Lines Parallel to the Same Line: Definition, Construction, Facts

Lines Parallel to the Same Line: Parallel lines can be found in numerous places, such as railroad tracks, street markings, and sidewalk margins, to name a few. Did you notice these lines never meet? We call such lines parallel lines.

Two straight lines, lying in the same plane, are said to be parallel if they do not meet or intersect, no matter how much they are extended to in either direction. So, two straight lines are parallel if they lie in the same plane and do not intersect even when produced to any extent on either side. This article will cover and learn everything about the line parallel to the same line, their properties, how to construct parallel lines, and so much more.

Parallel Lines

When two straight lines on the same plane do not intersect, they are called parallel lines. In figure (a), \(AB\) and \(C\) are parallel lines. We denote them by \(AB\parallel CD\) and say \(AB\) is parallel to \(CD\). In figure (b), \(PQ\) and \(RS\) are non-parallel lines as \(RS\) produced and \(PQ\) produced meet at point \(T\).

Parallel lines are indicated with equal numbers of arrowheads pointing in the same direction on each line. A different set of parallel lines are marked with a different number of arrowheads, as shown below.

Parallel Lines

Construction of Parallel Lines

Parallel lines can be constructed using a ruler and a set square, as shown below.

Step 1: Place the set square on the line \(AB\) and against a ruler.

Step 2: Slide the set square as shown and draw the line \(PQ\) parallel to \(AB\).

? Learn About Parallel and Perpendicular Lines

Construction of a Line Parallel to a Given Line Through a Point Outside It

Draw a line \(XY\). Mark a pint \(P\) outside it. Through \(P\), draw a line \(AB\) parallel to \(XY\) using a ruler and a pair of compasses only.

Steps of Construction:
1. Draw a given line \(XY\) and mark a point \(P\) outside it.

2. Take another point \(Q\) on \(XY\) and join \(Q\) and \(P\).

3. With \(Q\) as the centre and a convenient radius, draw an arc intersecting \(XY\) at \(R\) and \(QP\) at \(S\). Keeping the radius the same, draw another arc with centre \(P\), cutting \(PQ\) at \(P\).

4. With \(T\) as the centre, Mark the point \(U\) on the above arc such that arc \(TU=\) arc \(SR\). Join \(PU\) and produce it in both directions. Name this line as \(AB\).

Thus, \(AB\) is the line parallel to the given line \(XY\), passing through the point \(P\).

Transversal Lines Definition

When a line intersects two other lines, the line is called a transversal of those two lines. In the below-given figure, \(TS\) is the transversal, and there are \(8\) angles formed at the two intersections. These angles are named in pairs according to their relative positions.

Transversal Lines Definition

Properties

When a transversal cuts two parallel lines, then
1. The corresponding angles are equal
2. The alternate angles are equal
3. The sum of the interior angles on the same side of the transversal is \({180^ \circ }\)

In the given diagram, if \(AB\parallel CD\) and \(TS\) is a transversal, then,

Properties of Parallel Lines Cut by a Transversal

1. \(\angle a = \angle p,\angle b = \angle q,\angle s = \angle d\) and \(\angle r = \angle c\) (Corresponding angles are equal)
2. \(\angle a = \angle r,\angle b = \angle s,\angle d = \angle q\) and \(\angle c = \angle p\) (Alternate angles are equal)
3. \(\angle a = \angle s = {180^ \circ }\) and \(\angle b = \angle r = {180^ \circ }\) (Sum of co-interior angles is equal to \({180^ \circ }\))

Lines Parallel to the Same Line

If two lines are parallel to the same line, will they be parallel to each other?? Let us see whether they will be parallel or not.

In the below-given figure, line \(m\parallel l\) and line \(n\parallel l\). Now, let us draw a transversal line \(t\), for the lines \(l,m\) and \(n\).

Lines Parallel to the Same Line

Since, line \(n\parallel \) line \(l\), therefore,

\(\angle 1 = \angle 2\) (Corresponding angles)… \((i)\)

Similarly,

line \(m\parallel \) line \(l\), therefore,

\(\angle 3 = \angle 1\) ( Corresponding angles)… \((ii)\)

From equations \((i)\) and \((ii)\), we get,

\(\angle 1 = \angle 2 = \angle 3\)

Hence, \(\angle 2 = \angle 3\)

Since they are corresponding angles and are equal thus, we can say that line \(n\parallel \) line \(m\).

