In geometry, a line is a one-dimensional figure that has no initial point and no endpoint. Therefore, we represent a line with arrows on both ends to denote that it continues forever for both sides. But what are parallel lines? As the name suggests, these lines run parallel to each other. Parallel lines do not meet or intersect at any point. But why do we study parallel lines geometry, and what are their applications? Read on to learn more about line segments in detail below.
Here’s what we’ll cover:
- Parallel Lines Definition
- Parallel and Perpendicular Lines
- Parallel Lines Cut by a Transversal
- Properties of Parallel Lines
- Axioms and Theorems
- Applications of Parallel Lines in Real Life
Parallel lines definition: Two lines are said to be parallel if they do not intersect each other at any point in the plane. Parallel lines are equidistant from each other throughout.
Furthermore, in 3-D Euclidean space, parallel lines also maintain a constant separation between points closest to each other on the two lines in addition to never intersecting one another.
Symbol of parallel lines: The symbol representing parallel lines is ‘||.’
Note: Parallel lines are not always equal in length, but they are always at an equal distance from each other.
Parallel lines Examples from Real Life
If you observe your surroundings, you will encounter several parallel lines. Some common examples that you must watch next time are:
- The railway tracks run parallel to each other. They form two parallel lines or tracks for the wheels of the train to travel along on.
- Edges of a ruler are parallel to each other.
- Cricket stumps are parallel.
- Lines on a ruled paper are parallel.
- Zebra crossing has parallel white lines.
- The opposite boundaries of an eraser are parallel.
- Straight ceilings are parallel to floors.
- The queues in a bank are parallel to one other.
- The opposite sides of a rectangular or square-shaped object are parallel to each other.
Parallel and Perpendicular Lines
Parallel lines are different from perpendicular lines. The following table enumerates the key differences between parallel and perpendicular lines.
|Parallel Lines||Perpendicular Lines|
|The lines that never intersect and are equidistant are parallel lines.||The lines which make right angles at the point of intersection are perpendicular.|
|The slope of parallel lines is always equal.||The slopes of perpendicular lines are negative reciprocals of each other.|
|If two lines are parallel, the angle between them will be zero.||If two lines are perpendicular to each other, the angle between them will be 90.|
|The symbol for parallel lines is ||.||The symbol for perpendicular lines is .|
|Example: The two railings of a staircase are parallel to each other.||Example: The corner of two walls is perpendicular.|
Note: Although parallel lines have the same slope, they do not have the same equation. For example, if the equation of a line is, y = 3x + 2, this means the slope of this line is 3. So, if there is another straight line in the same plane with a slope of 3 will be parallel to the given line.
Two perpendicular lines are always intersecting lines. However, two intersecting lines are not always perpendicular to each other.
Parallel Lines Cut by a Transversal
In geometry, a transversal is a line that intersects two or more lines. When a transversal cuts two parallel lines, several angles are formed. The angles are related as follows:
Exterior angles – The angles that are situated outside of the line. Here, ∠1, ∠2, ∠7, and ∠8 are exterior angles.
Interior angles – The angles located inside of the lines are. Here ∠3, ∠4, ∠6, and ∠5 are interior angles.
Corresponding angles– The angles that occupy the same relative position at each intersection. Here,
- ∠1 and ∠5
- ∠2 and ∠6
- ∠4 and ∠8
- ∠3 and ∠7 are corresponding pairs of angles.
Alternate Interior angles- These angles are formed on the inside of two parallel lines when a transversal intersects them. The pairs of alternate interior angles are
- ∠3 and ∠5
- ∠4 and ∠6
Exterior Alternate angles– These angles are formed on either side of the transversal outside the two parallel. The pairs of alternate exterior angles are
- ∠1 and ∠7
- ∠2 and ∠8
Co-interior angles– These are also called interior angles on the same side or consecutive interior angles, or allied angles. Here, the co-interior angles are
- ∠4 and ∠5
- ∠3 and ∠6
Vertically opposite angles: The pairs of vertically opposite angles are
- ∠1, and ∠3
- ∠2 and ∠4
- ∠5 and ∠7
- ∠6 and ∠8
Properties of Parallel Lines
The basic properties of parallel lines are as follows:
- They do not intersect with each other.
