Construction of a Line Parallel to a Given Line: Parallel lines are considered two or more lines that lie in the same plane but never intersect or meet. In order to construct a parallel line to a given line that goes through a given point, we need to know few facts about parallel lines and copying angles. To do this, as a first step we will need to construct a transversal, creating several pairs of equal angles. A transversal is a line that crosses two parallel lines.

This article includes details about parallel lines and the steps for the construction of a line parallel to a given line.

In order to construct a parallel line to a given line that goes through a given point, we need to know few facts about parallel lines and copying angles.

Parallel Lines

Parallel lines are defined as two lines at an equal distance from each other and never meet.

Two or more lines that are the same distance apart and never join are called parallel lines.

Parallel lines are those that, no matter how far they are stretched, never meet. Parallel lines are given the symbol \(||.\)
In the figure, lines \(p\) and \(q\) are parallel lines.
Parallel lines \(p\) and \(q\) are symbolically written as \(p||q.\)
The symbol ∦ denotes non-parallel lines.

Rules for Parallel Lines

Some rules for parallel lines are listed below.

1. Parallel lines never meet or intersect at any point.
2. On a coordinate plane, parallel lines have the same slope.
3. Parallel lines don’t have to be perfectly straight all of the time. Curved lines can also be parallel.
4. When a transversal cuts two or more parallel lines, the angles that correspond to each other are the same.
5. When a transversal cuts two or more parallel lines, the vertically opposite angles are equal.
6. When a transversal cuts two or more parallel lines, the interior angles are equal on both sides.
7. When a transversal cuts two or more parallel lines, the exterior angles are equal on both sides.
8. When a transversal cuts two or more parallel lines, the pair of interior angles on the same side of the transversal is supplementary.

Construction of Parallel Line to a Given Line

To construct parallel lines, we require a ruler and a compass.

The following steps are followed to construct a parallel line to a given line,

Step-1: Let us consider the given line as \(m.\)
Step-2: Mark a point \(X\) on a line \(m,\) as shown in the figure.
Step-3: Choose the point \(Y,\) away from the line \(m.\) And join the points \(X\) and \(Y\) as shown in the figure.
Step-4: Consider \(X\) as the centre, and any suitable radius, draw an arc cutting the line segments \(XY\) at a point \(V\) and the line \(m\) at a point \(Z,\) respectively.
Step-5: With \(Y\) as the centre and any suitable radius, draw an arc cutting the line segments \(XY\) at a point \(U\) as shown below.
Step-6: With \(U\) a centre and the same radius, cut an arc \(PQ\) at a point \(A.\) And draw a line passing through the points \(A\) and \(Y.\)
Then, name the obtained line as \(n\) so that the line \(m\) is parallel to line \(n,\) i.e., \(m||n.\)

Slope of Two Parallel Lines

The slope of two parallel lines is equal.
If \({m_1}\) and \({m_2}\) are the slopes of two parallel lines, then \({m_1} = {m_2}.\)

If two lines have the same slope, they are said to be parallel.

Parallel Lines Equations

If \(ax + by + c = 0\) gives the equation of any line, then the equation of a line parallel to the above line is given by \(ax + by + d = 0,\) where \(d\) is a constant.

Solved Examples: Construction of a Line Parallel to a Given Line

Q.1. Find the slope of the line parallel to the line \(3x – 5y = 12.\)
Ans: We know that to find the line’s slope \(3x – 5y = 12,\) we need to convert the line into slope-intercept form \(y = mx + b.\)
The slope-intercept form of \(3x – 5y = 12\) is
\( – 5y = 12 – 3x\)
\( \Rightarrow y = \frac{{12 – 3x}}{{ – 5}}\)
\( \Rightarrow y = \frac{{ – 12}}{5} + \frac{{3x}}{5}\)
\( \Rightarrow y = \frac{3}{5}x + \frac{{ – 12}}{5}\)
The slope of the line \(3x – 5y = 12\) is \(m = \frac{3}{5}.\)
Therefore, the slope of every line parallel to line \(3x – 5y = 12\) will be \(m = \frac{3}{5}.\)

Q.2. Find the equation of a line that is parallel to \(y = \,- 3x + 5\) and passing through the point \(\left({1, – 6} \right).\)
Ans: The slope of the line \(y = \,- 3x + 5,\) is \(m = \,- 3.\) The line parallel to \(y = \, – 3x + 5\) also has a slope equal to \( – 3\) because they need to have equal slopes to be parallel.
We will use the point \(\left({1, – 6} \right)\) where this unknown line must pass through.
Substitute these values into the point-slope form and solve for \(y.\)
\(y – {y_1} = m\left({x – {x_1}} \right)\)
\( \Rightarrow y – \left({ – 6} \right) = \,- 3\left({x – 1} \right)\)
\( \Rightarrow y + 6 = \,- 3\left({x – 1} \right)\)
\( \Rightarrow y + 6 = \, – 3x + 3\)
\( \Rightarrow y = \,- 3x – 3\)
Therefore, the equation of a line that is parallel to \(y = \ – 3x + 5\) is \(y =\, – 3x – 3.\)

Q.3. Check whether the lines are parallel or perpendicular.
\(6y + 1 = 6x\) and \(x + 5 = y\)
Ans: Writing each equation in the slope-intercept form:
\(6y + 1 = 6x \Rightarrow 6y = 6x – 1 \Rightarrow y = x – \left({\frac{1}{6}}\right) \Rightarrow {\text{slope}} = 1\)
\(x + 5 = y \Rightarrow y = x + 5 \Rightarrow {\text{slope}} = 1\)
The slopes are the same.
Therefore, the lines are parallel.