Hence, we can say that if two lines are parallel to the same line, they will also be parallel to each other.

If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel.

In the below-given figure, a transversal \(AB\) intersects two lines \(PQ\)and \(RS\) at point \(B\) and \(C\) respectively.

Ray \(BE\) is the bisector of \(\angle ABQ\) and ray \(CG\) is the bisector of \(\angle BCS\) and \(BE\parallel CG\).

We have to prove \(PQ\parallel RS\).

It is given that ray \(BE\) is the bisector of  \(\angle ABQ\).

Therefore, \(\angle ABE = \frac{1}{2}\angle ABQ\)… \((i)\)

Similarly, ray \(CG\) is the bisector of \(\angle BCS\)

Therefore, \(\angle BCG = \frac{1}{2}\angle BCS\)… \((ii)\)

But \(BE\parallel CG\) and \(AD\) is the transversal.

Therefore, \(\angle ABE = \angle BCG\)…\((iii)\) (Corresponding angles are equal)

Now, substituting the values of equations \((i)\) and \((ii)\) in \((iii)\), we get,

\(\frac{1}{2}\angle ABQ = \frac{1}{2}\angle BCS\)

Or, \(\angle ABQ = \angle BCS\)

But they are corresponding angles formed by transversal \(AD\) with \(PQ\) and \(RS\) and are equal.

Therefore, \(PQ\parallel RS\).

Solved Examples

Q.1. State whether the following pairs of lines or rays appear to be parallel or intersecting.

Ans: If on extending the two lines on either side, they intersect or appear to intersect, they are intersecting lines; else, they are parallel lines. Thus,
a) Intersecting lines
b) Parallel lines
c) Intersecting lines

Q.2. From the below-given figure, write all pairs of parallel lines.

Ans: Two straight lines, lying in the same plane, are said to be parallel if they do not meet or intersect, no matter how much they are extended in either direction.
Thus, the pair of parallel lines is \(EF\parallel GH, EF\parallel IJ\), and \(GH\parallel IJ\).

Q.3. In the below-given figure, \(AB\parallel CD,\,CD\parallel EF\), and \(EA \bot AB\). If \(\angle BEF = {55^ \circ }\), find the value of \(x,y\) and \(z\).

Ans: Since, \(CD\parallel EF\), therefore,
\(\angle y + {55^ \circ } = {180^ \circ }\) (sum of co-interior angles is supplementary)
\(\angle y = {180^ \circ } – {55^ \circ } = {125^ \circ }\)
Now, \(AB\parallel CD\), therefore,
\(\angle x = \angle y\) (Corresponding angles are equal)
Thus, \(\angle x = {125^ \circ }\)
Now, since \(AB\parallel CD\) and \(CD\parallel EF\), therefore, \(AB\parallel EF\).
Hence, \(\angle EAB + \angle FEA = {180^ \circ }\) (Sum of co-interior angles are supplementary)
Therefore, \({90^ \circ } + z + {55^ \circ } = {180^ \circ }\)
\( \Rightarrow z + {145^ \circ } = {180^ \circ }\)
\( \Rightarrow z + {180^ \circ } = {145^ \circ }\)
\( \Rightarrow z = {35^ \circ }\)
Hence, the value of \(x = {125^ \circ },y = {125^ \circ }\) and \(z = {35^ \circ }\)

Q.4. In the below-given figure, lines \(l,m\), and \(n\) are parallel intersected by a transversal \(p\) at \(x,y\) and \(z\). Find \(\angle 1,\angle 2\) and \(\angle 3\).

