sense is called a pair of corresponding angles.
  6. A pair of angles in which one arm of each of the angles is on opposite sides of the transversal and whose other arms include segment PQ is called a pair of alternate interior angles.
  7. A pair of angles in which one arm of each of the angles is on opposite sides of the transversal and whose other arms are directed in opposite directions and do not include segment PQ is called alternate exterior angles in a transversal.
  8. If two straight lines parallel to each other are intersected by a transversal, then the pairs of corresponding angles, alternate interior angles and alternate exterior angles will be equal.
  9. If two straight lines parallel to each other are intersected by a transversal, then the sum of consecutive interior angles on the same side of the transversal as well as the sum of the consecutive exterior angles on the same side of the transversal is 180o.