### Theory:

Let $$l_1$$ and $$l_2$$ be two non-vertical lines.

The slope of line $$l_1$$ is $$m_1$$, and line $$l_2$$ is $$m_2$$.

Let the inclination of $$l_1$$ be $$\theta_1$$ and $$l_2$$ be $$\theta_2$$.

Assume $$l_1$$ and $$l_2$$ are parallel lines. If two lines are parallel, then their corresponding angles are equal.

$$\Rightarrow \theta_1 = \theta_2$$

$$\Rightarrow \tan \theta_1 = \tan \theta_2$$

$$\Rightarrow m_1 = m_2$$

The slopes are equal.

Conversely:

Assume slopes of two lines $$l_1$$ and $$l_2$$, are equal.

$$\Rightarrow m_1 = m_2$$

$$\Rightarrow \tan \theta_1 = \tan \theta_2$$

$$\Rightarrow \theta_1 = \theta_2$$

$$\Rightarrow$$ Corresponding angles are equal.

$$\Rightarrow$$ $$l_1$$ and $$l_2$$ are parallel.

Thus, the non-vertical lines are parallel if and only if their slopes are equal.
If two lines are parallel, then their slopes are equal. That is, $$m_1 = m_2$$.
Example:
If a line $$p$$ passing through the points $$(1, 8)$$ and $$(2, 13)$$ and a line $$q$$ passing through the points $$(0, -1)$$ and $$(1, 4)$$ are parallel?

Solution:

Let the points passing through the line $$p$$ be $$A$$ $$=$$ $$(1, 8)$$ and $$B = (2, 13)$$.

And, the points passing through the line $$q$$ be $$C = (0, -1)$$ and $$D = (1, 4)$$.

Two lines are parallel if their slopes are equal.

Let us find the slopes of $$p$$ and $$q$$.

Slope $$=$$ $\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$

Slope of $$p$$ $$=$$ $\frac{13-8}{2-1}=\frac{5}{1}=5$

Slope of $$q$$ $$=$$ $\frac{4+1}{1-0}=\frac{5}{1}=5$

Hence, the slope of $$p$$ $$=$$ slope of $$q$$.

Therefore, the lines $$p$$ and $$q$$ are parallel.
Important!
The quadrilateral is a parallelogram if the slopes of both pairs of opposite sides are equal.