Angle Between Two Lines (General Form)

Acute angle between a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.

For lines $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$, the slopes are:

$$m_1 = -\frac{a_1}{b_1}, \qquad m_2 = -\frac{a_2}{b_2}$$

Substituting into $\tan\theta = \left|\dfrac{m_1-m_2}{1+m_1m_2}\right|$:

$$m_1 - m_2 = -\frac{a_1}{b_1} + \frac{a_2}{b_2} = \frac{a_2b_1 - a_1b_2}{b_1b_2}$$ $$1 + m_1m_2 = 1 + \frac{a_1a_2}{b_1b_2} = \frac{b_1b_2 + a_1a_2}{b_1b_2}$$

The $b_1b_2$ cancels:

$$\boxed{\tan\theta = \left|\frac{a_1b_2 - a_2b_1}{a_1a_2 + b_1b_2}\right|}$$

Parallel condition: $a_1b_2 = a_2b_1$ (numerator vanishes).

Perpendicular condition: $a_1a_2 + b_1b_2 = 0$ (denominator vanishes).

This form avoids computing slopes explicitly and is more convenient when lines are given in general form.