Alternate Angle Property

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$\parallel$CD. 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$\parallel$CD.
Thus, equality of alternate angles $\Rightarrow$ parallelism.

Hence the theorem.

Recommended:
Angle Sum Property of triangle
Interior angles property
Isosceles Triangle Theorem