In this article, we will discuss how to find the angle between two lines.

Angle Between Two Lines

When two lines intersect each other, then two pairs of angles are formed that are opposite to each other. These angles are known as vertical angles.

If a line bisects another line, then we say that the lines are perpendicular to each other. In this case, the angles formed by intersection of lines are equal to 90^0.

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Formula to Calculate Angle Between Two Lines

Now, let us see what is the formula to calculate the angle between two lines when the lines are not perpendicular to each other.

tan \theta = \pm \frac {m_2 - m_1} {1 + m_1m_2}

Here, m_1 refers to the slope of the line 1 and m_2 refers to the slope of the line 2.

While finding the angle between two lines using the above formula keep the following things in mind:

  • The angle between the two intersecting lines PQ and RS is either acute or obtuse depending on the positive or negative value of \frac {m_2 - m_1} {1 + m_1m_2}.
  • When two straight lines intersect each other, then the angle between these lines is acute.
  • The formula tan \theta = \pm \frac {m_2 - m_1} {1 + m_1m_2} is not applicable to the two intersecting lines that are parallel to the y-axis. This is because of the fact that we cannot determine the slope of the lines parallel to y-axis.

The angle \theta between two intersecting lines is shown in the diagram below:

Angle between two lines

In the next section, we will solve couple of examples in which we will find angle between two lines using the formula discussed above.

Example 1

If P (3, 4), Q (2, 5) and R (-6, 8) are three points, then find the angle between the straight lines PQ and QR.

Solution

We will use the following formula to compute the angles between the lines PQ and QR.

tan \theta = \pm \frac {m_2 - m_1} {1 + m_1m_2}

You can see that in the above formula, we require m_1 and m_2. Let suppose line PQ is line 1, having a slope of m_1 and QR is line 2 which has a slope of m_2.

The coordinates of points P and Q are (3,4) and (2, 5) respectively. First, we will calculate m_1 using the following formula:

m _1 = \frac{y_2 - y_1} {x_2 - x_1}

m_1 = \frac{5 - 4} {2 - 3}

m_1 = -1

The coordinates of points Q and R are (2, 5) and (-6, 8) respectively. Now, we will compute the value of m_2 using the same formula.

m _2 = \frac{y_2 - y_1} {x_2 - x_1}

m _2 = \frac{8 - 5} {-6 - 2}

m_2 =- \frac{3}{8}

Now, we will put the value of m_1 and m_2 in the formula to get the required angle.

tan \theta = \pm \frac {m_2 - m_1} {1 + m_1m_2}

tan \theta = \pm \frac {-\frac{3}{8} - (-1)} {1 + (-1) (-\frac{3}{8})}

tan \theta = \pm \frac {-\frac{3}{8}+ 1} {1 + \frac{3}{8}}

tan \theta = \pm\frac{ \frac{5}{8}} {\frac{11}{8}}

  \theta = tan ^ {-1}\frac{5}{11}

= 24.44^0

Hence, the angle between the lines PQ and QR is of the measure 24.44 degrees.

 

Example 2

If A (2, 5), B (7, 6) and C (-3, 1) are three points, then find the angle between the straight lines AB and BC.

Solution

We will use the following formula to compute the angles between the lines AB and BC.

tan \theta = \pm \frac {m_2 - m_1} {1 + m_1m_2}

You can see that in the above formula, we require m_1 and m_2. Let suppose line AB is line 1, having a slope of m_1 and BC is line 2 which has a slope of m_2.

The coordinates of points A and B are (2, 5) and (7, 6) respectively. First, we will calculate m_1 using the following formula:

m _1 = \frac{y_2 - y_1} {x_2 - x_1}

m_1 = \frac{6 - 5} {7 - 2}

m_1 = \frac {1}{5}

The coordinates of points B and C are (7, 6) and (-3, 1) respectively. Now, we will compute the value of m_2 using the same formula.

m _2 = \frac{y_2 - y_1} {x_2 - x_1}

m _2 = \frac{1 - 6} {-3 - 7}

m_2 = \frac{5}{10} = \frac{1}{2}

Now, we will put the value of m_1 and m_2 in the formula to get the required angle.

