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Converse Consecutive Interior Angles Theorem

If two lines and a transversal form consecutive interior angles that are supplementary, then the two lines are parallel.
Two parallel lines cut by a transversal forming two pairs of congruent angles
Based on the characteristics of the diagram, the following relation holds true.

The theorem is named the Converse Consecutive Interior Angles Theorem because the same thing holds true but in opposite order.


The proof will be based on the given diagram, but it holds true for any pair of lines cut by a transversal. Consider only one pair of consecutive interior angles and one more angle.

Two parallel lines cut by a transversal forming two pairs of congruent angles
It needs to be proven that and are parallel lines. It is already given that is supplementary to
The diagram shows that and form a linear pair, which means that these angles are supplementary.
Notice that is common in both relationships. Subtracting the two equations gives a relation between and
The last equation tells that and have the same measure and therefore, they are congruent. The diagram also shows that and are corresponding angles. Given that relation, the Converse Corresponding Angles Theorem can be applied.

Converse Corresponding Angles Theorem

If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel.

Since and are corresponding congruent angles, then and are parallel lines. This completes the proof.