Parallel lines are straight lines that are equidistant from each other and have the same slope. So, when two or more lines do not meet at any point, we call them parallel.
Just observe the section-wise lines of each class during the morning assembly, you notice that these lines are at some distance from each other and they don’t meet at any point. This arrangement shows that lines are parallel, also, these lines meet at nowhere or infinity.
So, what other characteristics do parallel lines hold? Also, how do we categorise the properties of lines with the help of transversal lines, we will understand all about it in detail.
Parallel Lines and Transversal Lines
When a line intersects any two parallel lines, we call it a transversal. In this arrangement, many pairs of angles are formed. Among these, some angles are congruent (equal) while others are supplementary.
Now, let us observe the following figure to see the parallel lines labelled as l1 and l2 that intersect by a transversal:
Parallel Lines
In this figure, eight separate angles have been formed by the two parallel lines and a transversal. Each angle has been labelled using numbers.
Corresponding Angles:
In the given figure, there are four pairs of corresponding angles, that is, ∠1 = ∠5, ∠2 = ∠6, ∠4 = ∠8, and ∠3 = ∠7.
Alternate Interior Angles:
Angles that formed on the inside of two parallel lines that are intersected by a transversal are alternate interior angles. They are equal in measure. In this figure, ∠4 = ∠6 and ∠3 = ∠5.
Alternate Exterior Angles:
Alternate exterior angles are formed on either side of the transversal and they are equal in measure. In this figure, ∠1 = ∠7 and ∠2 = ∠8.
Consecutive Interior Angles:
Consecutive interior angles or co-interior angles are formed on the inside of the transversal and they are supplementary. Here, ∠3 + ∠6 = 180° and ∠4 + ∠5 = 180°.
Vertically Opposite Angles:
Angles formed when two straight lines intersect each other and they are equal in measure, we call them vertically opposite angles. Here, ∠1 = ∠3, ∠2 = ∠4, ∠5 = ∠7, and ∠6= ∠8.
Parallel lines Equation
The equation of a straight line is written in the slope-intercept form represented by the following equation:
y = mx + b
Here, 'm' is the slope and 'b' is the y-intercept.
Please note that the value of 'm' determines the slope or gradient and tells us how steep the line is. Also, it should be noted that the slope of any two parallel lines is always the same.
Assume that the slope of a line with the equation y = 4x + 3 is 4. Thus, any line that is parallel to y = 4x + 3 will also have the same slope, that is, 4.
We must remember that parallel lines have different y-intercepts and have no points in common.
Which Lines Appear to be Parallel?
Apart from the characteristics of parallel lines given above, when any two parallel lines are intersected by a transversal, we can understand the lines that appear to be parallel by the following properties:
Two lines are parallel when the corresponding angles so formed are equal.
Two lines are parallel when the alternate interior angles so formed are equal.
Two lines are parallel when the alternate exterior angles so formed are equal.
Two lines are parallel when the consecutive interior angles on the same side of the transversal are supplementary.
From the above text, we understand that parallel lines are equidistant and meet at infinity. Also, all the lines parallel to them will have the same slope, i.e., m.
FAQs on Characteristics of Parallel Lines
1. What are the fundamental characteristics of parallel lines in geometry?
Parallel lines are two or more lines on the same plane that never intersect, no matter how far they are extended in either direction. The two primary characteristics are:
They are always equidistant, meaning the perpendicular distance between them is constant at all points.
They are coplanar, which means they lie on the same two-dimensional surface.
2. What are some real-world examples that illustrate the concept of parallel lines?
You can find examples of parallel lines all around you. Some common examples include:
The two rails of a railway track.
The opposite edges of a ruler, door, or window.
The lines on a ruled notebook.
The stripes in a zebra crossing.
The shelves in a bookshelf.
3. What happens when a transversal line cuts across two parallel lines?
When a transversal intersects two parallel lines, it creates eight angles. These angles have special relationships:
Corresponding angles are equal.
Alternate interior angles are equal.
Alternate exterior angles are equal.
Consecutive interior angles (or allied angles) are supplementary, meaning they add up to 180 degrees.
4. Why is it important to understand the angle relationships created by a transversal?
Understanding these angle relationships is crucial for solving geometric problems. It forms the basis for proving many theorems in geometry. This principle is applied in practical fields like architecture to ensure walls are parallel, in engineering for structural integrity, and in cartography (map-making) to work with lines of latitude.
5. How are parallel lines represented using symbols in mathematics?
In mathematics, parallel lines are denoted by the symbol ||. For example, if line AB is parallel to line CD, it is written as AB || CD. This notation provides a clear and concise way to state the relationship between the two lines.
6. What is the difference between parallel lines and intersecting lines?
The key difference lies in their point of intersection. Parallel lines are coplanar and will never meet, maintaining a constant distance apart. In contrast, intersecting lines are also coplanar but cross each other at exactly one point. A special case of intersecting lines is perpendicular lines, which intersect at a 90-degree angle.
7. How can you determine if two lines are parallel using coordinate geometry?
In coordinate geometry, the defining characteristic of parallel lines is their slope. Two non-vertical lines are parallel if and only if they have the exact same slope. For example, the lines y = 2x + 3 and y = 2x - 5 are parallel because the slope (the coefficient of x) is 2 for both lines.
8. Can curved lines be parallel to each other?
Yes, the concept of being parallel can be extended to curves. Two curves are considered parallel if the perpendicular distance between them remains constant along their entire length. A common example is the concentric circles, which are curved lines that are all parallel to each other as they share the same centre and are always equidistant.
9. What is the distinction between parallel lines and skew lines?
This is a common point of confusion in three-dimensional geometry. While both parallel lines and skew lines never intersect, the key difference is the plane they lie on. Parallel lines are coplanar (on the same plane). Skew lines, however, are non-coplanar (on different planes) and are not parallel. Imagine the edge of a ceiling and an edge of a far wall; they will never meet but are not parallel.
