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,
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.
|
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
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