Q.4. In the given figure, \(a||b\) and l is a transversal. Find the values of x and y.

Ans: From the given figure, \(6x + y\) and \(x + 5y\) are corresponding angles.
\(6x + y = x + 5y\)
\(6x – x = 5y – y\)
\(5x = 4y\)
\(x = \frac{{4y}}{5}\)
Now, \(4x\) and \(6x + y\) are linear pair of angles, so,
\(4x + 6x + y = {180^ \circ }\)
\(10x + y = {180^ \circ }\)
\(\frac{{40y}}{5} + y = {180^ \circ }\)
\(\frac{{45y}}{5} = {180^ \circ }\)
\(45y = 180 \times 5 = {900^ \circ }\)
\(y = 20\)
\(x = \frac{{\left({4 \times 20} \right)}}{5} = 16\)
Therefore, the obtained values are \(x = 16\) and \(y = 20.\)

Q.5. If \(\angle A = 3x\) and \(\angle B = 2x\) and the two angles are interior angles on the same side of the transversal, then find the value of \(x\) and measures of each angle.
Ans: From the given \(\angle A = 3x\) and \(\angle B = 2x.\)
We know,
When a transversal intersects two lines, the two lines are parallel if and only if interior angles on the same side of the transversal supplementary.
So, \(\angle A + \angle B = {180^ \circ }\)
\( \Rightarrow 3x + 2x ={180^ \circ }\)
\( \Rightarrow 5x = {180^ \circ }\)
\(x = \frac{{{{180}^ \circ }}}{5} = {36^ \circ }\)
Therefore, the measure of \(\angle A = 3 \times {36^ \circ } = {108^ \circ }\) and \(\angle B = 2 \times {36^ \circ } = {72^ \circ }.\)

Summary

Two or more lines that are the same distance apart and never join are called parallel lines. Parallel lines are those that, no matter how far they are stretched, never meet. Parallel lines are given the symbol \(||.\) By comparing the slopes of two lines, we may determine if they are parallel using their equations. The lines are parallel if their slopes are the same. Knowing some basics about parallel lines and copying angles is required to construct a line parallel to a given line passing through a given point.

This article includes the steps for constructing the parallel line to a given line. It helps a lot in understanding about construction of parallel lines.

FAQs – Construction of a Line Parallel to a Given Line

Q.1. What are parallel lines?
Ans: Lines in a plane that are always the same distance apart are known as parallel lines. Parallel lines never meet one other.

Q.2. What are the rules for lines to be parallel?
Ans: A list of some rules for lines to be parallel when a transversal cut them
1. The angles that correspond to each other are the same.
2. The angles that are vertically opposite each other are equal.
3. The interior angles are equal on both sides.
4. The exterior angles are equal on both sides.
5. The pair of interior angles on the same side of the transversal is supplementary.

Q.3. What are the steps in the construction of a parallel line to a given line?
Ans: Steps in the construction of a parallel line to a given line are given below,
1. Consider a line \(m\) on a plane.
2. Mark the point \(A\) on a line \(m.\)
3. Mark a point \(P\) away from the line \(m.\)
4. Consider \(A\) as the centre, and any suitable radius draw an arc cutting the line segments \(AP\) at a point \(V\) and the line m at a point \(Z,\) respectively.
5. With \(P\) as the centre and any suitable radius, draw an arc cutting the line segments \(AP\) at a point \(U\) as shown below.
6. With \(U\) a centre and the same radius is taken earlier, cut an arc \(CD\) at a point \(E.\) And draw a line passing through the points \(E\) and \(P.\)
Then, name the obtained line as \(n\) so that the line \(m\) is parallel to line \(n,\) i.e., \(m||n.\)

Q.4. How many ways can we draw parallel lines to an existing line?
Ans: Drawing parallel lines to an existing line can be done in many ways.

Q.5. Do parallel lines have the same slope?
Ans: Yes, parallel lines have the same slope.

Q.6. How do you determine if two lines are parallel?
Ans: By comparing the slopes of two lines, we may determine if they are parallel using their equations. The lines are parallel if their slopes are the same, but their \(y\)-intercepts differ. The lines are not parallel if the slopes differ.

Q.7. What are the necessary conditions for the lines to be parallel?
Ans: The below are some of the conditions that must be met for the lines to be parallel if the transversal lines cut two different straight lines
1. When the two corresponding angles are equal, the two straight lines will be parallel to one another.
2. When the two alternate angles are equal, the two straight lines will be parallel to one another.
3. When two interior angles on the same side of a transversal are supplementary, the two straight lines will be parallel.

Learn about Lines Parallel to the Same Line here