Angle Between Two Lines

Fig 9.6 is a simple coordinate-plane sketch with two non-vertical lines, L₁ and L₂, crossing each other above the x-axis. L₁ is the shallower line (smaller slope), L₂ the steeper one (larger slope). Each line meets the x-axis at a distinct point, and at those crossing points the angle each line makes with the positive x-axis is marked: α₁ for L₁, α₂ for L₂. Both α₁ and α₂ are measured anticlockwise from the positive x-axis, so α₂ > α₁.

At the intersection point of L₁ and L₂, two adjacent angles are labelled: θ (the acute angle) and φ (the obtuse angle). The figure makes clear that θ = α₂ − α₁, because the exterior angle of a triangle equals the sum of the opposite interior angles — or, more directly, because the angle between the two lines is simply the difference of their inclinations.

The physical idea is simple: the steepness of a line is captured by its slope, and the angle between two lines can be expressed purely in terms of their slopes, without needing to draw the lines or measure angles with a protractor. The figure anchors the derivation that follows.

The obtuse angle $\phi$ is $180^\circ - \theta$, so $\tan\phi = -\tan\theta$.

The figure also illustrates why two answers appear when solving for an unknown slope: depending on which line is taken as L₁ and which as L₂, the acute angle formula gives either the slope of the steeper line or the shallower one. In Example 2, the two possible slopes (3 and $-\frac{1}{3}$) correspond to the two lines that make a $45^\circ$ angle with a given line of slope $\frac{1}{2}$ — one on each side of it. …

Fig. 9.7 is a geometric demonstration of why the angle-between-lines formula gives two possible slopes for the second line when the angle is fixed. The figure shows three lines that all pass through a single point (they are concurrent). One line, labelled L, has slope $\frac{1}{2}$. A second line, labelled L₂, has slope $3$. A third line, labelled L₁, has slope $-\frac{1}{3}$. Between L and L₂, and between L and L₁, a small $45^\circ$ arc is drawn. An '8' tick appears on the x‑axis (likely a coordinate label or a distance mark, though its exact meaning is not elaborated in the text).

The physical idea is this: if you know the slope of one line ($m_1 = \frac{1}{2}$) and the acute angle between it and another line ($\theta = 45^\circ$), there are two distinct lines that satisfy that condition — one on each side of the given line. The line L₂ makes a $45^\circ$ angle with L when measured in one direction; the line L₁ makes the same $45^\circ$ angle when measured in the opposite direction. Their slopes are different because the angle is measured from L to each line in opposite senses.

The textbook develops this with the formula for the acute angle $\theta$ between two lines of slopes $m_1$ and $m_2$:

Here $m_1$ is the known slope ($\frac{1}{2}$), $\theta = \frac{\pi}{4}$ ($45^\circ$), and $m_2$ is the unknown slope. Removing the absolute value gives two equations:

Solving the first yields $m_2 = 3$ (line L₂). Solving the second yields $m_2 = -\frac{1}{3}$ (line L₁). The figure makes this concrete: both lines L₂ and L₁ are at $45^\circ$ to L, but on opposite sides, so their slopes are not the same. …