# Angle Between Two Straight Lines

**THEOREM 1:**

**To find the angle between two given straight lines.**

**Solution:**

Let the two straight lines be AT1 and AT2 which meet at x-axis at T1 and T2 points respectively.

Let the equations of the two lines AT

_{1}and AT_{2}be
y = m

_{1}x + c_{1}and y_{2}= m_{2}x + c_{2}(1)
Hence tan AT

_{1}X = m_{1}and tan AT_{2}X = m_{2}
Now ∠T

_{1}AT_{2}= ∠AT_{1}X - ∠AT_{2}X
Hence tan T

_{1}AT_{2}= tan [AT_{1}X - AT_{2}X]
Hence the required angle = ∠T

_{1}AT_{2}
(2)

If equation (2) is a positive quantity then it is the tangent of the acute angle between the
two lines, if (2) is a negative quantity then it is the tangent of the obtuse angle.

**Example:**

Find the angle between two straight lines y = 6x - 11 and

**Solution:**

The equation of the first line is y = 6x - 11 which implies m

_{1}= 6.
The equation of the second line is hence

**THEOREM 2:**

**To find the condition that two straight lines can be parallel.**

Two straight lines are parallel when the angle between them is zero and hence the tangent
of this angle is zero. Hence in theorem 1 the equation (2) becomes

**m**which is the required result.

_{1}= m_{2}**Example:**

Find the equation of a straight line which passes through the point (4, -5) and which is
parallel to the straight line 3x + 4y + 5 = 0.

**Solution:**

Any straight line which is parallel to 3x + 4y + 5 = 0 will be 3x + 4y + c

_{2}= 0 because m_{1}= m_{2}.
Now given that this line 3x + 4y + c

_{2}= 0 passes through the point (4, -5) hence we get,
3. 4 + 4. (-5) + c

_{2}= 0 or c_{2}= 8
Putting the value of c

_{2}= 8 we get 3x + 4y + 8 = 0. (Answer)**THEOREM 3:**

**To find the condition that two straight lines can be perpendicular.**

Let the two straight lines be y = m

_{1}x + c_{1}and y_{2}= m_{2}x + c_{2}
If the angle between them is θ then by theorem 1,

(1)

If the two lines are perpendicular then θ = 90

^{ ο }and so tan θ = ∞.
Hence in equation (1)
which is only
possible if the denominator is zero

Therefore the condition of perpendicularity is

1 + m

_{1}m_{2}= 0 or**m**_{1}m_{2}= -1.
Hence the straight line y

_{2}= m_{2}x + c_{2}is perpendicular to y = m_{1}x + c_{1}if**Example:**

Find the equation of a straight line which passes through the point (4, -5) and is perpendicular
to the straight line 3x + 4y + 5 = 0.

**Solution:**

Let the equation of a straight line be y = m

_{1}x + c
Now given that this line passes through the point (4, -5) hence it becomes, -5 = 4m

_{1}+ c (1)
Given that the above line is perpendicular to the straight line 3x + 4y + 5 = 0 (2)

From (2) we get
where

Then by the condition of perpendicularity

**m**_{1}m_{2}= -1.
Hence

Or putting which in eq.
(1) we get,

Hence the required equation will be
Or 4x - 3y = 31 (Answer)

**Online Solution Angle between two straight lines Help:**

If you are stuck with a Solution Angle between two straight lines Homework problem and
need help, we have excellent tutors who can provide you with Homework Help. Our tutors
who provide Solution Angle between two straight lines help are highly qualified. Our
tutors have many years of industry experience and have had years of experience providing
Solution Angle between two straight lines Homework Help. Please do send us the Solution
Angle between two straight lines problems on which you need Help and we will forward then
to our tutors for review.

**Online Tutor Angle between two straight lines Parallel, Perpendicular:**

We have the best tutors in math in the industry. Our tutors can break down a complex
Angle between two straight lines Parallel, Perpendicular problem into its sub parts
and explain to you in detail how each step is performed. This approach of breaking down
a problem has been appreciated by majority of our students for learning Angle between two
straight lines Parallel, Perpendicular concepts. You will get one-to-one personalized
attention through our online tutoring which will make learning fun and easy. Our tutors
are highly qualified and hold advanced degrees. Please do send us a request for Angle
between two straight lines Parallel, Perpendicular tutoring and experience the quality
yourself.

**Other topics under Co-ordinate Geometry:**