Angle between two lines

When two non-parallel lines meet, they form four angles in two pairs of equal opposite angles. We take the acute angle as the angle between them. The formula expresses this angle via the slopes.

The angle formula

Suppose two lines have slopes $m_1, m_2$ and make angles $\theta_1, \theta_2$ with the positive $x$-axis. Then $m_1 = \tan\theta_1, m_2 = \tan\theta_2$. The angle between them is $\theta = \theta_2 - \theta_1$ (or its supplement). Using the tangent-difference formula: $\tan\theta = \frac{\tan\theta_2 - \tan\theta_1}{1 + \tan\theta_2 \tan\theta_1} = \frac{m_2 - m_1}{1 + m_1 m_2}.$

To always get the acute angle, take the absolute value: $\boxed{\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|.}$

Special cases

Parallel. Slopes equal: $m_1 = m_2$. The formula gives $\tan\theta = 0 \Rightarrow \theta = 0$.

Perpendicular. $\tan\theta = \infty$ requires the denominator to vanish: $1 + m_1 m_2 = 0$, i.e. $\boxed{m_1 m_2 = -1.}$

This fails when one line is vertical (slope undefined). For verticals, perpendicular means the other is horizontal.

In terms of general form

If lines are $A_1 x + B_1 y + C_1 = 0$ and $A_2 x + B_2 y + C_2 = 0$:

• Parallel: $A_1 B_2 = A_2 B_1$.
• Perpendicular: $A_1 A_2 + B_1 B_2 = 0$.
• Angle: $\tan\theta = \left|\dfrac{A_1 B_2 - A_2 B_1}{A_1 A_2 + B_1 B_2}\right|$.

Worked examples

Example 1. Find the angle between $y = 2x + 1$ and $y = -3x + 5$.

$m_1 = 2, m_2 = -3$. $\tan\theta = \left|\dfrac{2 - (-3)}{1 + 2 \cdot (-3)}\right| = \dfrac{5}{5} = 1 \Rightarrow \theta = 45°$.

Example 2. Find the angle between $3 x - 4 y + 7 = 0$ and $4 x + 3 y - 5 = 0$.

$A_1 A_2 + B_1 B_2 = 12 - 12 = 0$. So perpendicular: $\theta = 90°$.

Example 3. Find the equation of the line through $(2, 3)$ perpendicular to $4 x + 3 y + 7 = 0$.

The given line has slope $-4/3$, so the perpendicular slope is $3/4$. Equation: $y - 3 = \dfrac{3}{4}(x - 2) \Rightarrow 3 x - 4 y + 6 = 0$.

Example 4. Find the equations of the lines passing through $(2, 3)$ and making an angle of $45°$ with $y = 2 x + 5$.

$m_1 = 2$. $\tan 45° = \left|\dfrac{2 - m_2}{1 + 2 m_2}\right| = 1$. Case (a): $\dfrac{2 - m_2}{1 + 2 m_2} = 1 \Rightarrow 2 - m_2 = 1 + 2 m_2 \Rightarrow m_2 = 1/3$. Case (b): $\dfrac{2 - m_2}{1 + 2 m_2} = -1 \Rightarrow 2 - m_2 = -1 - 2 m_2 \Rightarrow m_2 = -3$.

Equations: $y - 3 = (1/3)(x - 2) \Rightarrow x - 3y + 7 = 0$; $y - 3 = -3(x - 2) \Rightarrow 3x + y - 9 = 0$.

Example 5. If the angle between $y = mx + 5$ and $y = x + 1$ is $\tan^{-1}(1/3)$, find $m$.

$\left|\dfrac{m - 1}{1 + m}\right| = 1/3$.

Case (a): $3(m - 1) = 1 + m \Rightarrow 2 m = 4 \Rightarrow m = 2$. Case (b): $3(m - 1) = -(1 + m) \Rightarrow 4 m = 2 \Rightarrow m = 1/2$.

Try it yourself

1. Angle between $y = x$ and $y = -x$.
2. Angle between $y = 2 x + 3$ and $y = 5$.
3. Find slope of a line perpendicular to $4 x + 5 y - 8 = 0$.
4. Are $2 x + 3 y - 1 = 0$ and $6 x + 9 y - 5 = 0$ parallel?
5. Find $k$ such that $3 x - k y + 2 = 0$ is perpendicular to $2 x + 5 y - 1 = 0$.
6. Find the equation of the line through $(1, 2)$ at $30°$ to the $x$-axis.
7. Find the equation of the line through origin perpendicular to $y = 3 x - 1$.
8. Two lines through $(2, 3)$ make an angle of $60°$ with $y = x$. Find them.
9. Show that the lines $x - 2 y + 3 = 0$ and $2 x + y - 5 = 0$ are perpendicular.
10. Find the equations of the angle bisectors of $3 x - 4 y + 7 = 0$ and $4 x - 3 y - 8 = 0$. (Hint: $\dfrac{3x - 4y + 7}{5} = \pm \dfrac{4x - 3y - 8}{5}$.)
11. Find the line through $(1, 1)$ which is perpendicular to the line joining $(2, 3)$ and $(5, 7)$.
12. Find $\theta$ between $3 x + 4 y - 7 = 0$ and $5 x - 12 y + 6 = 0$.

Pitfalls / Tricks

• Always take the absolute value in $\tan\theta$ to get the acute angle.
• $m_1 m_2 = -1$ only works when both slopes are finite.
• When one line is vertical, the angle equals $|90° - \theta_{\text{other}}|$.
• Insight. The formula $\tan\theta = |(m_1 - m_2)/(1 + m_1 m_2)|$ is essentially the tangent subtraction identity in disguise , derived from $\theta = \theta_1 - \theta_2$ on the unit circle.