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\parallelCD. 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\parallelCD.
Hence, supplementarity of consecutive interior angles \Rightarrow parallelism.
Hence the theorem.
Recommended:
Corresponding angles property
Alternate angles property
Angle sum property of triangle
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