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.

That is, 

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. 

That is, 

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. 

That is, 

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. 

Practice Problems

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.

Then, 

3k + 2  =  7

Subtract 2 from each side. 

3k  =  5

Divide each side 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 each side. 

5k  =  0

Divide each side 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

(10x/2) + (14y/2) + (5/2)  =  (0/2)

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. 

Then, ∠H  =  ∠F -------> ∠H  =  65°

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

Then, ∠D  =  ∠H -------> ∠D  =  65°

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

Then, ∠B  =  ∠D -------> ∠B  =  65°

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

Then, we have

∠F + ∠E  =  180°

Plug ∠F  =  65°

∠F + ∠E  =  180°

65° + ∠E  =  180°

∠E  =  115°

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

Then, ∠G  =  ∠E -------> ∠G  =  115°

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

Then, ∠C  =  ∠G -------> ∠C  =  115°

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

Then, ∠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. 

Then, we have

(2x + 20)°  =  ∠(3x - 10)°

2x + 20  =  3x - 10

Subtract 2x from each side. 

20  =  x - 10

Add 10 to each side. 

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. 

Then, we have

(3x + 20)° + 2x°  =  180°

3x + 20 + 2x  =  180

Simplify.

5x + 20  =  180

Subtract 20 from each side.

5x  =  160

Divide each side by 8.

x  =  32

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