Theorem: Two lines are parallel if and only if the two angles of any pair of alternate angles of any transversal are congruent (equal in measure).
Prerequisites:
Vertical angle theorem (proof)
Corresponding angles property (proof)
Proof:
First let us prove that parallelism implies equality of alternate angles:
In the given figure, let us assume that AB\parallelCD. Let \angle EMB = \theta. Then, by corresponding angles property,
\qquad\quad\angle END = \angle EMB = \theta\qquad\qquad\qquad\qquad\qquad\cdots\text{$(1)$}
Also, by vertical angle theorem,
\qquad\quad\angle FMA = \angle EMB = \theta\qquad\qquad\qquad\qquad\qquad\cdots\text{$(2)$}
Hence, by (1) and (2),
\qquad\quad\angle FMA = \angle END
Thus, parallelism \Rightarrow equality of alternate angles.
Conversely, let us assume that alternate angles are equal. Then,
\qquad\quad\begin{align}\angle END &= \angle FMA\qquad\qquad\qquad\qquad\qquad\text{(alternate angles)}\\
&= \angle EMB\qquad\qquad\qquad\qquad\qquad\text{(vertically opposite angles)}\\
&= \theta\end{align}
Hence, the corresponding angles, \angle END = \angle EMB = \theta. Thus, by corresponding angles property, line AB\parallelCD.
Thus, equality of alternate angles \Rightarrow parallelism.
Hence the theorem.
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