Angle Between Two Lines. Parallelism and Perpendicularity

The Angle Between Two Lines

Two intersecting lines form four angles. If they are not perpendicular, two of the angles are acute and equal, and the other two are obtuse and equal. If they are perpendicular, all four measure $90^\circ$. By convention, the angle between two lines is the acute one (between $0^\circ$ and $90^\circ$), or the right angle if they are perpendicular.

This formula comes from the difference of two angles of inclination: if $m_1=\tan\theta_1$ and $m_2=\tan\theta_2$, then $\alpha=\theta_1-\theta_2$ and the tangent of a difference is used.

If the line is given only by its slope, remember that $m=\tan\theta$, where $\theta$ is the angle it forms with the positive direction of the $x$-axis. So a line with slope $\sqrt{3}$ forms $60^\circ$ with the $x$-axis.

Parallel Lines

Why? The slope is the tangent of the angle of inclination. If two lines are tilted the same way, they go in the same direction and never meet. In the angle formula, $m_1=m_2$ gives $\tan\alpha=0$, that is, $\alpha=0^\circ$.

Perpendicular Lines

This agrees with the angle formula: $1+m_1m_2=0$ makes $\alpha=90^\circ$. If a line is horizontal ($m=0$), its perpendicular is vertical and has no slope.

Orthogonal Projection of a Point onto a Line

The orthogonal projection of a point $P$ onto a line $r$ is the point $H$ on $r$ closest to $P$: the foot of the perpendicular drawn from $P$.

It is found in three steps: perpendicular slope, perpendicular line through $P$, and intersection with $r$ (a 2×2 system).

Applications: Altitudes, Perpendicular Bisectors and Structures

Perpendicularity shows up whenever something must meet at a right angle: the altitude of a triangle, the perpendicular bisector of a segment (perpendicular through the midpoint), a crossbeam in a structure, or a new street that crosses another. The diagonals of a square or a rhombus are also perpendicular, and they cross at their midpoint.

Reflection: Reflected Ray and Symmetric Line

When a ray of light hits a flat mirror, the angle it forms with the normal (the perpendicular to the mirror at the point of impact) equals the angle of the reflected ray. This is the angle of incidence.

To reflect a line $r$ across another line $s$ (the symmetric line): find the point $P$ where they intersect, reflect a point of $r$ across $s$ (perpendicular through that point, foot of the perpendicular, and double the distance), and join the reflected point to $P$.

Extension: Lines with a Parameter

When a line depends on a parameter $k$, the required condition becomes an equation for $k$: parallelism ($m_1=m_2$), perpendicularity ($m_1m_2=-1$), or a given angle ($\tan\alpha$ equal to the formula).

The same method works to find lines that form a given angle with a known line: set up $\tan\alpha=\left|\dfrac{m-m_1}{1+mm_1}\right|$ with the slope $m$ as the unknown. For example, an equilateral triangle forces sides that form $60^\circ$ ($\tan60^\circ=\sqrt3$) with another side, and two lines result.

Extension: Families of Lines with a Fixed Point

An equation with a parameter, such as $(1+k)x+(2-k)y-3=0$, describes a family of lines: a different line for each value of $k$. Sometimes all of them pass through the same point.

Extension: Angle Bisectors of Two Lines

Two intersecting lines form four angles and have two angle bisectors. Every point on a bisector is the same distance from the two lines.

This always happens: the two bisectors are perpendicular to each other, because adjacent angles add up to $180^\circ$ and each bisector splits one of them in half (the sum of the two halves is $90^\circ$).

Summary

Go back to the beginning: for the new avenue to cross at a right angle, it is enough for the product of the slopes of the two streets to be $-1$.