(i) Condition for the lines to be parallel in terms of their slopes.

Let m_{1} and m_{2} be the slopes of two lines.

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

That is,

m_{1 }= m_{2}

(ii) Condition for the lines to be parallel in terms of their general form of equations.

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 + c_{1} = 0

ax + by + c_{2} = 0

(iii) Condition for the lines to be parallel in terms of their slope-intercept form of equations.

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 + b_{1}

y = mx + b_{2}

**(iv) Condition for the lines to be parallel in terms of angle of inclination.**

Let l_{1} and l_{2} 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.**

**(v) Condition for the lines to be parallel in terms of the perpendicular distance between them. **

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

**The figure shown below below illustrates the above situation.**

**(v) Condition for the lines to be parallel when they are cut by a transversal. **

**Corresponding Angles Converse : **

If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.

The diagram shown below illustrates this.

**Alternate Interior Angles Converse : **

If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.

The diagram shown below illustrates this.

**Consecutive Interior Angles Converse : **

If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.

The diagram given below illustrates this.

**Alternate Exterior Angles Converse : **

If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel.

The diagram given below illustrates this.

**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 diagram given below, if ∠1 ≅ ∠2, then prove m||n.

**Solution :**

∠1 ≅ ∠2 ∠2 ≅ ∠3 ∠1 ≅ ∠3 m||n |
Given Vertical angles theorem Transitive property of congruence Corresponding angles converse |

**Example 7 : **

In the diagram given below, find the value of x that makes j||k.

**Solution : **

Lines j and k will be parallel if the marked angles are supplementary.

x° + 4x° = 180°

5x = 180

x = 36

So, x = 36 makes j||k.

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