Condition for Straight Lines in Plane to be Parallel/Slope Form/Proof 3

Theorem

Let $L_1$ and $L_2$ be straight lines in the Cartesian plane.

Let the slope of $L_1$ and $L_2$ be $\mu_1$ and $\mu_2$ respectively.

Then $L_1$ is parallel to $L_2$ if and only if:

$\mu_1 = \mu_2$

Proof

Let $\psi$ be the angle between $L_1$ and $L_2$

When $L_1$ and $L_2$ are parallel:

$\psi = 0$

by definition.

From Angle between Straight Lines in Plane:

$\psi = \arctan \dfrac {m_1 - m_2} {1 + m_1 m_2}$

The result follows immediately.

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {III}$. Analytical Geometry: The Straight Line: The angle between two lines