Angle Between Two Pairs of Straight Lines

Understanding the angle between two pairs of straight lines is a fundamental concept in coordinate geometry. This concept is particularly important when dealing with intersecting lines and can be applied in various mathematical and real-world problems, such as finding the angle between roads or edges of objects.

Basic Concepts

Before diving into the angle between pairs of lines, let's establish some basic concepts:

• Straight Line Equation: A straight line in the plane can be represented by the linear equation $ax + by + c = 0$, where $a$, $b$, and $c$ are constants.
• Slope of a Line: The slope of a line, often denoted as $m$, is a measure of its steepness and is given by $m = -\frac{a}{b}$ for the line $ax + by + c = 0$.
• Angle Between Two Lines: The angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by the formula:

$$ \tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1m_2} \right| $$

Angle Between Two Pairs of Straight Lines

When dealing with pairs of straight lines, we are essentially finding the angle between two intersecting lines in each pair and then comparing those angles.

Formula for Angle Between Two Lines

Given two lines with slopes $m_1$ and $m_2$, the angle $\theta$ between them can be found using the formula mentioned above. If the lines are represented by their general equations:

$$ L_1: a_1x + b_1y + c_1 = 0 \quad \text{with slope} \quad m_1 = -\frac{a_1}{b_1} $$

$$ L_2: a_2x + b_2y + c_2 = 0 \quad \text{with slope} \quad m_2 = -\frac{a_2}{b_2} $$

Then the angle $\theta$ between $L_1$ and $L_2$ is given by:

$$ \tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1m_2} \right| = \left| \frac{\frac{a_1b_2 - a_2b_1}{b_1b_2}}{1 - \frac{a_1a_2}{b_1b_2}} \right| = \left| \frac{a_1b_2 - a_2b_1}{b_1b_2 + a_1a_2} \right| $$

Comparing Angles Between Two Pairs of Lines

Let's consider two pairs of lines, $(L_1, L_2)$ and $(L_3, L_4)$, with slopes $m_1$, $m_2$, $m_3$, and $m_4$ respectively. To compare the angles between these pairs, we calculate the angle between $L_1$ and $L_2$ and the angle between $L_3$ and $L_4$ using the formula above.

Example

Let's illustrate this with an example:

• Pair 1: $L_1: y = 2x + 3$ and $L_2: y = -\frac{1}{2}x + 1$
• Pair 2: $L_3: y = x - 2$ and $L_4: y = -3x + 4$

First, we find the slopes:

• $m_1 = 2$ for $L_1$
• $m_2 = -\frac{1}{2}$ for $L_2$
• $m_3 = 1$ for $L_3$
• $m_4 = -3$ for $L_4$

Now, we calculate the angles:

• Angle between $L_1$ and $L_2$: $\tan(\theta_1) = \left| \frac{2 - (-\frac{1}{2})}{1 + 2(-\frac{1}{2})} \right| = \left| \frac{\frac{5}{2}}{\frac{1}{2}} \right| = 5$
• Angle between $L_3$ and $L_4$: $\tan(\theta_2) = \left| \frac{1 - (-3)}{1 + 1(-3)} \right| = \left| \frac{4}{-2} \right| = 2$

Thus, $\theta_1$ and $\theta_2$ are the angles whose tangents are 5 and 2, respectively.

Differences and Important Points

Aspect	Pair of Lines 1 ($L_1$, $L_2$)	Pair of Lines 2 ($L_3$, $L_4$)
Slopes	$m_1 = 2$, $m_2 = -\frac{1}{2}$	$m_3 = 1$, $m_4 = -3$
Angle (in terms of tan)	$\tan(\theta_1) = 5$	$\tan(\theta_2) = 2$
Acute or Obtuse	Acute (since $\tan(\theta_1) > 0$)	Acute (since $\tan(\theta_2) > 0$)

Important Points to Remember

• The formula for the angle between two lines is based on their slopes.
• If the tangent of the angle is positive, the angle is acute; if negative, the angle is obtuse.
• When comparing angles between two pairs of lines, calculate each angle separately and then compare.
• The angle between two lines is always taken to be the smaller angle, i.e., the acute angle.

By understanding these concepts and applying the formulas correctly, one can determine the angle between two pairs of straight lines, which is a valuable skill in various mathematical applications.