PROPERTIES OF PARALLEL LINES

Property 1 :

Let m₁ and m₂ be the slopes of two lines.

If the two lines are parallel, then their slopes will be equal.

m₁ = m₂

Property 2 :

Let us consider the general form of equation of a straight line.

ax + by + c = 0

If the two lines are parallel, then their general forms of equations will differ only in the constant term and they will have the same coefficients of x and y.

ax + by + c₁ = 0

ax + by + c₂ = 0

Property 3 :

Let us consider the slope intercept form of equation of a straight line.

y = mx + b

If the two lines are parallel, then their slope-intercept form equations will will differ only in the "y"- intercept.

y = mx + b₁

 y = mx + b₂ 

Property 4 :

Let l₁ and l₂ be two lines.

If the two lines are parallel, the angle between them and the positive side of x-axis will be equal.

The figure given below illustrates the above situation.

Property 5 :

If the two lines are parallel, the perpendicular distance between them will be same at everywhere.

The figure given below illustrates the above situation.

Property 6 :

Let l₁ and l₂ be two parallel lines and the line m intersects the lines l₁ and l₂.

The figure shown below illustrates the above situation.

From the above figure, we can have the following important results.

Solving Problems Using Properties of Parallel Lines

Problem 1 :

The slopes of the two lines are 7 and (3k + 2). If the two lines are parallel, find the value of k.

Solution :

If two lines are parallel, then their slopes are equal.

3k + 2 = 7

Subtract 2 from both sides.

3k = 5

Divide both sides by 5.

k = 5/3

Problem 2 :

If the following equations of two lines are parallel, then find the value of k.

3x + 2y - 8 = 0

(5k + 3)x + 2y + 1 = 0

Solution :

If the two lines are parallel, then their general forms of equations will differ only in the constant term and they will have the same coefficients of x and y.

To find the value of k, equate the coefficients of x.

5k + 3 = 3

Subtract 3 from both sides.

5k  =  0

Divide both sides by 5.

k = 0

Problem 3 :

Find the equation  of a straight line is passing through (2, 3) and parallel to the line 2x - y + 7 = 0.

Solution :

Because the required line is parallel to 2x - y + 7 = 0, the equation of the required line and the equation of the given line 2x - y + 7 = 0 will differ only in the constant term.

Then, the equation of the required line is

2x - y + k = 0 ----(1)

The required line is passing through (2, 3).

Substitute x = 2 and y = 3 in (1).

2(2) - 3 + k = 0

4 - 3 + k = 0

1 + k = 0

k = -1

So, the equation of the required line is

(1)----> 2x - y - 1 = 0

Problem 4 :

Verify, whether the following equations of two lines are parallel.

3x + 2y - 7 = 0

y = -1.5x + 4

Solution :

In the equations of the given two lines, the equation of the second line is not in general form.

Let us write the equation of the second line in general form.

y = -1.5x + 4

1.5x + y - 4 = 0

Multiply by 2 on both sides,

3x + 2y - 8 = 0

Now, let us compare the equations of two lines,

3x + 2y - 7 = 0

3x + 2y - 8 = 0

The above two equations differ only in the constant term. 

So, the equations of the given two lines are parallel.

Problem 5 :

Verify, whether the following equations of two lines are parallel.

5x + 7y - 1 = 0

10x + 14y + 5 = 0

Solution :

In the equation of the second line 10x + 14y + 5 = 0, the coefficients of x and y have the common divisor 2.

So, divide the second equation by 2.

5x + 7y + 2.5 = 0

Now, let us compare the equations of two lines,

5x + 7y - 1 = 0

5x + 7y + 2.5 = 0

The above two equations differ only in the constant term.

So, the equations of the given two lines are parallel.

Problem 6 :

In the figure given below,  let the lines l₁ and l₂ be parallel and m is transversal. If ∠F = 65°, find the measure of each of the remaining angles.

Solution :

From the given figure,

∠F and ∠H are vertically opposite angles and they are equal.

∠H = ∠F ----> ∠H = 65°

∠H and ∠D are corresponding angles and they are equal.

∠D = ∠H ----> ∠D = 65°

∠D and ∠B are vertically opposite angles and they are equal.

∠B = ∠D ----> ∠B = 65°

∠F and ∠E are together form a straight angle.

∠F + ∠E = 180°

Substitute ∠F = 65°.

∠F + ∠E = 180°

65° + ∠E = 180°

∠E = 115°

∠E and ∠G are vertically opposite angles and they are equal.

∠G = ∠E ----> ∠G = 115°

∠G and ∠C are corresponding angles and they are equal.

∠C = ∠G ----> ∠C = 115°

∠C and ∠A are vertically opposite angles and they are equal.

∠A = ∠C ----> ∠A = 115°

Therefore,

∠A = ∠C = ∠E = ∠G = 115°

∠B = ∠D = ∠F = ∠H = 65°

Problem 7 :

In the figure given below,  let the lines l₁ and l₂ be parallel and t is transversal. Find the value of x.

Solution :

From the given figure, (2x + 20)° and (3x - 10)° are corresponding angles.

So, they are equal.

(2x + 20)° = (3x - 10)°

2x + 20 = 3x - 10

Subtract 2x from both sides.

20 = x - 10

Add 10 to both sides.

30 = x

Problem 8 :

In the figure given below,  let the lines l₁ and l₂ be parallel and t is transversal. Find the value of x.

Solution :

From the given figure, (3x + 20)° and 2x° are consecutive interior angles.

So, they are supplementary.

(3x + 20)° + 2x° = 180°

3x + 20 + 2x = 180

Simplify.

5x + 20 = 180

Subtract 20 from both sides.

5x = 160

Divide both sides by 8.

x = 32

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