Angle Between Two Lines

When two lines cross, they form two pairs of equal angles (one acute, one obtuse, adding to $180^\circ$). If the lines have slopes $m_1$ and $m_2$, the angle $\theta$ between them is given by the tangent formula:

The absolute-value bars are taken so that $\tan\theta \ge 0$, which returns the acute angle between the lines. (Dropping the bars gives a signed value whose sign distinguishes the acute from the obtuse angle; for board problems the acute angle is what is normally wanted.)

Two important cases fall straight out of this one formula:

• If $m_1 = m_2$, the numerator is $0$, so $\tan\theta = 0$ and $\theta = 0^\circ$ — the lines are parallel, exactly as Section 5 said.
• If $1 + m_1 m_2 = 0$, i.e. $m_1 m_2 = -1$, the denominator is $0$, so $\tan\theta$ is undefined and $\theta = 90^\circ$ — the lines are perpendicular, again matching Section 5. …