Ans: Since the parallel lines \(l\) and \(n\) are intersected by the transversal \(p\) at \(X\) and \(Z\), respectively.
Therefore,
\(\angle 2 = \angle 4\) (Corresponding angles are equal)
\(\angle 4 + {50^ \circ } = {180^ \circ }\) (Linear pair angles)
Thus,
\( \Rightarrow \angle 2 = {180^ \circ } – {50^ \circ }\)
\( \Rightarrow \angle 2 = {130^ \circ }\)
Now, since \(\angle 2\) and \(\angle 3\) are vertically opposite angles, therefore,
\(\angle 3 = \angle 2 = {130^ \circ }\)
Again, parallel lines \(m\) and \(n\) are intersected by the transversal \(p\) at \(Y\) and \(Z\), respectively.
Therefore,
\(\angle 1 = \angle 3\) (Alternate interior angles are equal)
\( \Rightarrow \angle 1 = {130^ \circ }\)
Thus, the measure of \(\angle 1,\,\angle 2\) and \(\angle 3\) is equal to \({180^ \circ }\).

Q.5. In the below-given figure, \(\angle MXQ = {135^ \circ }\) and \(\angle MYR = {40^ \circ }\), find \(\angle XMY\).

Ans: First, we must draw a line \(AB\) parallel to line \(PQ\), as shown below. Now, \(AB\parallel PQ\) and \(PQ\parallel RS\).

Therefore, \(AB\parallel RS\)
Now, \(AB\parallel PQ\) therefore, the sum of co-interior angles will be supplementary.
Thus, \(\angle QXM + \angle XMB = {180^ \circ }\)
\(\angle QXM = {135^ \circ }\) (Given)
So, \({135^ \circ } + \angle XMB = {180^ \circ }\)
\(\angle XMB = {180^ \circ } – {135^ \circ } = {45^ \circ }\)… \((i)\)
Again, \(AB\parallel RS\),
\(\angle BMY = \angle MYR\) (When two lines are parallel, then alternate interior angles are equal)[\angle BMY = \angle MYR] (When two lines are parallel, then alternate interior angles are equal)
Thus, \(\angle BMY = {40^ \circ }\)… \((ii)\)
Adding equations \((i)\) and \((ii)\), we get,
\(\angle XMB + \angle BMY = {45^ \circ } + {40^ \circ }\)
\(\angle XMY = {85^ \circ }\)
Hence, the value of \(\angle XMY\) is equal to \({85^ \circ }\).

Q.6. In the below-given figure, \(\angle 1 = {60^ \circ }\) and \(\angle 3 = \frac{2}{3}\) of a right angle. Prove that \(l\parallel m\).

Ans: Given, \(\angle 1 = {60^ \circ }\) and \(\angle 3 = \frac{2}{3}\) of a right angle
Therefore, \(\angle 3 = \frac{2}{3} \times {90^ \circ } = {60^ \circ }\)
Now, \(\angle 3 = \angle 2\) (Vertically opposite angles are equal)
Thus, \(\angle 2 = {60^ \circ }\)
Now, \(\angle 1 = \angle 2 = {60^ \circ }\) 
\(\angle 1\) and \(\angle 2\) are corresponding angles formed by the transversal \(n\) with lines \(l\) and \(m\) and are equal to \({60^ \circ }\).
Thus, we can say that \(l\parallel m\).

Summary

In this article, we first learnt about the parallel lines, how they are constructed, and then we learnt about the transversal and the angle formed by the transversal when it cuts the parallel lines. We also learnt the properties of parallel lines when cut by a transversal. Then we leant the proof about the lines parallel to the same line. In addition to this, we solved some examples to make our grip stronger over the concept.

FAQs

Q.1. What are parallel lines?
Ans: Two straight lines, lying on the same plane, are said to be parallel if they do not meet or intersect, no matter how much they are extended in either direction.

Q.2. What happens if two lines are parallel to the same line?
Ans: Lines that are parallel to the same line are also parallel to each other.

Q.3. Can we say a line is parallel to itself?
Ans: No, a line can not be parallel to itself.

Q.4. Which lines are parallel to each other?
Ans: The lines that lie on the same plane and do not meet or intersect are the lines that are parallel to each other.

Q.5. How do you prove that the given lines are parallel?
Ans: To prove that the given lines are parallel, then we have to prove either corresponding angles are equal or alternate angles are equal, or co-interior angles are supplementary.

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