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Rule

Consecutive Interior Angles Theorem

If two parallel lines are cut by a transversal, then the consecutive interior angles are supplementary angles.
Two parallel lines cut by a transversal
In the diagram, and are supplementary as well as and

Proof

Start by noticing that and form a linear pair as well as and

Two parallel lines cut by a transversal
Therefore, these pair of angles are supplementary.
On the other hand, since the lines and are parallel, by the Alternate Interior Angles Theorem, the alternate interior angles are congruent. This allows to write the following relations.
Next, in Equation I, substitute for
The last equation implies that and are supplementary. Similarly, in Equation II, substitute for
As before, the last equation implies that and are supplementary, which completes the proof.