# PROPERTIES OF PARALLEL AND PERPENDICULAR LINES

## 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,

m=  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.

 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°

## Properties of Perpendicular Lines

Property 1 :

Let m1 and m2 be the slopes of two lines.

If, the two lines are perpendicular, then the product of their slopes is equal to - 1

That is,

m1 x m2  =  -1

Property 2 :

Let us consider the general form of equation of a straight line ax + by + c  =  0.

If the two lines are perpendicular, then their general form of equations will differ as shown in the figure below. Property 3 :

Let us consider the slope intercept form of equation of a straight line  y  =  mx + b.

If the two lines are perpendicular, then their slope-intercept form equations will differ as given in the figure below ## 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 :

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

Solution :

If the given two lines are perpendicular, then the product of the slopes is equal to  -1.

7(3k + 2)  =  - 1

Use distributive property.

21k + 14  =  -1

Subtract 14 from each side.

21k  =  -15

Divide each side by 21.

k  =  -15/21

k  =  -5/7

Problem 4 :

The equations of the two perpendicular lines are

3x + 2y - 8  =  0

(5k+3) - 3y + 1  =  0

Find the value of 'k'.

Solution :

If the two lines are perpendicular, then the coefficient 'y' term in the first line is equal to the coefficient of 'x' term in the second line.

Then,

5k + 3  =  2

Subtract 3 from each side.

5k  =  -1

Divide each side by 5.

k  =  -1/5 Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

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