Contents

# Parallel|Definition & Meaning

## Definition

Two **lines** or line segments are said to be **parallel** to each other, if the **perpendicular distance** between their lines **remains same** throughout their length.

Two lines are called **parallel** to each other if we can prove that the **perpendicular distance** between them at all points is the **same,** they do **not intersect** each other at any point, they are pointing in the **same direction,** and they **never converge** or **diverge. **

All of the above **conditions** are basically describing the same thing in **different words.** The underlying **mathematical condition** or constraint remains the same. The following **figure** shows two line segments that are **parallel** to each other.

**Figure 1: Two Parallel Lines**

It can be seen clearly that both lines have the **same direction,** all points on both lines have **equal perpendicular distance** from the adjacent line, the are neither **converging** nor **diverging,** and, definitely they do not seem to have any **point of intersection** (at least not in the frame).

## Explanation of Parallel Lines

The Greek **Posidonius** is credited by **Proclus** with defining **parallel lines** as equally spaced lines. However, the modern concept of **parallelism** was formalized by **Euclid’s parallel postulate,** which focuses on parallel lines.

In the perspective of **geometry** in particular and mathematics in general, **parallel lines** may be classified as **co-planar straight lines** that **do not cross** each other at any point. In other words, any pair of **co-planer lines** that do not **intersect** are termed parallel lines. This concept can easily be extended to **planes.**

That is, any pair of planes in the same **three-dimensional space** that never cross each other are said to be **parallel planes.**

There is a predetermined **minimum separation** or perpendicular distance between parallel lines that they maintain from **minus infinity** to **plus infinity,** and they do not touch each other or **converge** at any point. In **three-dimensional Euclidean space,** a line and a plane are said to be parallel if they do not share a point.

On the other hand, the **intersection** of two **non-co-planar** lines results in **skew lines**.

The **parallel** lines are important because of the unique set of **deductions** and **geometrical laws** that they follow. They help us as reference objects in many **geometrical problems** and help simplify more **complex problems.** One example of this kind of geometry is **euclidean geometry,** and **parallelism** is a characteristic of **affine geometries.**

Similar parallelism qualities may be seen in lines in other geometries, such as **hyperbolic geometry.**

### Real-life Examples

**Parallelism** is very common in many **real-world applications.** The **figure** given below lists two such common **examples.**

**Figure 2: Real-Life Examples of Parallel Lines**

Here you can see on the **left** in the figure that there is a **ladder.** The **vertical supports** of the ladder are **parallel** to each other. If they were not parallel, they would not **support** each other, and the **structure** would **break.** The **rungs** of this ladder represent the **perpendicular distance** between the lines passing through the **supporting legs.**

These rungs are also **parallel** to each other. Notice that the distance between the parallel supports remains the same throughout the **length** of the **ladder,** which is proven by the fact that the **length** of the **rungs remains the same.**

The figure shows a **transmission line** on the **right side.** It can be noticed that the **hanging power lines** on the transmission supports are also **parallel** to each other. Two such lines are highlighted in **red** and **blue** color for clarity.

The **perpendicular distance** between these lines is kept constant and is depicted by the **cross arm length** of the **supporting tower** that, as we know, remains **constant.**

## Euclidean Postulates of Parallelism (Properties of Parallel Lines)

In this section, we present a more **mathematically rich perspective** of **parallelism** with respect to **straight lines.** We formally introduce parallelism and the **properties** of **parallel lines** in the following paragraphs. These properties can also be used to **verify** or **check** whether two **lines are parallel** or not.

**(a)** For two lines to be parallel, each point on one of the lines must **maintain** a **constant minimum distance** from the other line. That is, both lines should **maintain equal distance** at all points.

**(b)** For two lines to be parallel, there must not exist **any point** that **satisfies** **both line equations.** That is, there shouldn’t exist any **point of intersection** or the lines must **never converge.**

**(c)** If a **straight line crosses two parallel lines,** the **corresponding angles** created by this line with both of the parallel lines must be **congruent.** Congruent means that the angles will be **identical** to each other. This property is explained in the following figure.

