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# Slope of Parallel Lines

Parallel lines are the lines which does not intersect in any direction. Slope of a line is the value of the angle that a straight line makes with the positive direction of x-axis in the anticlockwise sense. Basically for any straight line y = mx + b, m is referred to as slope.

Slopes of two parallel lines are equal. If the slope of two parallel lines are $m_1$ and  $m_2$ then $m_1$ = $m_2$.

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## Slope of Parallel Lines Definition

Slope of a line in ratio of change in y axis to x axis that is the small change in entity in the y axis to the small change in entity in the x axis is called as slope. Parallel lines are the lines that lie in the same plane have the property that they never intersect each other. They never meet at a point. They are always the same distance apart.

Let y = m1x + c1 and y = m2x + c2 be two parallel lines then there slopes will be equal.

=> m1 = m2

In the above diagram, we can see the slope of two parallel lines. Both the slopes will have same value. Slope is the ratio of change in y to change in x. When we consider two parallel lines there ratios will be equivalent.

## Slope of Parallel Lines Property

Parallel lines have the value of the angle made with the x axis always same or always have the same slope.

Let the equation of two lines be y = (m1)x + c1 and y = (m2)x + c2 where, m1 and m2 are slopes of the lines. The two lines are parallel if and only if m1 = m2.

## Parallel Lines Slope Intercept Form

Two lines are parallel if they have same slope. The equation of the parallel line in slope-intercept form is y = mx + c, where m is the slope of the line.

### Intercept Theorem of Parallel Lines

If a transversal makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts.

Data:

AP || BQ || CR. The intercepts AB and BC are equal. PQ, QR are the intercepts on any other line.

To Prove Intercept Theorem:

PQ = QR

Construction:

Through A and B draw AE and BF parallel to PQR to cut BQ and CR at E and F respectively.

To prove that PQ = QR

Proof:

## Slope of Parallel Lines Examples

Given below are some of the examples on slope of parallel lines.

### Solved Examples

Question 1: Find the equation of a straight line parallel to y-axis and passing through the point (-3, 2).
Solution:

We know that the equation of a straight line parallel to y-axis is

x = a

Since it passes through the point (-3, 2), we get -3 = a i.e.

a = -3.

Substituting this values of a, we get

x = - 3 i.e. x + 3 = 0, which is the required equation.

Question 2: Given the equations of two lines 2y - 4x = 12 and y = 2x + 8. Check whether the given lines are parallel or not?
Solution:

In order to check that the lines are parallel or not, we have to calculate slope. First, convert the equation of lines into standard form y = mx + b to identify the slope.

Given the equation of the first line is

2y - 4x = 12

2y = 4x + 12

Dividing both sides by 2, we have

y = 2x + 6

Hence, slope of the first line is

m1 = 2.

Given the equation of the second line is

y = 2x + 8. It is already in the standard form.

Hence, slope of the second line is

m2 = 2.

Since the value of the slopes m1 = m2 = 2 are equal.

Hence, the given lines are parallel.(Answer)

Question 3: Given the equation of a line y = 5x + 8. What is the slope of the line parallel to given line?
Solution:

The equation of the line given is y = 5x + 8

It is in the standard form y = mx + b.

Hence, slope of the line is 5.

As we have to find the slope of the line parallel to the line y = 5x + 8.

We know that, the slopes of the two parallel lines must be the same.

Hence, the slope of required line is = 5 (Answer)

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