Proof

Alternate Interior Angles Theorem

If parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Consider the following diagram.

To prove that alternate interior angles are congruent, it will be shown that

∠ 3

and

∠ 5

are congruent.
Notice that, by definition,

∠ 1

and

∠ 3

are vertical angles. Thus,

m ∠ 1 = m ∠ 3 .

Notice also that, by definition,

∠ 1

and

∠ 5

are corresponding angles. Thus,

m ∠ 1 = m ∠ 5 .

By transitivity,

m ∠ 3 = m ∠ 5 .

Thus, it follows that

∠ 3 ≅ ∠ 5 .

The same reasoning applies to the other pair of alternate interior angles. Thererefore, when a pair of parallel lines is cut by a transversal, the pairs of alternate interior angles are congruent.
This can be summarized in the following two-column proof.

Statement	Reason
$∠ 1 ≅ ∠ 3$	Vertical Angles Theorem
$∠ 1 ≅ ∠ 5$	Corresponding Angles Theorem
$∠ 3 ≅ ∠ 5$	Transitive Property of Equality

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Proof

Alternate Interior Angles Theorem

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