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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\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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