**Property 1 : **

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}

**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 + c_{1} = 0

ax + by + c_{2} = 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 + b_{1}

y = mx + b_{2}

**Property 4 : **

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.**

**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 l_{1} and l_{2} be two parallel lines and the line m intersects the lines l_{1} and l_{2}.

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

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

Vertically opposite angles are equal. |
∠ 1 = ∠ 3 ∠ 2 = ∠ 4 ∠ 5 = ∠ 7 ∠ 6 = ∠ 8 |

Corresponding angles are equal. |
∠ 1 = ∠ 5 ∠ 2 = ∠ 6 ∠ 3 = ∠ 7 ∠ 4 = ∠ 8 |

Alternate interior angles are equal. |
∠ 3 = ∠ 5 ∠ 4 = ∠ 6 |

Consecutive interior angles are supplementary. |
∠ 3 + ∠ 6 = 180° ∠ 4 + ∠ 5 = 180° |

Same side exterior angles are supplementary. |
∠ 1 + ∠ 8 = 180° ∠ 2 + ∠ 7 = 180° |

**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 l_{1} and l_{2} 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 l_{1} and l_{2} 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 l_{1} and l_{2} 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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