An Introduction to Parallel Lines
No matter how far in either direction they may be extended, parallel lines are ones that are equally spaced apart from one another and never cross. For instance, parallel lines are represented by a rectangle's opposite sides. In terms of geometry, parallel lines are two separate lines that never cross each other and are located in the same plane. They may be vertical or horizontal. In this article, we'll talk about lines that are parallel to one another and the corresponding theorem, along with lots of examples that have been resolved.
Parallel Lines
Parallel lines are two straight lines that do not intersect on the same plane. Parallel lines AB and CD are shown in figure (a). They are identified as AB and CD, and we state that AB is parallel to CD. PQ and RS are non-parallel lines because they intersect at point T in figure (b), where produced lines RS and PQ meet.
Parallel Lines
Equal numbers of arrowheads pointing in the same direction are used to denote parallel lines. As seen in the illustration below, several parallel sets of lines are identified by various numbers of arrowheads.
Parallel Lines with Similar Arrows
Construction of a Line Through a Point Outside a Given Line Parallel to It
Create the line XY and mark some point P on it. Use a ruler, and a pair of compasses, and just by using point P, draw a line AB that is parallel to XY.
Method:
Step 1. Mark a point P outside the given line XY and draw the line.
Step 2. Join points Q and P at another point Q on the XY axis.
Construction of Parallel Lines
Step 3. Consider any radius and Q as the centre, draw an arc that crosses XY at R and QP at S. Draw a seond arc considering P as the centre, and cut PQ at T, maintaining the same radius as before.
Step 4. Mark the point U on the aforementioned arc so that arc TU=arc SR, using T as the centre. Bring PU together and create it from both ends. Abbreviate this line as AB.
Parallel Lines
As a result, line AB is the line that is parallel to the specified line XY and passes through point P.
Transversal Lines
The term "transversal of those two lines" refers to a line that crosses two other lines. Eight angles are created at the two crossings in the image below, where TS is the transversal. Depending on how they relate to one another, these angles are named in pairs.
Transversal Lines
Properties of Transversal Lines
If a transversal cuts two parallel lines, then
The corresponding angles are equal.
The alternative angles are equal.
The transversal's internal angles on the same side add up to $180^\circ $.
Theorem - Lines Parallel to the Same Line
If we have been given two lines and both these lines are parallel to a third line, then according to this theorem, the two given lines will also be parallel to each other. Now, let's use the below figure to test how to prove two lines are parallel.
Lines Parallel to the Same Line
We can infer from the given figure that line m and line n are parallel to line l. Lines m $\parallel $ line l and line n $\parallel $ line l are examples. Additionally, the transversal for the lines l, m, and n is "t."
We can then state that $\angle 1$ Equals $\angle 2$ and $\angle 1 = \angle 3$. Using the axioms of related angles.
Therefore, we can also state that $\angle 2$ and $\angle 3$ correspond to one another and are equal to one another.
Thus, $\angle 2 = \angle 3$ .
Line m is parallel to line n, according to the converse of the corresponding angle axioms.
That is, Line m $\parallel $ Line n.
Note that this feature can be applied to several lines as well.
Solved Examples
Q.1. Indicate which of the following pairs of lines or rays appears to be parallel or intersecting.
Parallel Lines and Transversal Lines
Solution: Two lines are intersecting if, when extended on either side, they cross or appear to cross; otherwise, they are parallel. Consequently,
Intersecting lines: The first pair of lines are intersecting lines because if both lines are extended then, they will surely meet at some point.
Parallel lines: The second pair is parallel lines as they will never cross each other even on extending on either of the sides.
Intersecting Lines: The third pair is also a pair of intersecting lines if we produce these lines they will intersect each other at some point for sure.
Q.2. Write down all pairs of parallel lines from the figure below.
Parallel and Transversal Lines
Solution: Regardless of how far apart they are in either direction, two straight lines that are located in the same plane are considered to be parallel if they do not cross or overlap.
$EF\parallel GH$, $EF\parallel IJ$, and $GH\parallel IJ$ are the parallel lines as a result.
Q.3. In the diagram below, $AB\parallel CD$, $CD\parallel EF$, and $EA \bot AB$. Find $x$, $y$ and $z$ values if $\angle BEF = 55^\circ $ .
Parallel Lines and Transversal Lines
Solution: As a result of $CD\parallel EF$, $\angle y + 55^\circ = 180^\circ $ (sum of co-interior angles is supplementary).
\[\angle y = 180^\circ -55^\circ = 125^\circ \]
As a result of \[AB\parallel CD\] , $x$ equals $y$. (Corresponding angles are equal)
Thus, \[\angle x = 125^\circ \] .