tan \theta = \pm \frac {m_2 - m_1} {1 + m_1m_2}

tan \theta = \pm \frac {\frac{1}{2} - \frac{1}{5}} {1 + \frac {5}{10}{1}{5}}

tan \theta = \pm \frac {\frac{3}{10}} {1 + \frac{1}{10}}

tan \theta = \pm\frac{ \frac{3}{10}} {\frac{11}{10}}

= tan \theta \frac{3}{11}

\theta = tan^{-1} \frac{3}{11}

\theta = 15.26^0

Hence, the angle between the lines AB and BC is of the measure 15.26 degrees.

 

Example 3

Find the angle between the lines of the following equations:

A : 4x - 8y = 8

B : 3x - y = 6

Solution

The equations of the lines A and B are given in the standard form. We have to convert these equations in slope intercept form to determine the slope.

 

A : 4x - 8y = 8

Divide both sides by 4 to get the following equation:

A : x - 2y = 2

-2y = -x + 2

Divide both sides by -2:

y = \frac{1}{2}x - 1

Hence, the slope of this equation is \frac{1}{2}

 

Now, we will convert the second equation in slope intercept form:

B : 3x - y = 6

B : 3x - 6 = y

The slope of equation B is 3.

m_1 is equal to \frac{1}{2} and m_2 is equal to 3.

tan \theta = \pm \frac {m_2 - m_1} {1 + m_1m_2}

tan \theta = \pm \frac {3 - \frac{1}{2}} {1 + \frac{1}{2} \cdot 3}

tan \theta = \pm \frac {3 - \frac{1}{2}} {1 + \frac{3}{2}}

tan \theta = \pm \frac {\frac{5}{2}} { \frac{5}{2}}

\theta = tan^{-1} 1

= 45^0

Hence, the angle between two lines A and B is 45 degrees.

Example 4

Find the angle between the lines of the following equations:

P : 2x - 3y = 6

Q : 2x - 5y = 7

Solution

The equations of the lines P and Q are given in the standard form. We have to convert these equations in slope intercept form to determine the slope.

 

P : 2x - 3y = 6

P : -3y = 6 - 2x

Divide both sides by -3:

y = \frac{2}{3} x - 2

Hence, the slope of this equation is \frac{2}{3}

 

Now, we will convert the second equation in slope intercept form:

Q : 2x - 5y = 7

Q: -5y = -2x + 7

Divide both sides by -5:

Q: y = \frac{2}{5}x - \frac{7}{5}

The slope of equation Q is \frac{2}{5}.

m_1 is equal to \frac{2}{3} and m_2 is equal to \frac{2}{5}.

tan \theta = \pm \frac {m_2 - m_1} {1 + m_1m_2}

Substitute the values of slopes in the above equation:

tan \theta = \pm \frac {\frac{2}{5} - \frac{2}{3}} {1 + \frac{2}{5} \cdot \frac{2}{3}}

tan \theta = \pm \frac {-\frac{4}{15}}} {1 + \frac{4}{15}}

tan \theta = \pm \frac {-\frac{4}{15}} { \frac{19}{15}}

= tan \theta -\frac{4}{19}

\theta = tan^{-1} \frac{4}{19}

= 11.89^0

Hence, the angle between two lines P and Q is 11.89 degrees.

 

Example 5

Find the angle between the lines of the following equations:

M : y = 3x + 4

N : y = 2x - 7

Solution

The equations of the lines M and N are already in the slope intercept form, i.e. y = mx + b. Hence, we can easily find the slopes of the lines like this:

Slope of the line M = m_1 = 3

Slope of the line N = m_2 = 2

Use the following formula to calculate the angle between the lines M and N.

tan \theta = \pm \frac {m_2 - m_1} {1 + m_1m_2}

Substitute the values of slopes in the above equation:

tan \theta = \pm \frac {2 - 3} {1 + 2 \cdot 3}

tan \theta = \pm \frac {-1} {1 + 6}

tan \theta = \pm  \frac{-1}{7}

= 8.13^0

Hence, the angle between two lines M and N is 8.13 degrees.

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.