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 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.

### Intercept Theorem 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: **

**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: **

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

** Solution: **

Let y = m_{1}x + c_{1} and y = m_{2}x + c_{2} be two parallel lines then there slopes will be equal.

=> m_{1} = m_{2}

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 = (m_{1})x + c_{1} and y = (m_{2})x + c_{2} where, m_{1} and m_{2} are slopes of the lines. The two lines are parallel if and only if m_{1} = m_{2}.

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.

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:**

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.

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

m_{1} = 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

m_{2} = 2.

Since the value of the slopes m_{1} = m_{2} = 2 are equal.

Hence, the given lines are parallel.(Answer)

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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