- Parallel lines do not meet, or they meet at infinity.
- They are always equidistant from each other.
How to Test If Two Lines are Parallel?
As shown above, when a transversal cuts two parallel lines at two distinct points, four angles are formed at a single point of intersection. In total, there are eight angles around two points of intersection.
The properties of parallel lines concerning transversals can be used for testing parallel lines. So, the two given lines will be parallel to each other if
- The corresponding angles are always equal to each other.
- Vertically opposite angles are equal.
- Alternate interior angles are equal.
- Alternate exterior angles are equal.
- Pair of co-interior angles on the same side of the transversal are supplementary, i.e., their sum is equal to 180 degrees.
A triangle has no parallel lines, while a hexagon has three pairs of parallel lines. Think about other shapes and write down their number of parallel lines.
Axioms and Theorems
There are several axioms and theorems for the parallel lines cut by a transversal. These theorems relate parallel lines and angles. They help determine whether a given pair of lines are parallel or not.
Corresponding Angle Axiom
If a transversal intersects two parallel lines, then the pair of corresponding angles are equal to each other.
From the above figure, the equal corresponding angle pairs are
- ∠1 = ∠5
- ∠2 = ∠6
- ∠4 = ∠8
- ∠3 = ∠7.
The converse of this hypothesis is also true. Therefore, if a pair of corresponding angles are equal, the lines will be parallel to each other.
If a transversal intersects two parallel lines, then the pair of alternate interior angles are equal.
From the above figure, the following pairs of alternate interior angles are equal:
∠3 = ∠5 and ∠4 = ∠6.
We can also prove this theorem as ∠4=∠2 and ∠1=∠3 (Vertically opposite angles).
Also, ∠1=∠5 and ∠2=∠6 (Corresponding Angles)
⇒∠3 = ∠5 and ∠4 = ∠6.
The converse is also true for the above theorem. So, if the pair of alternate interior angles are equal, then lines are parallel to one another.
If a transversal intersects two parallel lines, then the co-interior angles are supplementary to one another.
Thus, we can write as follows:
∠4 + ∠5 = 180°
∠3 + ∠6 = 180°
We can also prove this theorem as,
∠3=∠5 and ∠4=∠6 (Alternate interior angles)
∠3+ ∠4=180° and ∠5+∠6=180° (Linear pairs)
⇒∠4+ ∠5=180° and ∠3+∠6=180°
The converse is also true for the above theorem. So, if the pair of co-interior angles are supplementary, the lines are parallel.
Lines parallel to the same line are also parallel to each other.
Understanding Parallel Lines with Solved Examples
Example 1: In the following figure, find all other angles if ∠5 = 70 degrees.
Solution: Given ∠5 = 70°
So, ∠5 = ∠1 = 70° (corresponding angles)
∠1 = ∠4 = 70° ( vertically opposite angles)
∠5 = ∠7 = 70° ( vertically opposite angles)
Now, ∠1+ ∠2 = 180° (linear pair)
So, 70° + ∠ 2 = 180°
∠ 2 = 180° – 70°
∠ 2 = 110°
Now that we have found the value of ∠2, we find the remaining angles as follows
∠2 = ∠6 =110° (corresponding angles)
∠2 = ∠3 = 110° ( vertically opposite angles)
∠6 = ∠8 = 110° (vertically opposite angles)
Example 2: If AB is parallel to CD, find the value of x in the following figure if the value of y = 130°.
Solution: Given that AB is parallel to CD and y =130°
So, ∠ 4x+10 = y = 130° (Exterior alternate angles)
4x+10 = 130°
4x = 130 – 10
4x = 120
x = 30°