Conditions for Parallelism and Perpendicularity of Lines in Terms of Their Slopes

Fig 9.4 is a simple but essential diagram: it shows two straight lines, labelled $l_1$ and $l_2$, drawn on the $xy$-plane. Both lines are marked with double arrows to indicate they extend infinitely in both directions. The line $l_1$ is drawn in indigo, $l_2$ in blue. Each line makes an angle with the positive $x$-axis — the inclination of $l_1$ is labelled $\alpha$, and the inclination of $l_2$ is labelled $\beta$. The critical visual fact is that $\alpha$ and $\beta$ are drawn as equal angles. The $x$-axis itself is shown, and the two lines are clearly parallel — they never meet, and they slant in exactly the same direction.

The physical idea the figure teaches is that parallelism is equivalent to equal inclination. Inclination is the angle a line makes with the positive direction of the $x$-axis, measured anticlockwise from $0^\circ$ to $180^\circ$. If two lines are parallel, they must be tilted at the same angle relative to the horizontal — so their inclinations are equal. Conversely, if two lines have the same inclination, they point in the same direction and are therefore parallel. The diagram makes this one-to-one correspondence visually obvious before any algebra is introduced.

From this geometric observation, the textbook derives the algebraic condition for parallelism. Since the slope $m$ of a non-vertical line is defined as $m = \tan \theta$, where $\theta$ is its inclination, equal inclinations imply equal slopes:

Here $m_1 = \tan \alpha$ is the slope of $l_1$, $m_2 = \tan \beta$ is the slope of $l_2$, and the double arrow means the statement works both ways: if slopes are equal, the lines are parallel; if lines are parallel, their slopes are equal. The proof relies on the fact that the tangent function is one-to-one on the interval $(0^\circ, 180^\circ)$ — so $\tan \alpha = \tan \beta$ forces $\alpha = \beta$, and vice versa.

What Fig. 9.5 Shows

The figure depicts two straight lines, $l_1$ and $l_2$, drawn on a standard $xy$-coordinate plane. Both lines pass through the origin, crossing each other at a point above the origin — the intersection is marked with a small square to indicate a right angle. Line $l_1$ has a positive slope (it rises as we move right), while $l_2$ has a negative slope (it falls as we move right). The line $l_2$ is drawn in blue to distinguish it visually.

At the bottom, each line makes an angle with the positive $x$-axis. The angle for $l_1$ is labelled $\alpha$, and the angle for $l_2$ is labelled $\beta$. The key geometric fact shown in the diagram is that $\beta = \alpha + 90^\circ$ — the two inclinations differ by exactly $90^\circ$. This is the visual condition for perpendicularity.

The Physical Idea

The figure teaches a single, clean idea: two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. The diagram makes this relationship visible by showing that when one line's inclination is $\alpha$, the other's must be $\alpha + 90^\circ$ for them to meet at a right angle. The small square at the intersection is the standard geometric marker for a $90^\circ$ angle.

The Key Formula

From the geometry $\beta = \alpha + 90^\circ$, the textbook derives the slope condition using the tangent function:

Here:

• $m_1 = \tan \alpha$ is the slope of line $l_1$, where $\alpha$ is its inclination (angle with the positive $x$-axis, measured anticlockwise).
• $m_2 = \tan \beta$ is the slope of line $l_2$, where $\beta = \alpha + 90^\circ$ is its inclination.

The derivation works because:

So $m_2 = -\frac{1}{m_1}$, which rearranges to $m_1 m_2 = -1$.

What the Figure Does Not Show …