Key Points

The slope 'm' of a non-vertical line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.

If a line makes an angle $\theta$ with the positive direction of the x-axis, its slope 'm' is given by $m = \tan \theta$, where $\theta \neq 90^{\circ}$. The slope of a horizontal line is 0, and the slope of a vertical line is undefined.

Two non-vertical lines are parallel if and only if their slopes are equal. If line $L_1$ has slope $m_1$ and line $L_2$ has slope $m_2$, they are parallel if $m_1 = m_2$.

Two non-vertical lines are perpendicular if and only if the product of their slopes is -1. For lines with slopes $m_1$ and $m_2$, the condition is $m_1 \times m_2 = -1$.

The acute angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by the formula $\tan \theta = |\frac{m_2 - m_1}{1 + m_1 m_2}|$, provided $1 + m_1 m_2 \neq 0$.

Three points A, B, and C are collinear (lie on the same line) if and only if the slope of line segment AB is equal to the slope of line segment BC.

The equation of a line passing through a point $(x_0, y_0)$ with a slope 'm' is given by $y - y_0 = m(x - x_0)$.

The equation of a line passing through two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$.

The equation of a line with slope 'm' and y-intercept 'c' is $y = mx + c$. The line intersects the y-axis at the point $(0, c)$.

The equation of a line that makes an x-intercept 'a' and a y-intercept 'b' is given by $\frac{x}{a} + \frac{y}{b} = 1$. The line passes through $(a, 0)$ and $(0, b)$.

Any equation of the form $Ax + By + C = 0$, where A and B are not both zero, represents a straight line in the coordinate plane.

The perpendicular distance 'd' from a point $(x_1, y_1)$ to the line $Ax + By + C = 0$ is calculated using the formula $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$.

The distance 'd' between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is given by $d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$.