Interior Angles Property

Theorem: Two lines are parallel if and only if the two angles of any pair of consecutive interior angles of any transversal are supplementary (sum to 180°).

Prerequisites:
Corresponding angles property (proof)
Angle on a straight line property (proof)

Proof:

In the given figure, let the angle $\angle BMN = \alpha$ and $\angle MND = \beta$.
Let us first assume that the AB$\parallel$CD. Then by corresponding angles property,

$\qquad\quad\angle EMB = \angle MND = \beta$

Also, since AB is a straight line,

$\therefore\quad\;\;\angle BMN + \angle EMB = 180^o\\
\Rightarrow\quad\;\;\alpha + \beta = 180^o\\
\Rightarrow\quad\;\;\angle BMN + \angle MND = 180^o$

Hence, parallelism $\Rightarrow$ supplementarity of consecutive interior angles.

Conversely, let us assume that consecutive interior angles are supplementary.

$\therefore\quad\;\;\angle BMN + \angle MND = 180^o\\
\Rightarrow\quad\;\;\alpha + \beta = 180^o$

$\text{But,}\;\;\angle BMN + \angle EMB = 180^o\qquad\qquad\qquad\qquad\qquad\cdots\text{(angle on a straight line property)}$
$\begin{align}\therefore\quad\;\;\angle EMB &= 180^o - \alpha\\
&= \beta = \angle MND\end{align}$

Since the corresponding angles are equal, then by corresponding angles property, AB$\parallel$CD.
Hence, supplementarity of consecutive interior angles $\Rightarrow$ parallelism.

Hence the theorem.


Recommended:
Corresponding angles property
Alternate angles property
Angle sum property of triangle