Proof

Corresponding Angles Theorem

If parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

Consider the following diagram.

To prove that corresponding angles are congruent, it will be shown that

∠ 1

and

∠ 5

are congruent.
Notice that, by definition,

∠ 1

and

∠ 4

are supplementary angles, as are

∠ 4

and

∠ 5 .

Thus, the following equations can be written.

m ∠ 1 + m ∠ 4 = 18 0^{\circ} m ∠ 4 + m ∠ 5 = 18 0^{\circ}

By transitivity, the equations can be set equal to each other.

m ∠ 1 + m ∠ 4 = m ∠ 4 + m ∠ 5

It follows then that

m ∠ 1 = m ∠ 5 .

Thus,

∠ 1 ≅ ∠ 5 .

The same reasoning applies to all pairs of corresponding angles. Thererefore, when a pair of parallel lines is cut by a transversal, the pairs of corresponding angles are congruent.
This can be summarized in the following flowchart proof.

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Proof

Corresponding Angles Theorem

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