Therefore, $AB\parallel EF$ because \[AB\parallel CD\] and $CD\parallel EF$ .
Therefore, \[\angle EAB + \angle FEA = 180^\circ \] (Sum of co-interior angles are supplementary)
Thus, \[90^\circ + \angle z + 55^\circ = 180^\circ \] , \[\angle z + 145^\circ = 180^\circ \] . Hence, $\angle z = 35^\circ $ .
As a result, the values of \[x = 125^\circ \] , \[y = 125^\circ \] and $\angle z = 35^\circ $ .
Practice Questions
Q1. In the following figure, if \[y = 130^\circ \] and AB line parallel to CD, calculate the value of $x$ .
Parallel Lines AB and CD
Q2. Calculate angle ABC if PQ $\parallel $ RS.
Parallel Lines PQ and RS
Answers
$30^\circ $
$110^\circ $
Summary
In this article, we first discussed parallel lines and how they are made, then transversals and the angle that the transversal creates when it cuts parallel lines. In addition, we studied the characteristics of parallel lines when crossed by a transversal. The proof regarding the parallel lines to the same line was then given. In addition, we worked through various instances to further our understanding of the idea.
FAQs on Lines Parallel to the Same Line
1. What is the main principle of 'lines parallel to the same line' in geometry?
The main principle is a fundamental theorem which states that two lines that are parallel to the same third line are also parallel to each other. For instance, if line 'a' is parallel to line 'b', and line 'c' is also parallel to line 'b', then it logically follows that line 'a' must be parallel to line 'c'. This is often referred to as the transitive property of parallel lines.
2. How do you prove that lines parallel to the same line are parallel to each other?
To prove this theorem, you can use a transversal line that intersects all three lines. Here is a simple step-by-step explanation:
- Let lines l, m, and n be three lines such that l || n and m || n.
- Draw a transversal line 't' that intersects l, m, and n at points P, Q, and R respectively.
- Since l || n, the corresponding angles are equal (let's say ∠1 = ∠3).
- Similarly, since m || n, their corresponding angles are also equal (∠2 = ∠3).
- From these two steps, we can see that ∠1 = ∠3 and ∠2 = ∠3, which means ∠1 = ∠2.
- Since ∠1 and ∠2 are corresponding angles for lines l and m with transversal t, and they are equal, we can conclude that line l is parallel to line m.
3. What are the key conditions required to verify if two lines are parallel?
To verify that two lines are parallel when intersected by a transversal, you must check if any one of the following conditions is met:
- The pair of corresponding angles are equal.
- The pair of alternate interior angles are equal.
- The pair of alternate exterior angles are equal.
- The sum of consecutive interior angles (co-interior angles) on the same side of the transversal is 180 degrees (supplementary).
If any of these conditions hold true, the lines are parallel.
4. Why is the theorem about lines parallel to the same line important in geometry?
This theorem is crucial because it allows us to establish a relationship between multiple lines without directly comparing each pair. Instead of proving that line 'a' is parallel to 'c' separately, we can use a common line 'b' as a reference. This simplifies proofs involving complex geometric figures and is a foundational concept for constructing parallel lines and proving properties of quadrilaterals like parallelograms, rhombuses, and rectangles.
5. Can you give a real-world example of lines parallel to the same line?
A perfect real-world example is the shelves on a bookshelf. Each shelf is installed to be parallel to the floor (the same line/plane). Because each shelf is parallel to the floor, all the shelves are automatically parallel to each other. Another example is the railway tracks; two separate tracks running parallel to a central reference line will also be parallel to each other.
6. What is the difference between parallel lines and coincident lines?
This is a key distinction. Parallel lines are two distinct lines in a plane that never intersect, maintaining a constant distance from each other. In contrast, coincident lines are lines that lie directly on top of each other, sharing all their points. Essentially, they are the same line. So, while parallel lines have zero points in common, coincident lines have infinite points in common.
7. Can the property of lines being parallel to the same line extend to more than three lines?
Yes, absolutely. The property is infinitely extendable. If you have a set of lines (l1, l2, l3, l4, ...) and all of them are parallel to a single reference line (say, line 'm'), then all the lines in the set will be parallel to each other. This is a direct application of the transitive property of parallelism.
8. What is a transversal in the context of parallel lines?
A transversal is a line that intersects two or more other lines in the same plane at distinct points. It is the transversal that creates the various angles (like corresponding, alternate interior, etc.) that we use to prove whether the lines it crosses are parallel or not. It is a critical tool for studying the properties of parallel lines.
