PROPERTIES OF PARALLEL AND PERPENDICULAR LINES
About "Properties of parallel and perpendicular lines"
Here, we are going to see the properties of parallel and perpendicular lines.
First, let us look at the properties of parallel lines.
Properties of parallel lines
(i) 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
(ii) 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 form 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
(iii) 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
(iv) 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.
(v) If the two lines are parallel, the perpendicular distance between them will be same at everywhere.
The figure given below illustrates the above situation.
(vi) Let l₁ and l₂ be two parallel lines and the line m intersects the lines l₁ and l₂.
The figure given below illustrates the above situation.
From the above figure, we have the following important properties.
|
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 |
|
Alternate exterior angles are equal. |
< 1 = < 7 < 2 = < 8 |
|
Consecutive interior angles are supplementary. |
< 3 + < 6 = 180° < 4 + < 5 = 180° |
Properties of perpendicular lines
(i) 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
(ii) 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 given in the figure below.
(iii) 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
(iv) Let l₁ and l₂ be two lines.
If the two lines are perpendicular, the angle between them will be 90°
The figure given below illustrates the above situation.
Properties of parallel lines - 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 the given two lines are parallel, then their slopes are equal.
3k + 2 = 7
3k = 5
k = 5/3
Let us look at the next problem on "Properties of parallel and perpendicular lines"
Problem 2 :
The equations of the two parallel lines are
3x + 2y - 8 = 0
(5k+3) + 2y + 1 = 0
Find the value of "k"
Solution :
If the two lines are parallel, then their general form of equations will differ only in the constant term and they will have the same coefficients of "x" and "y".
If the given two lines are parallel, then their slopes are equal.
3k + 2 = 7
3k = 5
k = 5/3
Let us look at the next problem on "Properties of parallel and perpendicular lines"
Problem 3 :
Find the equation of a straight line is passing through (2, 3) and parallel to the line 2x - y + 7 = 0.
Solution :
Since 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.
Therefore, equation of the required line is 2x -y + k = 0 ------> (1)
The required line is passing through (2, 3).
So, we can plug x = 2 and y = 3 in the equation of the required line.
2(2) - 3 + k = 0
4 - 3 + k = 0
1 + k = 0
k = - 1
Hence, the equation of the required line is 2x -y - 1 = 0.
Let us look at the next problem on "Properties of parallel and perpendicular lines"
Problem 4 :
Verify, whether the two lines 3x + 2y - 7 = 0 and y = -1.5x + 4 are parallel.
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.
Hence, the equations of the given two lines are parallel.
Let us look at the next problem on "Properties of parallel and perpendicular lines"
Problem 5 :
Verify, whether the two lines 5x + 7y - 1 = 0 and 10x + 14y + 5 = 0 are parallel.
Solution :
In the equation of the second line 10x+14y + 5 = 0, the coefficients of "x" and "y" have the common divisor 2.
So, let us 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.
Hence, the equations of the given two lines are parallel.
Let us look at the next problem on "Properties of parallel and perpendicular lines"
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°
Let us look at the next problem on "Properties of parallel and perpendicular lines"
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
30 = x
Hence, x = 30°
Let us look at the next problem on "Properties of parallel and perpendicular lines"
Problem 8 :
In the figure given below, let the lines l₁ and l₂ be parallel and t is transversal. Find trhe 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°
5x + 20 = 180°
5x = 160°
x = 32°
Hence, x = 32°
Let us look at the next problem on "Properties of parallel and perpendicular lines"
Problem 9 :
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 parallel, then the product of the slopes is equal to -1.
7(3k + 2) = - 1
21k + 14 = -1
21k = -15
k = -15/21
k = -5/7
Let us look at the next problem on "Properties of parallel and perpendicular lines"
Problem 10 :
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.
So, we have
5k + 3 = 2
5k = -1
k = -1/5
Let us look at the next problem on "Properties of parallel and perpendicular lines"
Problem 11 :
Find the equation of a straight line is passing through (2, 3) and perpendicular to the line 2x - y + 7 = 0.
Solution :
Since the required line is perpendicular to 2x - y + 7 = 0,
then, equation of the required line is x + 2y + k = 0 ------> (1)
The required line is passing through (2, 3).
So, we can plug x = 2 and y = 3 in the equation of the required line.
2 + 2(3) + k = 0
2 + 6 + k = 0
8 + k = 0
k = - 8
Hence, the equation of the required line is x + 2y - 8 = 0.
Let us look at the next problem on "Properties of parallel and perpendicular lines"
Problem 12 :
Verify, whether the two lines 3x - 2y - 7 = 0 and y = - (2x/3) + 4 are perpendicular.
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 = - (2x/3) + 4
Multiply by 3 on both sides,
3y = - 2x + 12
2x + 3y - 12 = 0
Now, let us compare the equations of two lines,
3x - 2y - 7 = 0
2x + 3y - 12 = 0
When we look at the general form of equations of the above two lines, we get the following points.
(i) The sign of y- terms are different.
(ii) The coefficient of "x" term in the first equation is the coefficient of "y" term in the second equation.
(iii) The coefficient of "y" term in the first equation is the coefficient of "x" term in the second equation.
(iv) The above equations differ in constant terms.
Considering the above points, it is clear that the given two lines are perpendicular.
Hence, the equations of the given two lines are perpendicular.
Let us look at the next problem on "Properties of parallel and perpendicular lines"
Problem 13 :
Verify, whether the two lines 5x + 7y - 1 = 0 and 14x - 10y + 5 = 0 are perpendicular.
Solution :
In the equation of the second line 14x-10y + 5 = 0, the coefficients of "x" and "y" have the common divisor 2.
So, let us divide the second equation by 2
(14x/2) - (10y/2) + (5/2) = (0/2)
7x - 5y + 2.5 = 0
Now, let us compare the equations of two lines,
5x + 7y - 1 = 0
7x - 5y + 2.5 = 0
When we look at the general form of equations of the above two lines, we get the following points.
(i) The sign of y- terms are different.
(ii) The coefficient of "x" term in the first equation is the coefficient of "y" term in the second equation.
(iii) The coefficient of "y" term in the first equation is the coefficient of "x" term in the second equation.
(iv) The above equations differ in constant terms.
Considering the above points, it is clear that the given two lines are perpendicular.
Hence, the equations of the given two lines are perpendicular.
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