**(d)** For two lines to be parallel, they must have the **same slope.**

**Figure 3: Line Crossing two Parallel Lines**

Now in this figure, the **red** and **blue** lines are **parallel** if and only if the pair of **angles a, a’ and b, b’ are congruent (equal)**.

This means that if you draw a line such that it crosses two lines, as shown in the **figure,** and you somehow prove that such **corresponding angles are equal,** then it is a proof that the **lines are parallel.**

The first and third criteria are **“more complex”** than the second since they **require measurement,** although any of these related features might be used to locate parallel lines in **euclidean space.** As a result, in **euclidean geometry,** parallel lines are often represented by the **second characteristic.**

The effects of **Euclid’s Parallel Postulate** are the other features. The **same gradient between parallel lines** may serve as another characteristic of measurements **(slope).**

### Calculation of Distance Between Two Lines

There is a certain **distance between** the **two parallel lines** because parallel lines in a Euclidean plane are identical in length. Given the **equations** for **two parallel non-vertical lines,** by locating two points (one on each line) that are perpendicular to one another and figuring out their distances, it is possible to **determine** the **distance** between the two lines.

Let us say that **two lines** are **represented** in the **slope-intercept form** as follows:

**$ y = mx + u_1 $**

**$ y = mx + u_2 $**

Notice that the **slope** is kept the **same** (i.e., m) since the lines are **parallel.** The **distance** between these two lines is given by the **following formula:**

\[ d = \dfrac{ | u_1-u_2 | }{ \sqrt{ m^2 + 1 } } \]

If **two lines** are **represented** in the form of the **standard form** as follows:

**$ a x + b y + c_1 = 0 $**

**$ a x + b y + c_2 = 0 $**

Notice that for parallel lines, **a and b must remain the same**. The **distance** between these two lines is given by the **following formula:**

\[ d = \dfrac{ | c_1-c_2 | }{ \sqrt{ a^2 + b^2 } } \]

The following figure **summarizes** all these **formulae:**

**Figure 4: Distance Between Two Parallel Lines**

## Numerical Problems

**Part (a):** Find the **distance** between **parallel lines** represented by** 4x + 3y + 4 = 0** and **4x + 3y + 24 = 0.**

Part (b): Find the **distance** between **parallel lines** represented by **y = 10x + 2 and y = 10x +10**.

### Solution to Part (a)

**Given:**

**Line 1: 4x + 3y + 4 = 0**

**Line 2: 4x + 3y + 24 = 0**

**Comparing** with **standard** line **equation:**

**$ a $ = 4, $ b $ = 3, $ c_1 $ = 4, $ c_2 $ = 24**

Using the **formula:**

\[ d = \dfrac{ | c_1-c_2 | }{ \sqrt{ a^2 + b^2 } } \]

**Plugging** the values:

\[ d = \dfrac{ | 24-4 | }{ \sqrt{ 4^2 + 3^2 } } \]

\[ d = \dfrac{ | -20 | }{ \sqrt{ 25 } } \]

\[ d = \dfrac{ 20 }{ 5 } \]

\[ d = 4 \]

### Solution to Part (b)

Given:

**Line 1: y = 10x + 2**

**Line 2: y = 10x + 10**

**Comparing** with the **slope-intercept form:**

$ m $ = 10, $ u_1 $ = 4, $ u_2 $ = 24

Using the **formula:**

\[ d = \dfrac{ | u_1-u_2 | }{ \sqrt{ m^2 + 1 } } \]

\[ d = \dfrac{ | 2-10 | }{ \sqrt{ 10^2 + 1 } } \]

\[ d = \dfrac{ | -8 | }{ \sqrt{ 101 } } \]

\[ d = \dfrac{ 8 }{ 10.05 } \]

\[ d = 0.796 \]

*All images were created with GeoGebra.*